Quadratic Equations and Functions


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1 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)
2 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, or or 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 Rewrite the equation to fit the form k. The equation is now in the form k. Appl the square root propert. i
3 } Section. Square Root Propert and Completing the Square The solutions are i and i. Check: i Check: i i 7 0 i 7 0 i 7 0 i 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 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 onehalf 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
4 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 onehalf 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 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.
5 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 onehalf 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 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 Clear parentheses. Write the equation in the form a b c 0. Step : Divide both sides b the leading coefficient.
6 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 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) or Skill Practice Answers. z 9. and
7 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 SelfTests NetTutor a. Square root propert b. Completing the square eprofessors Videos
8 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 a.. v 0 7. m 0. p 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 p p 0. q q a a 0. a a p. n 7. w w. n p p p 9. nn 7 0. mm 0..
9 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 a b c d for a a 7 0. E mc for c c 7 0 a b c for b b 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.
10 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) P() 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.
11 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.
12 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 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. ;
13 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 straightline 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.
14 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 I0 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, 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.
15 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 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 9ft 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.
16 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 Solution: For each equation, first write the equation in standard form a b c 0. Then determine the discriminant. a. b. c. Equation Discriminant b ac 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.
17 Section. Quadratic Formula 7 d b ac 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 realvalued 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
18 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 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 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
19 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 Clear parentheses and write the equation in the form a b c 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.
20 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 Section. Boost our GRADE at mathzone.com! Stud Skills Eercise. Define the ke terms. Practice Eercises a. Quadratic formula b. Discriminant Practice Problems SelfTests NetTutor eprofessors 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 For Eercises 7 0, simplif the epressions 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?
21 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 t t 7 0. p p z z... k k w w 0 7. m m a a 0... p p 0. (Hint: Clear the fractions first.) w 7 w 0 (Hint: Clear the decimals first.) n n 0 a a 9 m m 0 h h t 0.7t 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 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.
22 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 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 n n 0. p m m k 7 0. nn nn. 9 For Eercises, find the  and intercepts of the quadratic function.. f. g. f. h
23 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 b 7 0 k k 0 7. a 7. b t t t t a a 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? 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?
24 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 found in Eercise 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.
25 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
26 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.
27 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 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
28 7 Chapter Quadratic Equations and Functions 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 SelfTests NetTutor eprofessors 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
29 Section. Equations in Quadratic Form 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 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 t... t t z z z Mied Eercises For Eercises, solve the equations t m 9m
30 0 Chapter Quadratic Equations and Functions t t. a t b a t b / / 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 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 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.
31 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 h() 0 a 0 0 g() 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() g() f() h() Figure 
32 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 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
33 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() ( ) g() ( ) f() h() ( ) 0 Figure 9 0 Skill Practice Answers. 0 g() 0 0 h() 0
34 Chapter Quadratic Equations and Functions Skill Practice. Refer to the graph of f h to determine the value of h 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
35 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() Skill Practice 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() ( ) g() f() 0 h() 0 0 0
36 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 onehalf 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() g() = f() = Figure  h() = Skill Practice. Graph the functions f, g, and h on the same coordinate sstem. f g h Skill Practice Answers 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.
37 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.
38 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 (, ) 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 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 g() ( )
39 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 c. Ais of smmetr: d. Maimum value: 0 Section. Boost our GRADE at mathzone.com! Stud Skills Eercise Practice Eercises Practice Problems SelfTests NetTutor eprofessors 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 a 7 a. tt. z z / / 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
40 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
41 Section. Graphs of Quadratic Functions V a W a r a A a b b b b 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
42 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
43 Section. Graphs of Quadratic Functions
44 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 m 9 7. n 7. q 7. A 7 7. B 77. F 7 7. G 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 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() ft. A 0m bridge over a crevasse is supported b a parabolic arch. The function defined b f 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)?
45 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 righthand 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 righthand 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 righthand side. This has the effect of adding 0 to the righthand side.
46 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 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.
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