Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In Chapter we revisit quadratic equations. In earlier chapters, we solved quadratic equations b factoring and appling the zero product rule. In this chapter, we present two additional techniques to solve quadratic equations. However, these techniques can be used to solve quadratic equations even if the equations are not factorable. Complete the word scramble to familiarize ourself with the ke terms for this chapter. As a hint, there is a clue for each word.. r a e q s u ( root propert). eitgcompln ( the square). dqatriuac ( formula). taiicidsrmnn. a o b r p l a a b ac (shape of a quadratic function). teerv (maimum or minimum point on the graph of a parabola) 7. s i a ( of smmetr)
Chapter Quadratic Equations and Functions Section. Concepts. Solving Quadratic Equations b Using the Square Root Propert. Solving Quadratic Equations b Completing the Square. Literal Equations Square Root Propert and Completing the Square. Solving Quadratic Equations b Using the Square Root Propert In Section. we learned how to solve a quadratic equation b factoring and appling the zero product rule. For eample, 9 9 0 9 0 0 9 or or 9 0 9 Set one side equal to zero. Factor. Set each factor equal to zero. The solutions are 9 and 9. It is important to note that the zero product rule can onl be used if the equation is factorable. In this section and Section., we will learn how to solve quadratic equations, factorable and nonfactorable. The first technique utilizes the square root propert. The Square Root Propert For an real number, k, if k, then k or k. Note: The solution ma also be written as k, read equals plus or minus the square root of k. Eample Solving Quadratic Equations b Using the Square Root Propert Use the square root propert to solve the equations. a. b. 7 0 c. w 0 Avoiding Mistakes: A common mistake is to forget the smbol when solving the equation k: k Solution: a. The equation is in the form k. Appl the square root propert. 9 The solutions are 9 and 9. Notice that this is the same solution obtained b factoring and appling the zero product rule. b. 7 0 7 Rewrite the equation to fit the form k. The equation is now in the form k. Appl the square root propert. i
} Section. Square Root Propert and Completing the Square The solutions are i and i. Check: i Check: i 7 0 7 0 i 7 0 i 7 0 i 7 0 i 7 0 7 0 7 0 7 7 0 7 7 0 c. w 0 The equation is in the form k, where w. w 0 w Appl the square root propert. Simplif the radical. w w Solve for w. The solutions are w and w. Skill Practice Use the square root propert to solve the equations.. a 00. 7 0. t. Solving Quadratic Equations b Completing the Square In Eample (c) we used the square root propert to solve an equation where the square of a binomial was equal to a constant. The square of a binomial is the factored form of a perfect square trinomial. For eample: Perfect Square Trinomial 0 t t 9 p p 9 w 0 square of a binomial constant Factored Form t p 7 For a perfect square trinomial with a leading coefficient of, the constant term is the square of one-half the linear term coefficient. For eample: 0 0 In general an epression of the form b n is a perfect square trinomial if n b. The process to create a perfect square trinomial is called completing the square. Skill Practice Answers. a 0. i. t
Chapter Quadratic Equations and Functions Eample Completing the Square Determine the value of n that makes the polnomial a perfect square trinomial. Then factor the epression as the square of a binomial. a. n b. n c. n d. 7 n Solution: The epressions are in the form b n. The value of n equals the square of one-half the linear term coefficient b. a. n n. Factored form b. n 9 n 9. Factored form c. n a b n. Factored form d. 7 n 7 9 a 7 b n 7 7 9 Factored form Skill Practice Determine the value of n that makes the polnomial a perfect square trinomial. Then factor.. 0 n. n. a a n 7. w 7 w n Skill Practice Answers. n 00; 0. n ;. n ; aa b 7. n 9 ; aw 7 b The process of completing the square can be used to write a quadratic equation a b c 0 a 0 in the form h k. Then the square root propert can be used to solve the equation. The following steps outline the procedure.
Section. Square Root Propert and Completing the Square Solving a Quadratic Equation in the Form a b c 0 (a 0) b Completing the Square and Appling the Square Root Propert. Divide both sides b a to make the leading coefficient.. Isolate the variable terms on one side of the equation.. Complete the square. (Add the square of one-half the linear term coefficient to both sides of the equation. Then factor the resulting perfect square trinomial.). Appl the square root propert and solve for. Eample Solving Quadratic Equations b Completing the Square and Appling the Square Root Propert Solve the quadratic equations b completing the square and appling the square root propert. a. 0 b. 0 0 Solution: a. 0 0 9 9 Step : Step : Step : Step : Since the leading coefficient a is equal to, we do not have to divide b a. We can proceed to step. Isolate the variable terms on one side. To complete the square, add 9 to both sides of the equation. Factor the perfect square trinomial. Appl the square root propert. i Simplif the radical. i Solve for. The solutions are imaginar numbers and can be written as i and i. b. 0 0 0 0 0 0 0 0 0 Clear parentheses. Write the equation in the form a b c 0. Step : Divide both sides b the leading coefficient.
Chapter Quadratic Equations and Functions 9 9 a b 0 9 a b 9 9 A 7 Step : Step : Step : Isolate the variable terms on one side. Add 9 to both sides. Factor the perfect square trinomial. Rewrite the righthand side with a common denominator. Appl the square root propert. Simplif the radical. 7 0 7 The solutions are rational numbers: and.the check is left to the reader. Skill Practice 7 Solve b completing the square and appling the square root propert.. z z 0 9. TIP: In general, if the solutions to a quadratic equation are rational numbers, the equation can be solved b factoring and using the zero product rule. Consider the equation from Eample (b). 0 0 0 0 0 0 0 0 or Skill Practice Answers. z 9. and
Section. Square Root Propert and Completing the Square 7. Literal Equations Eample Solving a Literal Equation Ignoring air resistance, the distance d (in meters) that an object falls in t sec is given b the equation d.9t where t 0 a. Solve the equation for t. Do not rationalize the denominator. b. Using the equation from part (a), determine the amount of time required for an object to fall 00 m. Round to the nearest second. Solution: a. d.9t d.9 t t A d.9 A d.9 Isolate the quadratic term. The equation is in the form t k. Appl the square root propert. Because t 0, reject the negative solution. b. t A d.9 A 00.9 t 0. Substitute d 00. The object will require approimatel 0. sec to fall 00 m. Skill Practice 0. The formula for the area of a circle is A pr, where r is the radius. a. Solve for r. (Do not rationalize the denominator.) b. Use the equation from part (a) to find the radius when the area is.7 cm. (Use. for p and round to decimal places.) Skill Practice Answers A 0a. r B p b. The radius is. cm. Section. Boost our GRADE at mathzone.com! Stud Skills Eercise. Define the ke terms. Practice Eercises Practice Problems Self-Tests NetTutor a. Square root propert b. Completing the square e-professors Videos
Chapter Quadratic Equations and Functions Concept : Solving Quadratic Equations b Using the Square Root Propert For Eercises 7, solve the equations b using the square root propert.. 00.. a.. v 0 7. m 0. p 9 9. 0. 0. 7 0. h.... a p 7. b 7 w 0 k 7 0 q t am b 0. Given the equation k, match the following statements. a. If k 7 0, then i. there will be one real solution. b. If k 0, then ii. there will be two real solutions. c. If k 0, then iii. there will be two imaginar solutions. 9. State two methods that can be used to solve the equation 0. Then solve the equation b using both methods. 0. State two methods that can be used to solve the equation 9 0. Then solve the equation b using both methods. Concept : Solving Quadratic Equations b Completing the Square For Eercises 0, find the value of n so that the epression is a perfect square trinomial. Then factor the trinomial.. n. n. t t n. v v n. c c n. 9 n 7. n. a 7a n 9. b 0. b n m 7 m n. Summarize the steps used in solving a quadratic equation b completing the square and appling the square root propert.. What tpes of quadratic equations can be solved b completing the square and appling the square root propert? For Eercises, solve the quadratic equation b completing the square and appling the square root propert.. t t 0. m m 0.. 7. p p 0. q q 0 9. 0 0 0.. a a 0. a a 7 0. 9 0 0.. p. n 7. w w. n p 9 9 0 p p 9. nn 7 0. mm 0..
Section. Square Root Propert and Completing the Square 9 Concept : Literal Equations. The distance (in feet) that an object falls in t sec is given b the equation d t, where t 0. a. Solve the equation for t. b. Using the equation from part (a), determine the amount of time required for an object to fall 0 ft.. The volume of a can that is in. tall is given b the equation V pr, where r is the radius of the can, measured in inches. a. Solve the equation for r. Do not rationalize the denominator. b. Using the equation from part (a), determine the radius of a can with volume of. in. Use. for p. For Eercises 0, solve for the indicated variable.. A pr for r r 7 0. 7. a b c d for a a 7 0. E mc for c c 7 0 a b c for b b 7 0 9. V 0. pr h for r r 7 0 V s h for s. A corner shelf is to be made from a triangular piece of plwood, as shown in the diagram. Find the distance that the shelf will etend along the walls. Assume that the walls are at right angles. Round the answer to a tenth of a foot. ft. A square has an area of 0 in. What are the lengths of the sides? (Round to decimal place.). The amount of mone A in an account with an interest rate r compounded annuall is given b A P r t where P is the initial principal and t is the number of ears the mone is invested. a. If a $0,000 investment grows to $, after ears, find the interest rate. b. If a $000 investment grows to $79.0 after ears, find the interest rate. c. Jamal wants to invest $000. He wants the mone to grow to at least $00 in ears to cover the cost of his son s first ear at college. What interest rate does Jamal need for his investment to grow to $00 in ears? Round to the nearest hundredth of a percent.
0 Chapter Quadratic Equations and Functions. The volume of a bo with a square bottom and a height of in. is given b V, where is the length (in inches) of the sides of the bottom of the bo. in. a. If the volume of the bo is 9 in., find the dimensions of the bo. b. Are there two possible answers to part (a)? Wh or wh not?. A tetbook compan has discovered that the profit for selling its books is given b P where is the number of tetbooks produced (in thousands) and P() is the corresponding profit (in thousands of dollars). The graph of the function is shown at right. a. Approimate the number of books required to make a profit of $0,000. [Hint: Let P 0. Then complete the square to solve for.] Round to one decimal place. b. Wh are there two answers to part (a)? Profit P() ($000) 0 0 0 0 0 P() 0 0 0 0 Number of Tetbooks (thousands). If we ignore air resistance, the distance (in feet) that an object travels in free fall can be approimated b dt t, where t is the time in seconds after the object was dropped. a. If the CN Tower in Toronto is ft high, how long will it take an object to fall from the top of the building? Round to one decimal place. b. If the Renaissance Tower in Dallas is ft high, how long will it take an object to fall from the top of the building? Round to one decimal place.
Section. Quadratic Formula Quadratic Formula. Derivation of the Quadratic Formula If we solve a general quadratic equation a b c 0 a 0 b completing the square and using the square root propert, the result is a formula that gives the solutions for in terms of a, b, and c. a b c 0 a a b a c a 0 a b a c a 0 Begin with a quadratic equation in standard form. Divide b the leading coefficient. Section. Concepts. Derivation of the Quadratic Formula. Solving Quadratic Equations b Using the Quadratic Formula. Using the Quadratic Formula in Applications. Discriminant. Mied Review: Methods to Solve a Quadratic Equation b a c a Isolate the terms containing. b a a b a b a b a b c a a b a b a b a b b a c a b ac a b a b ac B a b a b ac a b a b ac a b b ac a Add the square of the linear term coefficient to both sides of the equation. Factor the left side as a perfect square. Combine fractions on the right side b getting a common denominator. Appl the square root propert. Simplif the denominator. Subtract Combine fractions. from both sides. The solution to the equation a b c 0 for in terms of the coefficients a, b, and c is given b the quadratic formula. b a the solu- The Quadratic Formula For an quadratic equation of the form a b c 0 a 0 tions are b b ac a The quadratic formula gives us another technique to solve a quadratic equation. This method will work regardless of whether the equation is factorable or not factorable.
Chapter Quadratic Equations and Functions. Solving Quadratic Equations b Using the Quadratic Formula Eample Solving a Quadratic Equation b Using the Quadratic Formula Solve the quadratic equation b using the quadratic formula. TIP: Alwas remember to write the entire numerator over a. Solution: 0 a, b, c b b ac a 9 0 9 7 7 7 Write the equation in the form a b c 0. Identif a, b, and c. Appl the quadratic formula. Substitute a, b, and c. Simplif. 0 There are two rational solutions, and. Both solutions check in the original equation. Skill Practice Solve the equation b using the quadratic formula.. Eample Solving a Quadratic Equation b Using the Quadratic Formula Solve the quadratic equation b using the quadratic formula. Solution: 0 Write the equation in the form a b c 0. Skill Practice Answers. ;
Section. Quadratic Formula 0 0 a, b, and c b b ac a If the leading coefficient of the quadratic polnomial is negative, we suggest multipling both sides of the equation b. Although this is not mandator, it is generall easier to simplif the quadratic formula when the value of a is positive. Identif a, b, and c. Appl the quadratic formula. Substitute a, b, and c. Simplif. Simplif the radical. Factor the numerator. Simplif the fraction to lowest terms. There are two imaginar solutions, i and i. Skill Practice. i i i i i i Solve the equation b using the quadratic formula.. Using the Quadratic Formula in Applications Eample Using the Quadratic Formula in an Application A deliver truck travels south from Hartselle, Alabama, to Birmingham, Alabama, along Interstate. The truck then heads east to Atlanta, Georgia, along Interstate 0. The distance from Birmingham to Atlanta is mi less than twice the distance from Hartselle to Birmingham. If the straight-line distance from Hartselle to Atlanta is mi, find the distance from Hartselle to Birmingham and from Birmingham to Atlanta. (Round the answers to the nearest mile.) Skill Practice Answers 7.
Chapter Quadratic Equations and Functions Solution: The motorist travels due south and then due east. Therefore, the three cities form the vertices of a right triangle (Figure -). Hartselle I- mi Birmingham I-0 Figure - Atlanta Let represent the distance between Hartselle and Birmingham. Then represents the distance between Birmingham and Atlanta. Use the Pthagorean theorem to establish a relationship among the three sides of the triangle. Write the equation in the form a b c 0. Identif a, b, and c. Appl the quadratic formula. Simplif. Recall that represents the distance from Hartselle to Birmingham; therefore, to the nearest mile, the distance between Hartselle and Birmingham is 77 mi. The distance between Birmingham and Atlanta is 77 mi. Skill Practice 7, 7, 0 a b c 7, 7, 0,0 0, 0, 0, 0 7.97 mi or 70.7 mi We reject the negative solution because distance cannot be negative.. Steve and Tamm leave a campground, hiking on two different trails. Steve heads south and Tamm heads east. B lunchtime the are 9 mi apart. Steve walked mi more than twice as man miles as Tamm. Find the distance each person hiked. (Round to the nearest tenth of a mile.) Skill Practice Answers. Tamm hiked. mi, and Steve hiked. mi.
Section. Quadratic Formula Eample Analzing a Quadratic Function h(t) A model rocket is launched straight up from the side of a -ft cliff (Figure -). The initial velocit is ft/sec. The height of the rocket h(t) is given b 0 ht t t where h(t) is measured in feet and t is the time in seconds. Find the time(s) at which the rocket is 0 ft above the ground. Solution: 0 Figure - t ht t t 0 t t Substitute 0 for h(t). t t 0 Write the equation in the form at bt c 0. t t 0 Divide b. This makes the coefficients smaller, and it is less cumbersome to appl the quadratic formula. t 7t 0 The equation is not factorable. Appl the quadratic formula. t 7 7 Let a, b 7, and c. 7 t 7 t 7.7 0. The rocket will reach a height of 0 ft after approimatel 0. sec (on the wa up) and after.7 sec (on the wa down). Skill Practice. An object is launched from the top of a 9-ft building with an initial velocit of ft/sec. The height ht of the rocket is given b ht t t 9 Find the time it takes for the object to hit the ground. (Hint: h t 0 when the object hits the ground.). Discriminant The radicand within the quadratic formula is the epression b ac. This is called the discriminant. The discriminant can be used to determine the number of solutions to a quadratic equation as well as whether the solutions are rational, irrational, or imaginar numbers. Skill Practice Answers. sec 0.
Chapter Quadratic Equations and Functions Using the Discriminant to Determine the Number and Tpe of Solutions to a Quadratic Equation Consider the equation a b c 0, where a, b, and c are rational numbers and a 0. The epression b ac is called the discriminant. Furthermore,. If b ac 7 0, then there will be two real solutions. a. If b ac is a perfect square, the solutions will be rational numbers. b. If b ac is not a perfect square, the solutions will be irrational numbers.. If b ac 0, then there will be two imaginar solutions.. If b ac 0, then there will be one rational solution. Eample Using the Discriminant Use the discriminant to determine the tpe and number of solutions for each equation. a. 9 0 b. c. d... 0. Solution: For each equation, first write the equation in standard form a b c 0. Then determine the discriminant. a. b. c. Equation 9 0 0 0 Discriminant b ac 9 7 7 b ac b ac Solution Tpe and Number Because 7 0, there will be two imaginar solutions. Because 7 0 and is a perfect square, there will be two rational solutions. Because 7 0, but is not a perfect square, there will be two irrational solutions.
Section. Quadratic Formula 7 d... 0... 0. 0 b ac..0... 0 Because the discriminant equals 0, there will be onl one rational solution. Skill Practice Determine the discriminant. Use the discriminant to determine the tpe and number of solutions for the equation.. 0. t t 7. t t 9. 0 With the discriminant we can determine the number of real-valued solutions to the equation a b c 0, and thus the number of -intercepts to the function f a b c. The following illustrations show the graphical interpretation of the three cases of the discriminant. f Use 0 to find the value of the discriminant. b ac Since the discriminant is positive, there are two real solutions to the quadratic equation. Therefore, there are two -intercepts to the corresponding quadratic function, (, 0) and (, 0). f Use 0 to find the value of the discriminant. b ac Since the discriminant is negative, there are no real solutions to the quadratic equation. Therefore, there are no -intercepts to the corresponding quadratic function. f Use 0 to find the value of the discriminant. b ac 0 Since the discriminant is zero, there is one real solution to the quadratic equation. Therefore, there is one -intercept to the corresponding quadratic function, (, 0). Skill Practice Answers. ; two imaginar solutions. ; two rational solutions 7. 7; two irrational solutions. 0; one rational solution
Chapter Quadratic Equations and Functions Eample Finding the - and -Intercepts of a Quadratic Function Given: f a. Find the -intercept. b. Find the -intercept(s). Solution: a. The -intercept is given b f 0 0 0 The -intercept is located at 0,. b. The -intercepts are given b the real solutions to the equation In this case, we have f 0. f 0 a b 0 Multipl b to clear fractions. 7 i The equation is in the form a b c 0, where a, b, and c. Appl the quadratic formula. Simplif. These solutions are imaginar numbers. Because there are no real solutions to the equation f 0, the function has no -intercepts. The function is graphed in Figure -. Skill Practice 0 7 9. Find the - and -intercepts of the function given b Figure - f() 0 Skill Practice Answers 9. -intercepts:, 0., 0,, 0., 0; -intercept: a0, b f
Section. Quadratic Formula 9. Mied Review: Methods to Solve a Quadratic Equation Three methods have been presented to solve quadratic equations. Methods to Solve a Quadratic Equation Factor and use the zero product rule (Section.). Use the square root propert. Complete the square, if necessar (Section.). Use the quadratic formula (Section.). Using the zero product rule is often the simplest method, but it works onl if ou can factor the equation. The square root propert and the quadratic formula can be used to solve an quadratic equation. Before solving a quadratic equation, take a minute to analze it first. Each problem must be evaluated individuall before choosing the most efficient method to find its solutions. Eample 7 Solving Quadratic Equations b Using An Method Solve the quadratic equations b using an method. a. b. c. 9 p 0 Solution: a. 9 9 9 0 Clear parentheses and write the equation in the form a b c 0. 0 0 This equation is factorable. Factor. 0 or 0 Appl the zero product rule. or Solve for. This equation could have been solved b using an of the three methods, but factoring was the most efficient method. b. Clear fractions and write the equation in the form a b c 0. 0 a b c This equation does not factor. Use the quadratic formula. Identif a, b, and c. Appl the quadratic formula. Simplif.
70 Chapter Quadratic Equations and Functions Skill Practice Answers 0. t ; t 97.. i i Simplif the radical. i Reduce to lowest terms. i i This equation could also have been solved b completing the square and appling the square root propert. c. p 0 The equation is in the form k, where p. p Appl the square root propert. p Solve for p. This problem could have been solved b the quadratic formula, but that would have involved clearing parentheses and collecting like terms first. Skill Practice p Solve the quadratic equations b using an method. 0. t t t. 0.. 0.. Section. Boost our GRADE at mathzone.com! Stud Skills Eercise. Define the ke terms. Practice Eercises a. Quadratic formula b. Discriminant Practice Problems Self-Tests NetTutor e-professors Videos Review Eercises. Show b substitution that is a solution to 0. For Eercises, solve b completing the square and using the square root propert.. 9.. 0 0. 0 For Eercises 7 0, simplif the epressions. 0 0 7 7.. 9. 0. 7 0 7 Concept : Solving Quadratic Equations b Using the Quadratic Formula. What form should a quadratic equation be in so that the quadratic formula can be applied?
Section. Quadratic Formula 7. Describe the circumstances in which the square root propert can be used as a method for solving a quadratic equation.. Write the quadratic formula from memor. For Eercises 9, solve the equation b using the quadratic formula.. a a 0. b b 0. 7. t t 7 0. p p 0 9. 0. z z... k k. 0 0.. w w 0 7. m m. 9. 7 0. a a 0... p p 0. (Hint: Clear the fractions first.).. 0.0 0.0 0.0 0 w 7 w 0 (Hint: Clear the decimals first.) 7. 9 0 n n 0 a a 9 m m 0 h h 0 0. 0.7 0. 0. 0.t 0.7t 0. 0 9. 0.0 0.0 0.07 0 For Eercises 0, factor the epression. Then use the zero product rule and the quadratic formula to solve the equation. There should be three solutions to each equation. 0. a. Factor. 7. a. Factor. b. Solve. 7 0 b. Solve. 0. a. Factor.. a. Factor. b. Solve. 0 b. Solve. 0 0 0. The volume of a cube is 7 ft. Find the lengths of the sides.. The volume of a rectangular bo is ft. If the width is times longer than the height, and the length is 9 times longer than the height, find the dimensions of the bo. Concept : Using the Quadratic Formula in Applications. The hpotenuse of a right triangle measures in. One leg of the triangle is in. longer than the other leg. Find the lengths of the legs of the triangle. Round to one decimal place.
7 Chapter Quadratic Equations and Functions 7. The length of one leg of a right triangle is cm more than twice the length of the other leg. The hpotenuse measures cm. Find the lengths of the legs. Round to decimal place.. The hpotenuse of a right triangle is 0. m long. One leg is. m shorter than the other leg. Find the lengths of the legs. Round to decimal place. 9. The hpotenuse of a right triangle is 7 ft long. One leg is. ft longer than the other leg. Find the lengths of the legs. 0. The fatalit rate (in fatalities per 00 million vehicle miles driven) can be approimated for drivers ears old according to the function, F 0.00 0. 9.. Source: U.S. Department of Transportation a. Approimate the fatalit rate for drivers ears old. b. Approimate the fatalit rate for drivers 0 ears old. c. Approimate the fatalit rate for drivers 0 ears old. d. For what age(s) is the fatalit rate approimatel.?. Mitch throws a baseball straight up in the air from a cliff that is ft high. The initial velocit is ft/sec. The height (in feet) of the object after t sec is given b ht t t. Find the time at which the height of the object is ft.. An astronaut on the moon throws a rock into the air from the deck of a spacecraft that is m high. The initial velocit of the rock is. m/sec. The height (in meters) of the rock after t sec is given b ht 0.t.t. Find the time at which the height of the rock is m.. The braking distance (in feet) of a car going v mph is given b d v v 0 v v 0 a. How fast would a car be traveling if its braking distance were 0 ft? Round to the nearest mile per hour. b. How fast would a car be traveling if its braking distance were 00 ft? Round to the nearest mile per hour. Concept : Discriminant For Eercises : a. Write the equation in the form a b c 0, a 7 0. b. Find the value of the discriminant. c. Use the discriminant to determine the number and tpe of solutions... 9. 7. n n 0. p 0 9. 9m m k 7 0. nn nn. 9 For Eercises, find the - and -intercepts of the quadratic function.. f. g. f. h
Section. Quadratic Formula 7 Concept : Mied Review: Methods to Solve a Quadratic Equation For Eercises, solve the quadratic equations b using an method.. a a 0 7. z 7z 0. 0 9. 70. 0 7. b 7 0 k k 0 7. a 7. b 0 9 7. 7. 7. t t t t 77. 7. a a 79. 0. z 0z 0.. p 7 0. ww b b b b 0 k k 0 h 0 Sometimes students sh awa from completing the square and using the square root propert to solve a quadratic equation. However, sometimes this process leads to a simple solution. For Eercises, solve the equations two was. a. Solve the equation b completing the square and appling the square root propert. b. Solve the equation b appling the quadratic formula. c. Which technique was easier for ou?.. 0 7 Epanding Your Skills. An artist has been commissioned to make a stained glass window in the shape of a regular octagon. The octagon must fit inside an -in. square space. See the figure. in. in. a. Let represent the length of each side of the octagon. Verif that the legs of the small triangles formed b the corners of the square can be epressed as. b. Use the Pthagorean theorem to set up an equation in terms of that represents the relationship between the legs of the triangle and the hpotenuse. c. Simplif the equation b clearing parentheses and clearing fractions. d. Solve the resulting quadratic equation b using the quadratic formula. Use a calculator and round our answers to the nearest tenth of an inch. e. There are two solutions for. Which one is appropriate and wh?
7 Chapter Quadratic Equations and Functions Graphing Calculator Eercises 7. Graph Y. Compare the -intercepts with the solutions to the equation 0 found in Eercise.. Graph Y. Compare the -intercepts with the solutions to the equation 7 7 0 found in Eercise 0. 9. Graph Y 0. Compare the -intercepts with the solutions to the equation 0 0 found in Eercise. 90. Graph Y. Compare the -intercepts with the solutions to the equation 0 found in Eercise. 9. The recent population (in thousands) of Ecuador can be approimated b P t.t 0.t 97, where t 0 corresponds to the ear 97. a. Approimate the number of people in Ecuador in the ear 000. (Round to the nearest thousand.) b. If this trend continues, approimate the number of people in Ecuador in the ear 00. (Round to the nearest thousand.) c. In what ear after 97 did the population of Ecuador reach 0 million? Round to the nearest ear. (Hint: 0 million equals 0,000 thousands.) d. Use a graphing calculator to graph the function P on the window 0 0, 000,000. Use a Trace feature to approimate the ear in which the population in Ecuador first eceeded 0 million (0,000 thousands). Section. Concepts. Solving Equations b Using Substitution. Solving Equations Reducible to a Quadratic Equations in Quadratic Form. Solving Equations b Using Substitution We have learned to solve a variet of different tpes of equations, including linear, radical, rational, and polnomial equations. Sometimes, however, it is necessar to use a quadratic equation as a tool to solve other tpes of equations. In Eample, we will solve the equation 9 0. Notice that the terms in the equation are written in descending order b degree. Furthermore, the first two terms have the same base,, and the eponent on the first term is eactl double the eponent on the second term. The third term is a constant. An equation in this pattern is called quadratic in form. eponent is double rd term is constant 9 0. To solve this equation we will use substitution as demonstrated in Eample.
Section. Equations in Quadratic Form 7 Eample Solve the equation. Solving an Equation in Quadratic Form 9 0 Solution: 9 0 u u 0 u or u Substitute u. u u 9 0 If the substitution u is made, the equation becomes quadratic in the variable u. The equation is in the form au bu c 0. The equation is factorable. Appl the zero product rule. Reverse substitute. or or 9 or 9 or or Write the equations in the form k. Appl the square root propert. The solutions are,,,. Substituting these values in the original equation verifies that these are all valid solutions. Avoiding Mistakes: When using substitution, it is critical to reverse substitute to solve the equation in terms of the original variable. Skill Practice Solve the equation.. t 0 t 0 0 For an equation written in descending order, notice that u was set equal to the variable factor on the middle term. This is generall the case. Eample Solve the equation. Solving an Equation in Quadratic Form p p Solution: p p p p 0 Set the equation equal to zero. Skill Practice Answers. t, t, t, t
7 Chapter Quadratic Equations and Functions p p 0 Make the substitution Substitute u p. Then the equation is in the form au bu c 0. u u 0 u u 0 u or Reverse substitute. u u p. The equation is factorable. Appl the zero product rule. p or p p or p p or p p or p The equations are radical equations. Cube both sides. Check: p Check: p p p p p The solutions are p and p. Skill Practice Solve the equation.. Eample Solve the equation. Solving a Quadratic Equation in Quadratic Form 0 Solution: The equation can be solved b first isolating the radical and then squaring both sides (this is left as an eercise see Eercise ). However, this equation is also quadratic in form. B writing as /, we see that the eponent on the first term is eactl double the eponent on the middle term. / 0 u u 0 Let u /. u u 0 Factor. u or u Solve for u. / or / Back substitute. Skill Practice Answers., 7 or The solution Solve each equation for. Notice that the principal square root of a number cannot equal.
Section. Equations in Quadratic Form 77 Skill Practice. z z 0. Solving Equations Reducible to a Quadratic Some equations are reducible to a quadratic equation. In Eample, we solve a polnomial equation b factoring. The resulting factors are quadratic. Eample Solve the equation. Solving a Polnomial Equation 7 0 Solution: 0 or 0 7 0 0 A or or This is a polnomial equation. Factor. Set each factor equal to zero. Notice that the two equations are quadratic. Each can be solved b the square root propert. Appl the square root propert. or i Simplif the radicals. The equation has four solutions:,, i, i. Skill Practice. 9 0 Eample Solving a Rational Equation Solve the equation. Solution: This is a rational equation. The LCD is. a b Multipl both sides b the LCD. Clear fractions. Skill Practice Answers. z., i
7 Chapter Quadratic Equations and Functions 0 0 9 0 Appl the distributive propert. The equation is quadratic. Write the equation in descending order. Each coefficient in the equation is divisible b. Therefore, if we divide both sides b, the coefficients in the equation are smaller. This will make it easier to appl the quadratic formula. Appl the quadratic formula. The values and are both defined in the original equation. Skill Practice Answers. t Skill Practice. t t t Section. Boost our GRADE at mathzone.com! Stud Skills Eercise Practice Eercises. What is meant b an equation in quadratic form? Practice Problems Self-Tests NetTutor e-professors Videos Review Eercises For Eercises 7, solve the quadratic equations... a. b 7 n n.. k 7k 7. 0 Concept : Solving Equations b Using Substitution. a. Solve the quadratic equation b factoring. u 0u 0 b. Solve the equation b using substitution. 0 0
Section. Equations in Quadratic Form 79 9. a. Solve the quadratic equation b factoring. u u 0 b. Solve the equation b using substitution. w w w w 0 0. a. Solve the quadratic equation b factoring. u u 0 b. Solve the equation b using substitution. 0. a. Solve the quadratic equation b factoring. u u 0 b. Solve the equation b using substitution. p p p p 0 For Eercises, solve the equation b using substitution.... m m 0.. t 7t 0 7.. 9. 0. 0.. a b a b 0. n 7n 0 p p 0 p p t t 0 9 a b a b 0. In Eample, we solved the equation 0 b using substitution. Now solve this equation b first isolating the radical and then squaring both sides. Don t forget to check the potential solutions in the original equation. Do ou obtain the same solution as in Eample? Concept : Solving Equations Reducible to a Quadratic For Eercises, solve the equations.. t t 0. w w 0 7.. 7 9. 0 0. t... t t 9 7 9. z z z Mied Eercises For Eercises, solve the equations.. 0. 7. 0. t 0 0 9. m 9m 0 0. 7 0
0 Chapter Quadratic Equations and Functions. 9 0. 7 0. 0. t t. a t b a t b 0. 7. / / 0. / / 9. a b a b 0 0 m m 0 0. c c 0. a a a 0 Hint: Factor b grouping first.. b 9b b 9 0. 0 0. 0 Graphing Calculator Eercises. a. Solve the equation 0. b. How man solutions are real and how man solutions are imaginar? c. How man -intercepts do ou anticipate for the function defined b? d. Graph Y on a standard viewing window.. a. Solve the equation 0. b. How man solutions are real and how man solutions are imaginar? c. How man -intercepts do ou anticipate for the function defined b? d. Graph Y on a standard viewing window. 7. a. Solve the equation 0. b. How man solutions are real and how man solutions are imaginar? c. How man -intercepts do ou anticipate for the function defined b? d. Graph Y on a standard viewing window.. a. Solve the equation 0 9 0. b. How man solutions are real and how man solutions are imaginar? c. How man -intercepts do ou anticipate for the function defined b 0 9? d. Graph Y 0 9 on a standard viewing window.
Section. Graphs of Quadratic Functions Graphs of Quadratic Functions In Section., we defined a quadratic function as a function of the form f a b c a 0. We also learned that the graph of a quadratic function is a parabola. In this section we will learn how to graph parabolas. A parabola opens upward if a 7 0 (Figure -) and opens downward if a 0 (Figure -). If a parabola opens upward, the verte is the lowest point on the graph. If a parabola opens downward, the verte is the highest point on the graph. The ais of smmetr is the vertical line that passes through the verte. a 0 0 0 0 h() 0 a 0 0 g() 0 0 0 Figure - Figure - Section. Concepts. Quadratic Functions of the Form f() k. Quadratic Functions of the Form f() ( h). Quadratic Functions of the Form f() a. Quadratic Functions of the Form f () a( h) k. Quadratic Functions of the Form f() k One technique for graphing a function is to plot a sufficient number of points on the function until the general shape and defining characteristics can be determined. Then sketch a curve through the points. Eample Graphing Quadratic Functions of the Form f() k Graph the functions f, g, and h on the same coordinate sstem. f g h Solution: Several function values for f, g, and h are shown in Table - for selected values of. The corresponding graphs are pictured in Figure -. Table - f() g() h() 9 0 7 0 0 9 0 7 g() f() h() 0 0 0 0 Figure -
Chapter Quadratic Equations and Functions Notice that the graphs of g and h take on the same shape as f. However, the -values of g are greater than the -values of f. Hence the graph of g is the same as the graph of f shifted up unit. Likewise the -values of h are less than those of f. The graph of h is the same as the graph of f shifted down units. Skill Practice. Refer to the graph of f k to determine the value of k. 0 0 0 0 The functions in Eample illustrate the following properties of quadratic functions of the form. f k Graphs of f() k If k 7 0, then the graph of f k is the same as the graph of shifted up k units. If k 0, then the graph of f k is the same as the graph of shifted down 0k 0 units. Calculator Connections Tr eperimenting with a graphing calculator b graphing functions of the form k for several values of k. Skill Practice Answers. k Eample Sketch the functions defined b Solution: a. m m Graphing Quadratic Functions of the Form f() k a. m b. n 7 0 Because k, the graph is obtained b shifting the graph of down 0 units (Figure -7). 0 m() 0 0 Figure -7 m() 0
Section. Graphs of Quadratic Functions b. n 7 Because k 7, the graph is obtained b shifting the graph of up units (Figure -). Skill Practice Graph the functions. n() 7. g. h Figure - n() 7 7. Quadratic Functions of the Form f() ( h) The graph of f k represents a vertical shift (up or down) of the function. Eample shows that functions of the form f h represent a horizontal shift (left or right) of the function. Eample Graphing Quadratic Functions of the Form f() ( h) Graph the functions f, g, and h on the same coordinate sstem. f g h Solution: Several function values for f, g, and h are shown in Table - for selected values of. The corresponding graphs are pictured in Figure -9. Table - f() g() ( ) h() ( ) 9 9 0 9 0 0 9 0 9 9 0 0 g() ( ) f() h() ( ) 0 Figure -9 0 Skill Practice Answers. 0 g() 0 0 h() 0
Chapter Quadratic Equations and Functions Skill Practice. Refer to the graph of f h to determine the value of h. 0 0 0 0 Eample illustrates the following properties of quadratic functions of the form f h. Graphs of f() ( h) If h 7 0, then the graph of f h is the same as the graph of shifted h units to the right. If h 0, then the graph of f h is the same as the graph of shifted 0h 0 units to the left. From Eample we have h and g g 0 shifted units shifted 0 unit to the right to the left Eample Graphing Functions of the Form f() ( h) Sketch the functions p and q. a. p 7 b. q. Solution: a. p 7 Because h 7 7 0, shift the graph of to the right 7 units (Figure -0). 0 p() 0 0 p() ( 7) Figure -0 0 Skill Practice Answers. h
Section. Graphs of Quadratic Functions b. q. q. Because h. 0, shift the graph of to the left. units (Figure -). 0 q() 0 q() (.) 0 Figure - 0 Skill Practice Graph the functions.. g. h. Quadratic Functions of the Form f() a Eamples and investigate functions of the form f a a 0. Eample Graphing Functions of the Form f() a Graph the functions f, g, and h on the same coordinate sstem. f g h Solution: Several function values for f, g, and h are shown in Table - for selected values of. The corresponding graphs are pictured in Figure -. Table - f() g() h() 9 9 0 0 0 0 9 Skill Practice 9 0 0 7. Graph the functions f, g, and h on the same coordinate sstem. f g h g() f() h() 0 Figure - 0 Skill Practice Answers. 7. g() ( ) h() ( ) 0 0 0 0 g() f() 0 h() 0 0 0
Chapter Quadratic Equations and Functions In Eample, the function values defined b g are twice those of f. The graph of g is the same as the graph of f stretched verticall b a factor of (the graph appears narrower than f ). In Eample, the function values defined b h are one-half those of f. The graph of h is the same as the graph of f shrunk verticall b a factor of (the graph appears wider than f ). Eample Graphing Functions of the Form f () a Graph the functions f, g, and h on the same coordinate sstem. f g h Solution: Several function values for f, g, and h are shown in Table - for selected values of. The corresponding graphs are pictured in Figure -. Table - f() g() h() 0 0 0 0 9 7 9 7 7 g() = f() = Figure - h() = Skill Practice. Graph the functions f, g, and h on the same coordinate sstem. f g h Skill Practice Answers. 0 0 0 0 f() = g() = h() = Eample illustrates that if the coefficient of the square term is negative, the parabola opens down. The graph of g is the same as the graph of f with a vertical stretch b a factor of 0 0. The graph of h is the same as the graph of f with a vertical shrink b a factor of 0 0.
Section. Graphs of Quadratic Functions 7 Graphs of f() a. If a 7 0, the parabola opens upward. Furthermore, If 0 a, then the graph of f a is the same as the graph of with a vertical shrink b a factor of a. If a 7, then the graph of f a is the same as the graph of with a vertical stretch b a factor of a.. If a 0, the parabola opens downward. Furthermore, If 0 0a 0, then the graph of f a is the same as the graph of with a vertical shrink b a factor of 0a 0. If 0a 0 7, then the graph of f a is the same as the graph of with a vertical stretch b a factor of 0a 0.. Quadratic Functions of the Form f() a( h) k We can summarize our findings from Eamples b graphing functions of the form f a h k a 0. The graph of has its verte at the origin 0, 0. The graph of f a h k is the same as the graph of shifted to the right or left h units and shifted up or down k units. Therefore, the verte is shifted from (0, 0) to (h, k). The ais of smmetr is the vertical line through the verte, that is, the line h. Graphs of f() a( h) k. The verte is located at h, k.. The ais of smmetr is the line h.. If a 7 0, the parabola opens upward, and k is the minimum value of the function.. If a 0, the parabola opens downward, and k is the maimum value of the function. a 0 a 0 f() f() (h, k) Minimum point Maimum point (h, k) a 0 h h Eample 7 Given the function defined b Graphing a Function of the Form f( ) a( h) k f a. Identif the verte. b. Sketch the function. c. Identif the ais of smmetr. d. Identif the maimum or minimum value of the function.
Chapter Quadratic Equations and Functions Calculator Connections Some graphing calculators have Minimum and Maimum features that enable the user to approimate the minimum and maimum values of a function. Otherwise Zoom and Trace can be used. Solution: a. The function is in the form f a h k, where a, h, and k. Therefore, the verte is,. b. The graph of f is the same as the graph of shifted to the right units, shifted up units, and stretched verticall b a factor of (Figure -). c. The ais of smmetr is the line. d. Because a 7 0, the function opens upward. Therefore, the minimum function value is. Notice that the minimum value is the minimum -value on the graph. Skill Practice 9. Given the function defined b g a. Identif the verte. b. Sketch the graph. c. Identif the ais of smmetr. d. Identif the maimum or minimum value of the function. f() 0 (, ) 0 0 0 Figure - f() ( ) Eample Given the function defined b Graphing a Function of the Form f() a( h) k g 7 a. Identif the verte. b. Sketch the function. c. Identif the ais of smmetr. d. Identif the maimum or minimum value of the function. Solution: a. g 7 a 7 b Skill Practice Answers 9a. Verte: (, ) b. 0 0 0 c. Ais of smmetr: d. Minimum value: 0 The function is in the form g a h k, where a, h, and k 7. Therefore, the verte is, 7. b. The graph of g is the same as the graph of shifted to the left units, shifted down 7 units, and opening downward (Figure -). c. The ais of smmetr is the line. d. The parabola opens downward, so the maimum function value is 7. 7 (, ) 0 g() 0 0 Figure - 0 7 g() ( )
Section. Graphs of Quadratic Functions 9 Skill Practice 0. Given the function defined b h a. Identif the verte. b. Sketch the graph. c. Identif the ais of smmetr. d. Identif the maimum or minimum value of the function. Skill Practice Answers 0a. Verte: (, ) b. 0 0 0 c. Ais of smmetr: d. Maimum value: 0 Section. Boost our GRADE at mathzone.com! Stud Skills Eercise Practice Eercises Practice Problems Self-Tests NetTutor e-professors Videos. Define the ke terms. a. Parabola b. Verte c. Ais of smmetr d. Maimum value e. Minimum value Review Eercises For Eercises, solve the equations.. 0.. a 7 a. tt. z z 9 0 7. / / 0. m m 7 Concept : Quadratic Functions of the Form f () k 9. Describe how the value of k affects the graphs of functions of the form f k. For Eercises 0 9, graph the functions. 0. g. f. p 7 7
90 Chapter Quadratic Equations and Functions. q. T(). S 7 7. M 7. n. P 7 9. Q 7 Concept : Quadratic Functions of the Form f () ( h) 0. Describe how the value of h affects the graphs of functions of the form f h. For Eercises 0, graph the functions.. r. h. k. L 7 7 7 7 7 7
Section. Graphs of Quadratic Functions 9.. 7.. V a W a r a A a b b b b 7 7 7 7 9. M. 0. N. 7 7 Concept : Quadratic Functions of the Form f () a. Describe how the value of a affects the graph of a function of the form f a, where a 0.. How do ou determine whether the graph of a function defined b h a b c a 0 opens up or down? For Eercises 0, graph the functions.. f. g. h. f
9 Chapter Quadratic Equations and Functions 7. c. 9. v 0. f g Concept : Quadratic Functions of the Form f () a( h) k For Eercises, match the function with its graph.. f. g. k. h. t. m 7. p. n a. b. c. d. e. f. g. h. For Eercises 9, graph the parabola and the ais of smmetr. Label the coordinates of the verte, and write the equation of the ais of smmetr. 9. 0... 7 7 7 7 7
Section. Graphs of Quadratic Functions 9.... 7 7 7 7 7 7.. 9. 0. 7 7 7... 9 7 7 7... 7 7 9 7
9 Chapter Quadratic Equations and Functions For Eercises 7 7, write the coordinates of the verte and determine if the verte is a maimum point or a minimum point. Then write the maimum or minimum value. 7. f 9. g 7 9. p 70. h 7. k 7. 7 0 m 9 7. n 7. q 7. A 7 7. B 77. F 7 7. G 7 79. True or false: The function defined b g has a maimum value but no minimum value. 0. True or false: The function defined b f has a maimum value but no minimum value.. True or false: If the verte, represents a minimum point, then the minimum value is.. True or false: If the verte, represents a maimum point, then the maimum value is.. A suspension bridge is 0 ft long. Its supporting cable hangs in a shape that resembles a parabola. The function defined b H 90 0 0 where 0 0 approimates the height of the supporting cable a distance of ft from the end of the bridge (see figure). a. What is the location of the verte of the parabolic cable? b. What is the minimum height of the cable? c. How high are the towers at either end of the supporting cable? H() 0 0 0 ft. A 0-m bridge over a crevasse is supported b a parabolic arch. The function defined b f 0. 00 where 0 0 approimates the height of the supporting arch meters from the end of the bridge (see figure). f() 0 m 0 0 a. What is the location of the verte of the arch? b. What is the maimum height of the arch (relative to the origin)?
Section. Verte of a Parabola and Applications 9 Graphing Calculator Eercises For Eercises, verif the maimum and minimum points found in Eercises 7 70, b graphing each function on the calculator.. Y. Y 9 Eercise 7 7 Eercise 7. Y. Y 7 0 Eercise 70 Eercise 9 Verte of a Parabola and Applications. Writing a Quadratic Function in the Form f() a( h) k A quadratic function can be epressed as f a b c a 0. However, b completing the square, we can write the function in the form f a h k. In this form, the verte is easil recognized as (h, k). The process to complete the square in this contet is similar to the steps outlined in Section.. However, all algebraic steps are performed on the right-hand side of the equation. Eample Writing a Quadratic Function in the Form f() a( h) k (a 0) Given: f a. Write the function in the form f a h k. b. Identif the verte, ais of smmetr, and minimum function value. Solution: a. f Rather than dividing b the leading coefficient on both sides, we will factor out the leading coefficient from the variable terms on the right-hand side. Net, complete the square on the epression within the parentheses:. Section. Concepts. Writing a Quadratic Function in the Form f() a( h) k. Verte Formula. Determining the Verte and Intercepts of a Quadratic Function. Verte of a Parabola: Applications Avoiding Mistakes: Do not factor out the leading coefficient from the constant term. Rather than add to both sides of the function, we add and subtract within the parentheses on the right-hand side. This has the effect of adding 0 to the right-hand side.
9 Chapter Quadratic Equations and Functions Use the associative propert of addition to regroup terms and isolate the perfect square trinomial within the parentheses. Factor and simplif. b. f The verte is,. The ais of smmetr is. Because a 7 0, the parabola opens upward. The minimum value is (Figure -). 0 (, ) Skill Practice 0. Given: f Figure - a. Write the equation in the form f a h k. b. Identif the verte, ais of smmetr, and minimum value of the function. 0 f() ( ) 0 Eample Analzing a Quadratic Function Given: f a. Write the function in the form f a h k. b. Find the verte, ais of smmetr, and maimum function value. c. Find the - and -intercepts. d. Graph the function. Skill Practice Answers a. f 7 b. Verte:, 7; ais of smmetr: ; minimum value: 7 Solution: a. f To find the verte, write the function in the form f a h k. 9 9 9 9 Factor the leading coefficient from the variable terms. Add and subtract the quantit 9 within the parentheses. To remove the term 9 from the parentheses, we must first appl the distributive propert. When 9 is removed from the parentheses, it carries with it a factor of. Factor and simplif.
Section. Verte of a Parabola and Applications 97 b. f The verte is (, ). The ais of smmetr is. Because a 0, the parabola opens downward and the maimum value is. c. The -intercept is given b f 0 0 0. The -intercept is 0,. To find the -intercept(s), find the real solutions to the equation f 0. f 0 Substitute f 0. 0 Factor. 0 or The -intercepts are (, 0) and (, 0). d. Using the information from parts (a) (c), sketch the graph (Figure -7). 0 (, ) f() Figure -7 Skill Practice. Given: g a. Write the equation in the form g a h k. b. Identif the verte, ais of smmetr, and maimum value of the function. c. Determine the - and -intercepts. d. Graph the function.. Verte Formula Completing the square and writing a quadratic function in the form f a h k a 0 is one method to find the verte of a parabola. Another method is to use the verte formula. The verte formula can be derived b completing the square on the function defined b f a b c a 0. f a b c a 0 a a b a b c a a b a b a b a b c a a b a b b ab aa a b c Factor a from the variable terms. Add and subtract b a b a within the parentheses. Appl the distributive propert and remove the term b a from the parentheses. Skill Practice Answers a. g b. Verte: (, ); ais of smmetr: ; maimum value: c. -intercepts: (, 0) and (, 0); -intercept: (0, ) d. 0 0 0 0
9 Chapter Quadratic Equations and Functions a a b a b a a b a b a a b a b a a b a b b a c c b a ac a b a a c a b a bd ac b a ac b a Factor the trinomial and simplif. Appl the commutative propert of addition to reverse the last two terms. Obtain a common denominator. f a h k The function is in the form f a h k, where Hence, the verte is h b a a b a and k ac b, b a ac b a Although the -coordinate of the verte is given as ac b a, it is usuall easier to determine the -coordinate of the verte first and then find b evaluating the function at b a. The Verte Formula For f a b c a 0, the verte is given b a b a ac b, a b or a b a, fa b a bb. Determining the Verte and Intercepts of a Quadratic Function Eample Determining the Verte and Intercepts of a Quadratic Function Given: h a. Use the verte formula to find the verte. b. Find the - and -intercepts. c. Sketch the function.
Section. Verte of a Parabola and Applications 99 Solution: a. h a b c Identif a, b, and c. b The -coordinate of the verte is. a The -coordinate of the verte is h. The verte is (, ). b. The -intercept is given b h0 0 0. The -intercept is (0, ). To find the -intercept(s), find the real solutions to the equation h 0. h 0 0 i i This quadratic equation is not factorable. Appl the quadratic formula: a, b, c. The solutions to the equation h 0 are not real numbers. Therefore, there are no -intercepts. c. Skill Practice 7 (0, ) Figure - h() (, ) Because a 0, the parabola opens up. TIP: The location of the verte and the direction that the parabola opens can be used to determine whether the function has an -intercepts. Given h(), the verte (, ) is above the -ais. Furthermore, because a 7 0, the parabola opens upward. Therefore, it is not possible for the function h to cross the -ais (Figure -).. Given: f a. Use the verte formula to find the verte of the parabola. b. Determine the - and -intercepts. c. Sketch the graph. Skill Practice Answers a. Verte:, b. -intercepts: none; -intercepts: (0, ) c. 0 0 0 0
00 Chapter Quadratic Equations and Functions. Verte of a Parabola: Applications Eample Appling a Quadratic Function The crew from Etravaganza Entertainment launches fireworks at an angle of 0 from the horizontal. The height of one particular tpe of displa can be approimated b the following function: ht t t where h(t) is measured in feet and t is measured in seconds. a. How long will it take the fireworks to reach their maimum height? Round to the nearest second. b. Find the maimum height. Round to the nearest foot. Calculator Connections Some graphing calculators have Minimum and Maimum features that enable the user to approimate the minimum and maimum values of a function. Otherwise, Zoom and Trace can be used. Solution: ht t t a b c 0 The -coordinate of the verte is b a.9 The -coordinate of the verte is approimatel This parabola opens downward; therefore, the maimum height of the fireworks will occur at the verte of the parabola. Identif a, b, and c, and appl the verte formula. h.9.9.9 7 The verte is (.9, 7). a. The fireworks will reach their maimum height in.9 sec. b. The maimum height is 7 ft. Skill Practice. An object is launched into the air with an initial velocit of ft/sec from the top of a building ft high. The height h(t) of the object after t seconds is given b h t t t a. Find the time it takes for the object to reach its maimum height. b. Find the maimum height. Eample Appling a Quadratic Function Skill Practice Answers a.. sec b. ft A group of students start a small compan that produces CDs. The weekl profit (in dollars) is given b the function P 0 00 where represents the number of CDs produced. a. Find the -intercepts of the profit function, and interpret the meaning of the -intercepts in the contet of this problem. b. Find the -intercept of the profit function, and interpret its meaning in the contet of this problem.
Section. Verte of a Parabola and Applications 0 c. Find the verte of the profit function, and interpret its meaning in the contet of this problem. d. Sketch the profit function. Solution: a. P 0 00 The -intercepts are the real solutions of the equation P 0. 0 0 00 0 0 00 0 0 0 0 or 0 Solve b factoring. The -intercepts are (0, 0) and (0, 0). The -intercepts represent points where the profit is zero. These are called break-even points. The break-even points occur when 0 CDs are produced and also when 0 CDs are produced. b. The -intercept is P0 0 00 00 00. The -intercept is 0, 00. The -intercept indicates that if no CDs are produced, the compan has a $00 loss. c. The -coordinate of the verte is b a 0 0 The -coordinate is P0 0 00 00 00. The verte is (0, 00). A maimum weekl profit of $00 is obtained when 0 CDs are produced. d. Using the information from parts (a) (c), sketch the profit function (Figure -9). 00 Profit 0 00 0 0 0 0 00 Skill Practice 00 Number of CDs Produced Figure -9. The weekl profit function for a tutoring service is given b P 0 00, where is the number of hours that the tutors are scheduled to work and P represents profit in dollars. a. Find the -intercepts of the profit function, and interpret their meaning in the contet of this problem. b. Find the -intercept of the profit function, and interpret its meaning in the contet of this problem. c. Find the verte of the profit function, and interpret its meaning in the contet of this problem. d. Sketch the profit function. Skill Practice Answers a. -intercepts: (, 0) and (, 0). The -intercepts represent the breakeven points, where the profit is 0. b. -intercept: 0, 00. If the service does no tutoring, it will have a $00 loss. c. Verte: (, 00). A maimum weekl profit of $00 is obtained when hr of tutoring is scheduled. d. Profit 00 00 00 0 00 0 0 0 00 00 Number of Tutoring Hours
0 Chapter Quadratic Equations and Functions Eample Appling a Quadratic Function The average number of visits to office-based phsicians is a function of the age of the patient. N 0.00 0.0. where is a patient s age in ears and N() is the average number of doctor visits per ear (Figure -0). Find the age for which the number of visits to office-based phsicians is a minimum. Solution: Number of Visits N() 7 0 0 Average Number of Visits to a Doctor s Office per Year Based on Age of Patient 0 N 0.00 0.0. b a 0.0. 0.00 0 0 0 0 0 70 7 0 90 Age (ears) Figure -0 (Source: U.S. National Center for Health Statistics.) Find the -coordinate of the verte. The average number of visits to office-based phsicians is lowest for people approimatel. ears old. Skill Practice Answers. The maimum price for the stock occurred after 9 months. Skill Practice. In a recent ear, an Internet compan started up, became successful, and then quickl went bankrupt. The price of a share of the compan s stock is given b the function S. 0..7 where is the number of months that the stock was on the market. Find the number of months at which the stock price was a maimum. Section. Boost our GRADE at mathzone.com! Stud Skills Eercise. Write the verte formula. Practice Eercises Practice Problems Self-Tests NetTutor e-professors Videos
Section. Verte of a Parabola and Applications 0 Review Eercises. How does the graph of f compare with the graph of?. How does the graph of p compare with the graph of?. How does the graph of Q compare with the graph of?. How does the graph of r 7 compare with the graph of?. How does the graph of s compare with the graph of? 7. How does the graph of t 0 compare with the graph of?. Find the coordinates of the verte of the graph of g. For Eercises 9, find the value of n to complete the square. 9. n 0. n. 7 n. a a n... t m b. 9 b n 7 m n t n p p n Concept : Writing a Quadratic Function in the Form f() a( h) k For Eercises 7 0, write the function in the form f a h k b completing the square. Then identif the verte. 7. g. h 9. n 0. f 9. p.. k 7 0. m.. g 7. F 0. q f G 7 9. P 0. Q Concept : Verte Formula For Eercises, find the verte b using the verte formula.. Q 7. T 7.. s 9. N. r M 7. m. n 9. k 0. h. f. g 9. A. B
0 Chapter Quadratic Equations and Functions Concept : Determining the Verte and Intercepts of a Quadratic Function For Eercises 0 a. Find the verte. b. Find the -intercept. c. Find the -intercept(s), if the eist. d. Use this information to graph the function.. 9. 7 0 7. 0 0 0. 9. 9 9 0. 7 9 7 0 0 0 7 Concept : Verte of a Parabola: Applications. Mia sells used MP plaers. The average cost to package MP plaers is given b the equation C 0 00, where is the number of MP plaers she packages in a week. How man plaers must she package to minimize her average cost?. Ben sells used ipods. The average cost to package ipods is given b the equation C 0 00, where is the number of ipods packaged per month. Determine the number of ipods that Ben needs to package to minimize the average cost.. The pressure in an automobile tire can affect its wear. Both overinflated and underinflated tires can lead to poor performance and poor mileage. For one particular tire, the function P represents the number of miles that a tire lasts (in thousands) for a given pressure. P 0.7. 0 where is the tire pressure in pounds per square inch (psi). a. Find the tire pressure that will ield the maimum mileage. Round to the nearest pound per square inch. b. What is the maimum number of miles that a tire can last? Round to the nearest thousand.
Section. Verte of a Parabola and Applications 0. A baseball plaer throws a ball, and the height of the ball (in feet) can be approimated b 0.0 0.77 where is the horizontal position of the ball measured in feet from the origin. a. For what value of will the ball reach its highest point? Round to the nearest foot. b. What is the maimum height of the ball? Round to the nearest tenth of a foot. 0 0. For a fund-raising activit, a charitable organization produces cookbooks to sell in the communit. The profit (in dollars) depends on the number of cookbooks produced,, according to a. How much profit is made when 00 cookbooks are produced? b. Find the -intercept of the profit function, and interpret its meaning in the contet of this problem. c. How man cookbooks must be produced for the organization to break even? (Hint: Find the -intercepts.) d. Find the verte. e. Sketch the function. p where 0. 0 0, f. How man cookbooks must be produced to maimize profit? What is the maimum profit? Profit ($) p() 00 00 000 00 00 00 00 0 00 00 00 00. A jewelr maker sells bracelets at art shows. The profit (in dollars) depends on the number of bracelets produced,, according to p where 0. 0 0 a. How much profit does the jeweler make when 0 bracelets are produced? b. Find the -intercept of the profit function, and interpret its meaning in the contet of this problem. c. How man bracelets must be produced for the jeweler to break even? Round to the nearest whole unit. 00 00 00 00 00 00 Number of Cookbooks Produced
0 Chapter Quadratic Equations and Functions d. Find the verte. e. Sketch the function. f. How man bracelets must be produced to maimize profit? What is the maimum profit? 7. Gas mileage depends in part on the speed of the car. The gas mileage of a subcompact car is given b the function m 0.0. 9, where 0 70 represents the speed in miles per hour and m() is given in miles per gallon. At what speed will the car get the maimum gas mileage? Profit ($) p() 000 000 000 000 0 000 00 00 00 00 000 000 Number of Bracelets Produced. Gas mileage depends in part on the speed of the car. The gas mileage of a luur car is given b the function L 0.0., where 70 represents the speed in miles per hour and L() is given in miles per gallon. At what speed will the car get the maimum gas mileage? 9. Tetanus bacillus bacterium is cultured to produce tetanus toin used in an inactive form for the tetanus vaccine. The amount of toin produced per batch increases with time and then becomes unstable. The amount of toin (in grams) as a function of time t (in hours) can be approimated b the following function. bt t t How man hours will it take to produce the maimum ield? 0. The bacterium Pseudomonas aeruginosa is cultured with an initial population of 0 active organisms. The population of active bacteria increases up to a point, and then due to a limited food suppl and an increase of waste products, the population of living organisms decreases. Over the first hr, the population can be approimated b the following function. Pt 7.7t,00t 0,000 where 0 t Find the time required for the population to reach its maimum value. Epanding Your Skills. A farmer wants to fence a rectangular corral adjacent to the side of a barn; however, she has onl 00 ft of fencing and wants to enclose the largest possible area. See the figure. a. If represents the length of the corral and represents the width, eplain wh the dimensions of the corral are subject to the constraint 00. corral b. The area of the corral is given b A. Use the constraint equation from part (a) to epress A as a function of, where 0 00. c. Use the function from part (b) to find the dimensions of the corral that will ield the maimum area. [Hint: Find the verte of the function from part (b).]
Section. Verte of a Parabola and Applications 07. A veterinarian wants to construct two equal-sized pens of maimum area out of 0 ft of fencing. See the figure. a. If represents the length of each pen and represents the width of each pen, eplain wh the dimensions of the pens are subject to the constraint 0. b. The area of each individual pen is given b A. Use the constraint equation from part (a) to epress A as a function of, where 0 0. c. Use the function from part (b) to find the dimensions of an individual pen that will ield the maimum area. [Hint: Find the verte of the function from part (b).] Graphing Calculator Eercises For Eercises, graph the functions in Eercises 0 on a graphing calculator. Use the Ma or Min feature or Zoom and Trace to approimate the verte.. Y (Eercise ). Y 9 7 0 (Eercise ). Y (Eercise 7). Y 9 (Eercise ) 7. Y (Eercise 9). Y (Eercise 0) 9 9
0 Chapter Quadratic Equations and Functions Chapter SUMMARY Ke Concepts The square root propert states that If Section. k then k Square Root Propert and Completing the Square Eamples Eample i (square root propert) Follow these steps to solve a quadratic equation in the form a b c 0 a 0 b completing the square and appling the square root propert:. Divide both sides b a to make the leading coefficient.. Isolate the variable terms on one side of the equation.. Complete the square: Add the square of one-half the linear term coefficient to both sides of the equation. Then factor the resulting perfect square trinomial.. Appl the square root propert and solve for. Eample 0 0 Note: c d 9 9 9 9 0 9 a b 9 9 9 A or 9
Summar 09 Section. Quadratic Formula Ke Concepts The solutions to a quadratic equation a b c 0 a 0 are given b the quadratic formula b b ac a The discriminant of a quadratic equation a b c 0 is b ac. If a, b, and c are rational numbers, then. If b ac 7 0, then there will be two real solutions. Moreover, a. If b ac is a perfect square, the solutions will be rational numbers. b. If b ac is not a perfect square, the solutions will be irrational numbers.. If b ac 0, then there will be two imaginar solutions.. If b ac 0, then there will be one rational solution. Eample Eample i i 0 a b c The discriminant is. Therefore, there will be two imaginar solutions. Three methods to solve a quadratic equation are. Factoring and appling the zero product rule.. Completing the square and appling the square root propert.. Using the quadratic formula. Section. Equations in Quadratic Form Ke Concepts Substitution can be used to solve equations that are in quadratic form. Eamples Eample 0 Let u. Therefore, u u u 0 u u 0 u or u or or 7 Cube both sides.
0 Chapter Quadratic Equations and Functions Section. Graphs of Quadratic Functions Ke Concepts A quadratic function of the form f k shifts the graph of up k units if k 7 0 and down 0k 0 units if k 0. Eamples Eample f() A quadratic function of the form f h shifts the graph of right h units if h 7 0 and left 0h 0 units if h 0. Eample f() ( ) The graph of a quadratic function of the form f a is a parabola that opens up when a 7 0 and opens down when a 0. If 0a 0 7, the graph of is stretched verticall b a factor of 0a 0. If 0 0a 0, the graph of is shrunk verticall b a factor of 0a 0. Eample f() A quadratic function of the form f a h k has verte (h, k). If a 7 0, the verte represents the minimum point. If a 0, the verte represents the maimum point. Eample 7 f() ( ) minimum point (, )
Review Eercises Section. Verte of a Parabola and Applications Ke Concepts Completing the square is a technique used to write a quadratic function f a b c a 0 in the form f a h k for the purpose of identifing the verte (h, k). The verte formula finds the verte of a quadratic function f a b c a 0. The verte is a b a ac b, a b or a b a, f a b a bb Eamples Eample f The verte is,. Because a 7 0, the parabola opens upward and the verte, is a minimum point. Eample f a b c f a b a b a b The verte is,. Because a 0, the parabola opens downward and the verte, is a maimum point. Chapter Review Eercises Section. For Eercises, solve the equations b using the square root propert.... a. b 9 0. Use the square root propert to find the length of the sides of a square whose area is in.. Use the square root propert to find the length of the sides of a square whose area is 0 in. Round the answer to the nearest tenth of an inch.. 7. 9 7.. m 9. The length of each side of an equilateral triangle is 0 in. Find the height of the triangle. Round the answer to the nearest tenth of an inch. 0 in. h 0 in. 0 in. For Eercises, find the value of n so that the epression is a perfect square trinomial. Then factor the trinomial.. n. 9 n.. z n z n
Chapter Quadratic Equations and Functions For Eercises, solve the equation b completing the square and appling the square root propert.. w w 0 7. 0. 9. b 7 b 0.. t t 0 Section.. Eplain how the discriminant can determine the tpe and number of solutions to a quadratic equation with rational coefficients. For Eercises, determine the tpe (rational, irrational, or imaginar) and number of solutions for the equations b using the discriminant.... z 7z. a a 0 7. 0b b. 0 For Eercises 9, solve the equations b using the quadratic formula. 9. 0 0. m m 0 a. Find the landing distance for a plane traveling 0 ft/sec at touchdown. b. If the landing speed is too fast, the pilot ma run out of runwa. If the speed is too slow, the plane ma stall. Find the maimum initial landing speed of a plane for a runwa that is 000 ft long. Round to decimal place.. The recent population (in thousands) of Kena can be approimated b Pt.t.t,, where t is the number of ears since 97. a. If this trend continues, approimate the number of people in Kena in the ear 0. b. In what ear after 97 will the population of Kena reach 0 million? (Hint: 0 million equals 0,000 thousand.) Section. For Eercises, solve the equations b using substitution, if necessar.. 0.. n n 0 0. aa 0 a.. m m 0. b. k 0.k 0.0 b. 0. 0 For Eercises 7 0, solve using an method. 7.. b b 7.. 9. 0. t t 0 p p 0 t t t m m m 9. 0. a. 0. The landing distance that a certain plane will travel on a runwa is determined b the initial landing speed at the instant the plane touches down. The function D relates landing distance in feet to initial landing speed s: Ds for s 0 0 s s where s is in feet per second.. 0
Review Eercises Section. For Eercises 0, graph the functions.. g. f 7 7 7 7. h. k 9 7 7 7 7. m. n For Eercises, write the coordinates of the verte of the parabola and determine if the verte is a maimum point or a minimum point. Then write the maimum or the minimum value... For Eercises, write the equation of the ais of smmetr of the parabola.. a a b. Section. For Eercises, write the function in the form f a h k b completing the square. Then write the coordinates of the verte.. z 7. 7.. t s 7 7 w a b 9 b p 0 q For Eercises 9 7, find the coordinates of the verte of each function b using the verte formula. 9. f 7 9. p 0. q 7 9 7 7 70. g 7. m 7. n 7 7. For the quadratic equation a. Write the coordinates of the verte. b. Find the - and -intercepts.
Chapter Quadratic Equations and Functions c. Use this information to sketch a graph of the parabola. 7. The height of a projectile fired verticall into the air from the ground is given b the equation ht t 9t, where t represents the number of seconds that the projectile has been in the air. How long will it take the projectile to reach its maimum height? 7. The weekl profit for a small catering service is given b P 00, where is the number of meals prepared. Find the number of meals that should be prepared to obtain the maimum profit. Chapter Test For Eercises, solve the equation b using the square root propert... p.. Find the value of n so that the epression is a perfect square trinomial. Then factor the trinomial d 7d n. For Eercises, solve the equation b completing the square and appling the square root propert.. 0. 7 For Eercises 7 a. Write the equation in standard form a b c 0. b. Identif a, b, and c. c. Find the discriminant. d. Determine the number and tpe (rational, irrational, or imaginar) of solutions. 7.. For Eercises 9 0, solve the equation b using the quadratic formula. 9. 0. m 0. The base of a triangle is ft less than twice the height. The area of the triangle is ft. Find the base and the height. Round the answers to the nearest tenth of a foot.. A circular garden has an area of approimatel 0 ft. Find the radius. Round the answer to the nearest tenth of a foot. For Eercises, solve the equation b using substitution, if necessar.. 0. / /.. p p 7. For Eercises, find the - and -intercepts of the function. Then match the function with its graph.. f 9. 0. 0 0 k 9 p. q a. b. 0 0 0 0 0 0 0 0 0 0 0 0 70 0
Test c. d. 0 0 0 0. A child launches a to rocket from the ground. The height of the rocket can be determined b its horizontal distance from the launch pad b where and h() are in feet. 0 0 0 h() ft h() h 0 feet How man feet from the launch pad will the rocket hit the ground?. The recent population (in millions) of India can be approimated b Pt 0.t.t 00, where t 0 corresponds to the ear 97. a. If this trend continues, approimate the number of people in India in the ear 0. b. Approimate the ear in which the population of India reached billion (000 million). (Round to the nearest ear.) 7. Given the function defined b f a. Identif the verte of the parabola. b. Does this parabola open upward or downward? c. Does the verte represent the maimum or minimum point of the function? d. What is the maimum or minimum value of the function f? e. What is the ais of smmetr for this parabola?. For the function defined b g() 0, find the verte b using two methods. a. Complete the square to write g() in the form g() a( h) k. Identif the verte. b. Use the verte formula to find the verte. 9. A farmer has 00 ft of fencing with which to enclose a rectangular field. The field is situated such that one of its sides is adjacent to a river and requires no fencing. The area of the field (in square feet) can be modeled b A 00 where is the length of the side parallel to the river (measured in feet).. Eplain the relationship between the graphs of and.. Eplain the relationship between the graphs of and ( ).. Eplain the relationship between the graphs of and. Use the function to determine the maimum area that can be enclosed.
Chapter Quadratic Equations and Functions Chapters Cumulative Review Eercises. Given: A,,,, 0 and B,,, a. Find A B. b. Find A B.. Perform the indicated operations and simplif.. Simplif completel. 0 a b. Solve the equation. 0. Solve the equation. 0. What number would have to be added to the quantit 0 to make it a perfect square trinomial?. Factor completel. 0. Perform the indicated operations. Write the answer in scientific notation:. a. Factor completel..0 0.0 0 9 b. Divide b using long division. Identif the quotient and remainder. 9. Graph the line. 0. Multipl. 7. Simplif.. Jacques invests a total of $0,000 in two mutual funds. After ear, one fund produced % growth, and the other lost %. Find the amount invested in each fund if the total investment grew b $900. 9. Solve the sstem of equations. 9 9 9 0. An object is fired straight up into the air from an initial height of ft with an initial velocit of 0 ft/sec. The height in feet is given b ht t 0t where t is the time in seconds after launch. a. Find the height of the object after sec. b. Find the height of the object after 7 sec. c. Find the time required for the object to hit the ground.. a. Find the -intercepts of the function defined b g 9 0. b. What is the -intercept of g? 7. Michael Jordan was the NBA leading scorer for 0 of seasons between 97 and 99. In his 99 season, he scored a total of 7 points consisting of -point free throws, -point field goals, and -point field goals. He scored more -point shots than he did free throws. The number of -point shots was less than the number of -point shots. Determine the number of free throws, -point shots, and -point shots scored b Michael Jordan during his 99 season.. Eplain wh this relation is not a function.
Cumulative Review Eercises 7 9. Graph the function defined b f.. Solve.. Solve for f. 7. Solve.. Simplif. p q f t t t t 0. The quantit varies directl as and inversel as z. If when 0 and z 0, find when and z.. The total number of flights (including passenger flights and cargo flights) at a large airport can be approimated b F 00,000 0.00, where is the number of passengers. a. Is this function linear, quadratic, constant, or other? b. Find the -intercept and interpret its meaning in the contet of this problem. c. What is the slope of the function and what does the slope mean in the contet of this problem?. Given the function defined b g, find the function values (if the eist) over the set of real numbers. a. g 7 b. g(0) c. g(). Let m and n. Find a. The domain of m b. The domain of n 9. Given: the function defined b f() ( ) a. Write the coordinates of the verte. b. Does the graph of the function open upward or downward? c. Write the coordinates of the -intercept. d. Find the -intercepts, if possible. e. Sketch the function. 0 0 0 0 0. Find the verte of the parabola. f. Consider the function f graphed here. Find a. The domain b. The range c. f d. f() e. f(0) f. For what value(s) of is f 0?