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1 QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a Quadratic Function.09 Maimum and Minimum Values..... Finding X and Y intercepts of Parabolas 4 Finding the Range and Domain....5 Properties of Parabolas Graphing Parabolas...9 Intervals Finding the Equation of a Parabola Finding a Quadratic Equation.. 3 Word Problems with Quadratics

4 Here is a more complicated eample = 0 Here is the quadratic equation in standard form a b c + + = = 0 Begin by multiplying the entire equation by ½. 7 3 = 3 Subtract to both sides. b = = = = 4 6 Now we need to create a perfect square binomial. We need to find the missing number, so we need to evaluate b = Once we evaluate b over squared, we add the result to both sides of the equation. 7 5 = 4 6 Now, on the left side of the equal sign is the perfect square in factored form. When b over is evaluated, the result is -7/4. That is the number that goes in the binomial on the left. 7 5 = ± 4 6 Now take the square root of both sides of the equation. Do not forget to use ± on the right side. 7 5 = ± Simplify the radical if possible = ± = + and = = = 4 4 = 3 and = Now add 7/4 to both sides. And simplify the solutions if possible. In this eample, once everything is divided by, the leading coefficient is one. This means the completing the square method can now be used. This is a complicated problem because the b term is a fraction. Which means all those rules regarding fractions will be coming into play. Be mindful of such problems in the future. Being able to complete the square is VITAL as it will be used many times in the future!!! 96

5 For following problems, you are to find the number needed to create a perfect square binomial. A) 6 + B) 3 + C) 7 + D) + + E) 4 + F) + 3 Find all solutions to the following quadratic equations by completing the square. A) ( 3) = B) ( ) + 5 = 7 C) 4 6 = 0 D) = 0 E) = 0 F) + 3 = 0 G) = H) 6 7 = 0 I) 9 + = 0 J) = K) = L) + =

6 M) = 0 N) = 0 O) = 0 Do not forget to check for etraneous roots for the following problems. P) = Q) a a + = 0 R) = + S) 5 = T) 6 + = + U) =

7 The Quadratic Formula The solutions to any quadratic equation in the form found by using the quadratic formula. + + = 0, where a 0, can be a b c ± = b b 4ac a The quadratic formula can be derived when solving the general quadratic equation, a + b + c = 0, by completing the square. Beginning with the general quadratic equation formula. a b c + + = 0, derive the quadratic There are times when completing the square can be complicated. This happens in particular, when the leading coefficient is a number other than and the b term is odd. If the b term is odd it will be a bit more complicated to evaluate b. In these cases, it is easier to just use the quadratic formula. Since it has been proven that the quadratic formula is actually derived by completing the square, the solutions will be identical. 99

8 Solve each of the following quadratic equations by using the quadratic formula. A) = B) = C) = 0 D) 4 7 = 0 E) = F) a + 5a = 0 G) 5t t 0 + = H) = 0 I) a + a + = 0 + = 4 K) J) ( )( ) = L) =

9 M) ( + 3) = 5 N) 5 = 0 O) = 0 P) + = 0 Q) 6 + = + R) 3 = S) + + = + T) = U) m m + = 0 4 0

10 The Discriminant ± = b b 4ac a Within the quadratic formula lies the discriminant. The discriminant, about your solutions. b 4ac, tells you If b 4ac > 0 you will have two real solutions Ifb 4ac = 0 you will have one real solution If b 4ac < 0 you will have no real solutions Think about what this means in terms of the graph of the function. The quadratic formula allows you to find roots of a quadratic equation. If the quadratic is given to you as a function, then you are finding the zeros of the function. The only difference is an equation verses a function. In other words, one of them has f ( ) in front of it. In either case, whether you are being asked to find the roots of the equation, or the zeros of the function, you are solving for. So, finding the roots of an equation involve the same procedures as finding the zeros of a function. As we have discussed in class, the zeros of a function are the intercepts of the graph of that function. Therefore, as an etension, you can conclude the following. If If If b b b 4ac > 0 you will have two intercepts 4ac = 0 you will have one intercept 4ac < 0 you will have no intercepts If the graph of a quadratic function is a parabola, how is it possible to have only one intercept? How can you have a parabola with no intercepts? 0

11 Determine the nature of the roots of each of the following quadratic equations without solving. Tell whether you have two or one real solutions, or no real solutions. A) 4 0 = B) ( ) 5 = + C) + + = D) = E) = F) = G) = H) ( ) ( ) = 0 I) = J) = K) + 3 = 0 L) = 0 M) = N) = O) + = What do you notice about the quadratic equations that have one real solution? 03

13 Solve each of the following equations. A) ( ) ( ) + 4 = 0 B) 7 + = 0 C) y y + 36 = 0 D) ( 3a) 3( 3a) + 36 = 0 4 E)( ) ( ) = 0 F) = 0 G) = 0 H) ( a 3) 4( a 3) =

14 The Standard Form of a Parabola In this class we will be graphing quadratic functions using the standard form of a parabola. y = a( h) + k This is the standard form of a parabola, where a, h and k represent real number constants. In Algebra I, students are taught to graph a parabola in the form y = a + b + c, by first b finding the verte. This meant the student had to first evaluate to find the value of the a verte. The student would then proceed to evaluate to find the y value of the verte. In other words, first find the value of the verte, then plug it back into the problem to find the y value. From here the student would set up a table to find any additional coordinates. The results would then be graphed. From this point forward, you will be given the function in the form y a b c = + +, and rewrite it in the standard form, ( ) f b a y = a h + k. Once in this form, it is easier to identify the verte, -intercepts and any other information you need to graph the function. Once the function is in standard form, the verte is given by ( h, k ). The a value determines if the function opens up or down, and gives some indication as to how wide or narrow the graph will be. The h value dictates how the function shifts horizontally. Remember, in the standard form of a parabola you see h, but in the verte you see h. This means the value of the verte is the opposite of what you see when it is standard form. Remember P.L.N.R., Positive Left Negative Right, and it will help you with the shift. The k value of the function is the y value of the verte, thereby making the graph of the function shift vertically. As stated earlier, the process of completing the square is vital in mathematics. This is the method used to put the equation of a parabola in standard form. As you complete the square of the function y = a + b + c, the dependant variable, y, must be kept in the problem. The y variable will keep track of the a term for you since the completing the square method cannot be used if the leading coefficient is any number other that one. The y variable holds on to the number for you, then gives it back at the end. The following eample shows this happening. 06

15 Eample = Remember to keep y in the problem. y y 8 4 = + y 4 7 = + = 7 4 To put into standard form, we will need to complete the square, which means the coefficient for the term must be one, so divide the whole equation by -. Division by - yields this equation Subtract 7 to both sides of the equation b 4 = = ( ) = 4 To find the missing term evaluate binomial on the right. b to create a perfect square y + = + y ( ) 3 = y ( ) = + 3 Add the result, 4, to both sides of the equation Simplify Now we need to begin isolating the y, so add 3 to both sides of the equation y = ( ) + 3 ( ) y = 6 To isolate the y, multiply the entire equation by -. Notice the y was just holding on to - throughout the entire process. The equation for this parabola is now in standard form. As previously stated, this skill is vital. In the future, it will be necessary to complete the square twice within the same problem. This is the procedure used when putting the equation of a circle, ellipse, or hyperbola into standard form. In each case, you will be required to complete the square for two different variables in order to write the equations in standard form. 07

16 Write each of the following quadratic functions in standard form. A) = B) = + 6 C) y = D) = E) = F) = 6 9 G) = 5 6 H) = I) = + 3 J) = + K) = 5 + L) y = 3 M) = N) 3 = O) y =

17 Working with the Standard Form of a Quadratic Function Answer each of the following questions regarding the standard form of a parabola. y = a( h) + k Where a, h and k represent constants. Which constant is responsible for a vertical shift of the function? Which constant is responsible for a horizontal shift of the function? Which constant will cause the function to open up or down? What is the most notable difference between the graphs of the following functions? y = 4( ) + 3 and y = ( ) Once the quadratic function is in y = a( h) + k form, what makes up the verte of the parabola? The highest point of a parabola that opens down is the. The lowest point of a parabola that opens up is the. What is the domain of any quadratic function? What would a parabola that has only one -intercept look like? 09

18 Referring to a parabola in y = a( h) + k form, answer the following. If a < 0, the range of the function is the interval. If a > 0, the range of the function is the interval. Eplain how it is possible to have a parabola with no -intercepts. Is it possible to have a parabola with no y-intercepts? Why or why not? A standard parabola can have two intercepts. Can a parabola have two y intercepts as well? Why or why not. Eplain, in your own words, what an ais of symmetry is? What are the two things you need to know to find the range of a quadratic function? 0

19 Find the verte of each of the following. These are the same problems you wrote in standard form earlier on page 08, so it is not necessary to go through that process again. However, be sure to write the standard form of the quadratic function below each one. A) = B) = + 6 C) y = D) = E) = F) = 6 9 G) = 5 6 H) = I) = + 3 J) = + K) = 5 + L) y = 3 M) = N) 3 = O) y = + 3

21 Answer each of the following. Find the maimum value of the function: f( ) = Find the maimum value of the function: f( ) = For what value of does the function f( ) = achieve its maimum value? Find the minimum value of the function: f( ) = Find the minimum value of the function: f( ) =

23 Finding the Range and Domain We know that the domain for any quadratic function is all real numbers, so that won t be a problem. The difficulty can arise from finding the range of the function. To find the range of a quadratic function, we need to know how it opens, and the y value of the verte. As it was discussed earlier, the y value of the verte is the maimum or minimum value of a parabola, depending on how it opens; so it is the absolute highest or lowest point on the graph of the function. The following is true for a quadratic function in standard form y = a( h) + k. If a > 0, the range is [ k, ). If a < 0, the range is (, k]. State the range and domain for each of the following. A) = B) = C) y = 6 D) = + 3 E) y = F) y = G) = + 3 H) 3 = + I) 4 3 =

24 Properties of Parabolas For each of the following, find the standard form of the function; describe how the graph opens; identify the y-intercept, verte, -intercepts (if any), and the range and domain. A) y = + 4 Standard Form: Opens: Y-intercept: Verte: X-intercepts: Range: Domain: B) y = 3 Standard Form: Opens: Y-intercept: Verte: X-intercepts: Range: Domain: C) y = + 9 Standard Form: Opens: Y-intercept: Verte: X-intercepts: Range: Domain: 6

25 D) = 6 + Standard Form: Opens: Y-intercept: Verte: X-intercepts: Range: Domain: E) = + 4 Standard Form: Opens: Y-intercept: Verte: X-intercepts: Range: Domain: F) 5 y = + 4 Standard Form: Opens: Y-intercept: Verte: X-intercepts: Range: Domain: 7

26 G) = 0 5 Standard Form: Opens: Y-intercept: Verte: X-intercepts: Range: Domain: H) = 0 0 Standard Form: Opens: Y-intercept: Verte: X-intercepts: Range: Domain: I) 4 47 = Standard Form: Opens: Y-intercept: Verte: X-intercepts: Range: Domain: 8

27 Graphing Parabolas Graph each of the following parabolas. Make sure you find all required information, label the verte, y-intercept, and -intercepts (if they eist). A) y = 4 Standard Form: Opens: Y-intercept: Verte: X-intercepts: Range: Domain: B) = 6 9 Standard Form: Opens: Y-intercept: Verte: X-intercepts: Range: Domain: 9

28 C) = Standard Form: Opens: Y-intercept: Verte: X-intercepts: Range: Domain: D) = 4 Standard Form: Opens: Y-intercept: Verte: X-intercepts: Range: Domain: 0

29 Intervals In this section, we will determine the interval in which the value of a function is increasing, decreasing, constant, positive, or negative. In order to do this, the graph of the function is read from the left to the right. Increasing To determine the interval in which the value of the function is increasing, read the graph of the function going from left to right. If the graph is rising going from left to right, the value of the function is increasing. Decreasing To determine the interval in which the value of the function is decreasing, read the graph of the function going from left to right. If the graph is falling going from left to right, the value of the function is decreasing. Constant To determine the interval in which the value of the function is constant, read the graph of the function going from left to right. If the graph is neither rising nor falling going from left to right, the value of the function is constant. In other words, we are looking for a horizontal line. Positive To determine the interval in which the value of a function is positive, the intercepts of the function must be known. If any portion of the graph is above the ais, this is the interval in which the value of the function is positive. Negative To determine the interval in which the value of a function is negative, the intercepts of the function must be known. If any portion of the graph is below the ais, this is the interval in which the value of the function is negative. In the case of a parabola, it will not be necessary to see the graph to do this. Increasing/Decreasing The interval in which the value of a quadratic function is increasing or decreasing is based on how the parabola opens, and the value of the verte. Positive/Negative The interval in which the value of a quadratic function is positive or negative is based on the intercepts of the graph of the function. It is also very important to keep in mind the direction in which the parabola opens. If for eample, it is determined that a parabola opens down and has no intercepts, there is no interval in which the value of the function is positive. This happens because the verte would be below the ais.

30 Here is an eample of determine intervals. Notice that only the values of the coordinates were used to describe the interval, and only parenthesis are used. Given the above graph of a function, determine the intervals in which the value of the function is increasing, decreasing, constant, positive and negative. The value of the function is increasing in the intervals: (-0,-6.), (-4,-0.8), and (4,6) The value of the function is decreasing in the intervals: (-6.,-4) and (6,) The value of the function is constant in the interval: (-0.8, 4) The value of the function is positive in the intervals: (-8,--6) and (-, 9) The value of the function is negative in the intervals: (-0,-8), (-6,-) and (9,)

31 The reason parenthesis must be used when describing these intervals and not brackets is in the meaning of each symbol when using interval notation. A bracket means inclusive. Notice for eample the intervals in which the value of the function is increasing and decreasing. The value of -6. is used in both instances. If there had been brackets by the -6., it would be stating that the value of the function is both increasing and decreasing at the same time when is -6.. When asked to determine the interval in which the value of the function is increasing, the y values of the function are not used to describe these intervals. The reason these values are not used is because of the question itself. The y coordinates of a function are the values of the function. The question is asking where this is happening, not what the actual values are. A couple of guidelines to intervals Make sure to only use the values of the function to describe the interval. Never use brackets when answering these types of questions. Determine the interval in which the value of the function = is increasing. Determine the interval in which the value of the function = + 8 is increasing. Determine the interval in which the value of the function = is decreasing. 3

32 Determine the interval in which the value of the function = is decreasing. Determine the interval in which the value of the function = is negative. Determine the interval in which the value of the function = + 8 is positive. Determine the interval in which the value of the function = + is negative. Determine the interval in which the value of the function = is positive. 4

33 Finding the Equation of a Parabola You will be required to find the equation of a parabola with the given information. Find the equation of a parabola that has intercepts of ( ) and ( ) ( ) and ( ) 3,0,0. Given intercepts of -3 and 3,0,0. = 3 = ( + 3) ( ) If the intercepts are -3 and, then the roots of the equation are -3 and. You must now set each root equal to zero. For the first root, you will add 3 to both sides of the equal sign. For the second root, you will subtract to both sides of the equal sign. + 6 Now multiply the results together, and you have a quadratic epression. = + 6 Set your epression equal to y, and you have the equation of a parabola. In the previous eample, all you need to do is set each root equal to, and work backwards. Imagine solving a quadratic equation in the reverse. You have the solutions, and you work backwards to find out what the original problem is. Now, keep in mind, there are an infinite number of parabolas that can have those two particular intercepts. If you wanted, for eample, to find the equation of a parabola that opens down and has intercepts of -3 and, you would multiply your resultant epression, + 6, by (-). You could also multiply your epression by any other constant, and the parabola would still have the same - intercepts. Find the equation of a parabola that opens up, and has the following intercepts. A) (-3,0) and (4,0) B) (-,0) and (-3,0) C) (,0) and (5,0) Find the equation of a parabola that opens down, and has the following intercepts. A) (-,0) and (6,0) B) (,0) and (7,0) C) (5,0) 5

34 Find the equation of a parabola that has verte of (, 4), and passes through (,3 ). y = a( h) + k Begin with the standard form of a quadratic function. ( ( ) ) ( 4) y = a + Substitute the values of h and k in for the verte of the parabola. ( ) y = a + 4 Simplify, now all you need is the value of a to complete the equation. ( ) a ( ) ( ) 3 = + 4 ( ) 3 = a = 9a 4 7 = 9a a = 7 9 y = 7 ( + ) 9 4 Since you have the values of h and k, you need to find the value of a, so substitute the values of and y for the point (,3 ) into the equation. Simplify and solve for a. Once you have found the value of a, rewrite the completed equation. You have just found the equation of a parabola that has verte of (, 4) through (,3 )., and passes When you encounter these types of problems the most common mistake people make, is they forget to find the value of a. Find the equation of a parabola that has a verte of (-3,) and contains the point (4,7). Find the equation of a parabola that has a verte of (4,5) and contains the point (-,-). 6

35 Find the equation of a parabola that has a verte of (-,-3) and contains the point (4,) Find the equation of a parabola that has a verte of (0,3) and passes the ais at (7,0). Find the equation of a parabola that has a verte of (3,-) and has a y intercept of (0,-8). Find the equation of a parabola that has a verte of (5,0) and has a y intercept of (0,-). On the net page you will learn to find the equation of a parabola given 3 points on the parabola. 7

36 Find the equation of a parabola that passes through (,0), (,-) and (3,0). (,0 ), (, ), ( 3,0) Given y = a + b + c Use the general form of a parabola 0 = a + b + c = 4a + b + c 0 = 9a + 3b + c Eq a + b + c = 0 Eq 4a + b + c = Eq 9a + 3b + c = 0 3 Substitute the values of and y into the general form of a parabola for each set of coordinates setting up 3 equations. You have now created a system of equations in 3 variables. Eq a + b + c = 0 Eq 4a + b + c = Eq a + b + c = 0 Eq 9a + 3b + c = 0 3 One of these equations will be used twice. It doesn t matter which. In this case, equation will be used twice. Eq a b c = 0 + Eq 4a + b + c = Eq a b c = 0 + Eq 9a + 3b + c = 0 Eq4 3a + b = Eq5 8a + b = 0 Eq 3a + b = 4 Eq 8a + b = Multiplying equation by - and combining the equations yielded two new equations. You must get rid of the same variable each time. Now we have a system of equations in two variables. Eq 6a b = 4 + Eq 8a + b = 0 5 a = a = Multiply equation 4 by -, and add the result to equation 5. This yields a numerical value for the variable a. a = a = b = 4 Substitute into equation 5 Substitute into equation ( ) 8 + b = 0 ( ) ( ) c = 0 b = 8 3+ c = 0 b = 4 c = 3 Once the value of the first variable is found, substitute that number, in this case, into either equation 4 or 5 and solve for the remaining variable. Now that the value of two of the variables is known, go back to equation, substitute and find the value of the third variable. a = b = 4 c = 3 The value of all three variables has now been found. = Substitute these values into the general form of a parabola, and the equation of the parabola that passes through (,0), (,-) and (3,0) is found. 8

37 A) Find the equation of a parabola that passes through (,6), (,5) and (0,5). B) Find the equation of a parabola that passes through (0,6), (,) and (5,). 9

38 C) Find the equation of a parabola that passes through (3,-0), (4,0) and (6,8). D) Find the equation of a parabola that passes through (0,6), (-6,0) and (,6). 30

39 Finding a Quadratic Equation Find a quadratic equation that has the following roots. A) 7, 8 B) 4, C) 5 3 0, 8 D) 3 + i, 3 i E) + 5 i, 5 i F) 3 +, 3 G) + 3, 3 H) +, 3 3 I) i, i J) 3 3, 5 5 K) + i, i L) i, 3 5i 3

40 Word Problems with Quadratics Solve each of the following. A) Find two consecutive odd integers whose product is 99. B) The sum of two numbers is 40. Find the greatest possible product of the two numbers. C) Find the number of units that produce a maimum revenue is the revenue in dollars, and is the number of units sold. R = , where R D) The profit a company makes is given by the model P = , where P is the profit the company earns and is the amount spent on advertisement in hundreds of dollars. What amount should the company spend on advertising in order to maimize profits. E) The more epensive a product, the less you can sell. The relationship between the price, p, and the quantity, q, of sold products is given by the following formula q = 30 p. The revenue is given by R = pq. For what price will you have the maimum revenue? 3

41 F) Two cyclists meet at an intersection. The first cyclist travels east, while the second south. The eastbound cyclist is traveling 7 miles per hour faster than the southbound cyclist. After one hour, the two are 3 miles apart. How far have each of the cyclists traveled? G) The base of a right triangle is 6 cm more than twice the length of its leg. Find the length of the two legs of the right triangle if the hypotenuse is 39 cm. H) A fireman has a 5 foot ladder. He needs to reach a window that is feet high. How far away from the base of the wall should the ladder be placed? I) Find two consecutive even integers whose product is 4. 33

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