Quadratic Functions and Transformations If a, the graph is a stretch or compression of the parent function b a factor of 0 a 0. 0 0 0 0 0 a a 7 The graph is a vertical The graph is a vertical compression of the parent function. stretch of the parent function Vertical compression a Vertical a stretch What is the graph of = ( + 3) +? Step Write the function in verte form: = [ - (-3)] + Step The verte is ( - 3, ). Step 3 The ais of smmetr is = -3. Step Because a =, the graph of this function is a vertical stretch b of the parent function. In addition to sliding the graph of the parent function 3 units left and units up, ou must change the shape of the graph. Plot a few points near the verte to help ou sketch the graph. ( 3) Slide units Slide 3 units Vertical stretch 5 0 3 0 Eercises Graph each function. Identif the verte and ais of smmetr.. = ( - ) + 3. = ( + ) - 3. = ( + ) + (, 3); (, ); 8 = = 8 O. = ( - ) + 3 5. = ( + ) -. = 0.9( + ) + O O (, ); = 8 O (, 3); = O (, ); = 8 O (, ); = Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching Standard Form of a Quadratic Function The graph of a quadratic function, = a + b + c, where a 0, is a parabola. The ais of smmetr is the line = - b a. The -coordinate of the verte is - a b. The -coordinate of the verte is = f ( - a), b or the -value when = - a b. The -intercept is (0, c). What is the graph of = - 8 + 5? b = - a = - (-8) = 8 () = Find the equation of the ais of smmetr. -coordinate of verte: - a b b f ( - a) = f () = () - 8() + 5 Find the -value when =. = 8 - + 5 = -3 -coordinate of verte: - 3 The verte is (, - 3). -intercept: (0, 5) The -intercept is at (0, c) = (0, 5). 5 (0, 5) 3 O 3 Eercises 5 (, 3) (, 5) Because a is positive, the graph opens upward, and the verte is at the bottom of the graph. Plot the verte and draw the ais of smmetr. Plot (0, 5) and its corresponding point on the other side of the ais of smmetr. Graph each parabola. Label the verte and the ais of smmetr.. = -3 + - 9. = - - 8-5 O (, ) (0, 9) (, 9) 3. = - 8 +. = - - - 7 (, ) (0, ) O 8 (, 7) = (, ) 3 9 3 (, 5) (0, 5) ( 3, ) = 3 O (, 7) (0, 7) Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Standard Form of a Quadratic Function Standard form of a quadratic function is = a + b + c. Verte form of a quadratic function is = a( - h) + k. For a parabola in verte form, the coordinates of the verte are (h, k). What is the verte form of = 3 - + 50? = a + b + c = 3 - + 50 b = -, a = 3 Find b and a. b -coordinate = - a = - - (3) = Simplif. -coordinate = 3() - () + 50 = 3( - ) + = Simplif. Verif that the equation is in standard form. For an equation in standard form, the -coordinate of the verte can be found b using = - a b. Substitute. Substitute into the standard form to find the -coordinate. Substitute for h and for k into the verte form. Once the conversion to verte form is complete, check b multipling. = 3( - 8 + ) + = 3 - + 50 The result is the standard form of the equation. Eercises Write each function in verte form. Check our answers. 5. = - - 3. = - + + 7. = + 3-0 = ( ) = ( ) + 0 = ( + 3 ) 9 8. = - 9 9. = + 0. = + 5 + = ( 9 ) 8. = + 8-3. = 3 + 9 3. = - + + Write each function in standard form. = ( + ) = ( + 5 ) 9 = ( + ) 7 = 3 ( + ) 7 = ( ) + 3. = ( - 3) + 5. = ( - ) - 3. = -3( + ) + = + 0 = = 3 7 Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching Factoring Quadratic Epressions What is - 5 - in factored form? a =, b = -5, and c = - Find a, b, and c; the are the coefficients of each term. ac = - and b = -5 We are looking for factors with product ac and sum b. Factors of,,,, 3, 8 3, 8,, Sum of factors 3 3 0 0 5 5 The factors 3 and -8 are the combination whose sum is -5. + 3 3-8 - 3 Rewrite the middle term using the factors ou found. 3( + ) - ( + ) Find common factors b grouping the terms in pairs. (3 - )( + ) Rewrite using the Distributive Propert. Check (3 - )( + ) You can check our answer b multipling the factors together. + 3-8 - - 5 - Remember that not all quadratic epressions are factorable. Eercises Factor each epression.. + + 8 ( + )( + ). - + 3 ( 3)( ) 3. - + ( )( ). - + 5 5. - 7 - ( + )( ). + + 5 7. - 5 - ( + )( 7) 8. 7-9 - ( )( 5) ( + 5)( + 3) (7 + )( 3) 9. - - 7 ( 9)( + 8) 0. + 9 + 7 ( + 7)( + ). + + 3 ( + )( + 8). - 8 + 9 ( 7)( 7) 3. - 3-0 ( 5)( + ). + 9 + 5. 9 - + (3 )(3 ). - 0 + 9 7. + - ( + )( ) 8. + 7 + 0 9. - 8 + ( )( ) 0. - 5-3 ( + )( + ) ( )( 9) ( + 5)( + ) ( + )( 3). - + 5 ( )( 5). 3 + - 8 (3 )( + ) Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Factoring Quadratic Epressions a + ab + b = (a + b) a - ab + b = (a - b) a - b = (a + b)(a - b) Factoring perfect square trinomials Factoring a difference of two squares What is 5-0 + in factored form? There are three terms. Therefore, the epression ma be a perfect square trinomial. a = 5 and b = Find a and b. a = 5 and b = Take square roots to find a and b. Check that the choice of a and b gives the correct middle term. ab = # 5 # = 0 Write the factored form. a - ab + b = (a - b) 5-0 + = (5 - ) Check (5 - ) You can check our answer b multipling the factors together. Eercises (5 - )(5 - ) Rewrite the square in epanded form. 5-0 - 0 + 5-0 + Factor each epression. Distribute. Simplif. 3. - + 3. + 30 + 5 5. 9 - + ( ) ( + 5) (3 ). - 7. 9 - + 9 8. 5 - ( + 8)( 8) (3 7) (5 + )(5 ) 9. 7-30. 9 + + 9 3. - 3 + 3(3 + )(3 ) (7 + 3) ( ) 3. 9-33. 8-8 3. 8 + + 9 (3 + )(3 ) ( + 3)( 3) (9 + 7) 35. 5-00 + 0 3. - + 9 37. - - - 9 5(5 ) ( + )( ) ( + 3) Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching Quadratic Equations There are several was to solve quadratic equations. If ou can factor the quadratic epression in a quadratic equation written in standard form, ou can use the Zero-Product Propert. If ab = 0 then a = 0 or b = 0. What are the solutions of the quadratic equation + = 5? + = 5 Write the equation. + - 5 = 0 Rewrite in standard form, a + b + c = 0. ( - 5)( + 3) = 0 Factor the quadratic epression (the nonzero side). - 5 = 0 or + 3 = 0 Use the Zero-Product Propert. = 5 or = -3 Solve for. = 5 or = -3 Check the solutions: = 5 : ( 5 ) + ( 5 ) 5 = -3: (-3) + (-3) 5 5 + 5 5 8-3 5 5 = 5 5 = 5 Both solutions check. The solutions are = 5 and = -3. Eercises Solve each equation b factoring. Check our answers.. - 0 + = 0, 8. + = 3 9, 7 3. + 9 =,. - + = 0 5. = 7 +,. = -5 +, 3 7. - 7 = - 3, 8. + 0 = 0 5, 0 9. + =, 0. 3-5 + = 0 3,. = -5-3,. + = 0 5, Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Quadratic Equations Some quadratic equations are difficult or impossible to solve b factoring. You can use a graphing calculator to find the points where the graph of a function intersects the -ais. At these points f () = 0, so is a zero of the function. The values r and r are the zeros of the function = ( r )( r ). The graph of the function intersects the -ais at = r, or (r, 0), and = r, or (r, 0). What are the solutions of the quadratic equation 3 = + 7? Step Rewrite the equation in standard form, a + b + c = 0. 3 - - 7 = 0 Step Enter the equation as Y in our calculator. Plot Plot Plot3 \Y=3X X 7 \Y= \Y3= \Y= \Y5= \Y= \Y7= Step 3 Step Graph Y. Choose the standard window and see if the zeros of the function Y are visible on the screen. If the are not visible, zoom out and determine a better viewing window. In this case, the zeros are visible in the standard window. Use the ZERO option in the CALC feature. For the first zero, choose bounds of and and a guess of.5. The screen displa gives the first zero as = -.3039. Zero X =.3039 Y = 0 Similarl, the screen displa gives the second zero as =.898053. The solutions to two decimal places are = -.3 and =.90. Zero X =.898053 Y = 0 Eercises Solve the equation b graphing. Give each answer to at most two decimal places. 3. = 5.,.. = 5 + 0.9, 5.9 5. + 7 = 3 7., 0.. + = 5.79,.79 7. + 3 + = 0., 0.38 8. = +., 3. 9. 3-5 + 9 = 8 0.3,.3 0. = + 3.35, 0.85. - = -7.59,.. - = 8 + 8.83,.7 Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching Completing the Square Completing a perfect square trinomial allows ou to factor the completed trinomial as the square of a binomial. Start with the epression + b. Add ( b ). Now the epression is + b + ( b ), which can be factored into the square of a binomial: + b + ( b ) = ( + b ). To complete the square for an epression a + ab, first factor out a. Then find the value that completes the square for the factored epression. What value completes the square for - + 0? Think Write the epression in the form a( + b). Write - + 0 = -( - 5) Find b. b = - 5 = - 5 Add ( b ) to the inner epression to complete the square. Factor the perfect square trinomial. Find the value that completes the square. -c - 5 + a - 5 b d = -a - 5 + 5 b -( - 5 ) -( 5 ) = - 5 Eercises What value completes the square for each epression?. +. - 3. +. - 0 5. 5 00 + 5. - 9 7. - 7 8. 3 + 9. - + 8 3 9 0. 5 + 80 30. -7 + 7. -3-5 75 Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Completing the Square You can easil graph a quadratic function if ou first write it in verte form. Complete the square to change a function in standard form into a function in verte form. What is = - + in verte form? Think Write an epression using the terms that contain. - Write Find b. b = - = -3 Add ( b ) to the epression to complete the square. Subtract 9 from the epression so that the equation is unchanged. Factor the perfect square trinomial. Add the remaining constant terms. - + (-3) = - + 9 = - + 9 + - 9 = ( - 3) + - 9 = ( - 3) + 5 Eercises Rewrite each equation in verte form. 3. = + + 3 ( + ). = - + 3 5. = - + - 0 ( ). = - - 3 7. = + 8 + 3 ( + ) 3 8. = - - - 9. = - + 0-8 ( 5) + 7 0. = + - 8 ( 3) + ( ) ( + 3) + 5 ( + ) 9. = + - 3 ( + ) 5. = 3 - + 8 3( ) Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching The Quadratic Formula You can solve some quadratic equations b factoring or completing the square. You can solve an quadratic equation a + b + c = 0 b using the Quadratic Formula: = - b { b - ac a Notice the { smbol in the formula. Whenever b - ac is not zero, the Quadratic Formula will result in two solutions. What are the solutions for + 3 =? Use the Quadratic Formula. + 3 - = 0 Write the equation in standard form: a + b + c = 0 a = ; b = 3; c = - a is the coefficient of, b is the coefficient of, c is the constant term. = - b { b - ac a = - (3) { (3) - ()(-) () = - 3 { Write the Quadratic Formula. Substitute for a, 3 for b, and for c. Simplif. = - 3 + or - 3 - Write the solutions separatel. Check our results on our calculator. Replace in the original equation with - 3 + and - 3 -. Both values for give a result of. The solutions check. ( 3 + ()) X.85078059 X + 3X ( 3 ()) X.35078059 X + 3X Eercises What are the solutions for each equation? Use the Quadratic Formula.. - + 7-3 = 0 7 + 37 or 7 37. + = 0 3 + 9 or 3 9 3. = + 3 + 0 or 0. + 8 = 3 9 5. + = 5-7 or. - 0 + 3 = 0 5 + 7 or 5 7 Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
The Quadratic Formula There are three possible outcomes when ou take the square root of a real number n: n 7 0 S two real values (one positive and one negative) = 0 S one real value (0) 0 S no real values Now consider the quadratic formula: = - b { b - ac a. The value under the radical smbol determines the number of real solutions that eist for the equation a + b + c = 0: b - ac 7 0 S two real solutions = 0 S one real solution 0 S no real solutions The value under the radical, b ac, is called the discriminant. What is the number of real solutions of -3 + 7 =? -3 + 7 = -3 + 7 - = 0 Write in standard form. a = -3, b = 7, c = - Find the values of a, b, and c. b - ac Write the discriminant. (7) - (-3)(-) Substitute for a, b, and c. 9 - Simplif. The discriminant, 5, is positive. The equation has two real roots. Eercises 5 What is the value of the discriminant and what is the number of real solutions for each equation? 7. + - = 0 8. - + 3-0 = 0 9. + + 5 = 0 9; two 9; two ; none 0. = 8-8. - + 7 + = 0. - 5 + 5 = 0 0; one 5; two 0; one 3. + 7 = 5. + 5 = 5. + 5 = 3 3; none 9; two ; none. 9 = - 3 7. + + 3 = 0 8. = + 8 0; one ; none ; two Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching Quadratic Sstems You used graphing and substitution to solve sstems of linear equations. You can use these same methods to solve sstems involving quadratic equations. What is the solution of the sstem of equations? e = - - 8 = - 3 = - 3 Write one equation. - - 8 = - 3 - - 5 = 0 Substitute - - 8 for in the linear equation. Write in standard form. ( + )( - 5) = 0 Factor the quadratic epression. + = 0 or - 5 = 0 = - or = 5 Solve for. Use the Zero-Product Propert. Because the solutions to the sstem of equations are ordered pairs of the form (, ), solve for b substituting each value of into the linear equation. You can use either equation, but the linear equation is easier. = -: = - 3 = (-) - 3 = -5 S (-, -5) = 5: = - 3 = (5) - 3 = 7 S (5, 7) The solutions are (-, -5) and (5, 7). Check these b graphing the sstem and identifing the points of intersection. Eercises Solve each sstem.. e = + 3-5 = 3 - (, 7), (, ). e = - + 5 - = - + (, 3), (5, ) 3. e = - - 5 = 3 + (, ), (3, 0). e = + 3-7 = - - ( 5, 3), (, 3) 5. e = - 5 + = 5-7 (, ), (, 3). e = - - + 3 = - (, 5), (, 0) Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Quadratic Sstems To solve a sstem of linear inequalities, ou graph each inequalit and find the region where the graphs overlap. You can also use this technique to solve a sstem of quadratic inequalities. What is the solution of this sstem of inequalities? e - + 7 - - 8 Step Graph the equation = - +. Use a dashed boundar line because the points on the curve are not part of the solution. Choose a point on one side of the curve and check if it satisfies the inequalit. - + 0? -() + () Check the point (, 0). 8 0 The inequalit is true. Step Points below the curve satisf the inequalit, so shade that region. Graph the equation = - - 8. Use a dashed boundar line because the points on the curve are not part of the solution. Choose a point on one side of the curve and check if it satisfies the inequalit. 7 - - 8 0 7? () - () - 8 Check the point (, 0). 0 7-8 The inequalit is true. 8 Step 3 Eercises Points above the curve satisf the inequalit, so shade that region. The solution to the sstem of both inequalities is the set of points satisfing both inequalities. In other words, the solution is the region where the graphs overlap. The region contains no boundar points. 8 Solve each sstem b graphing. 7. e - + 7-8 8 8. e 7 - - - - + 8 8 9. e - - + 8 7 - + 8 0. e 7 - - + 7 8 Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching A New Look at Parabolas What is the graph of the equation = -? Label the verte, focus, and directri on our graph. Step Identif information from the given equation. = - a 0 a is negative. opens downward When a is negative, the focus: (0, -c) parabola has these directri: = c characteristics. Step Find c. 0 a 0 = c 0-0 = c (c) 0-0 = (c) c = True for all parabolas. Substitute - for a. Solve for c. Step 3 Find the verte, the focus, and the equation of the directri. (0, 0) The parabola is of the form = a, so the verte is at the origin. (0,- ) The focus is alwas (0, -c). = The directri is at = c. Step Locate two more points on the parabola. = - () Substitute for. = - (,- ) Solve for. = - (-) Substitute - for. Step 5 Graph the parabola using the information ou found. O = - ( -,- ) Solve for. (0, ) Eercises Graph each equation. Label the verte, focus, and directri on each graph.. =. = - 3. = - 3 3 3 3 (0, ) O 3 3 3 3 3 O 3 3 (0, ) 3 (, 0) O 3 Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
A New Look at Parabolas You can analze the equation of a parabola b analzing transformations of the parent function =. What are the vertices, foci, and directrices of the parabolas = and = -? The equation = - is a transformation of =. Both parabolas have vertices at (0, 0). Both equations can be written in the form = c. The focus is (0, c), and the directri is = -c. To find the focus (0, c) of =, find the value of c. You know that = = = # so c =. The focus is ( 0, ) and the directri is = -. In the transformed equation = -, ou know that the equation opens downward. To find the focus 3 3 O 3 3 (0, c) of = -, identif the value of c: = - = The focus is ( 0, - ) and the directri is =. ( - ). Eercises Identif the verte, focus, and directri of the parabola.. = (0, 0); (0, 5. = -3 8); = 8 (0, 0); (0, ); =. = 8 (0, 0); (0, ); = 7. = (0, 0); (0, ); = Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching Circles in the Coordinate Plane When working with circles, begin b writing the equation in standard form: ( - h) + ( - k) = r Unlike equations of parabolas, which include either or, the equation of a circle will include both and. What is the center and radius of the circle with the equation + - + =? ( - ) + ( + ) = Rearrange and group the terms b variable. b = - b = = 3 To complete each square, find b for each group. ( b ) = (-) = ( b ) = (3) = 9 Find ( b ) for each epression. ( - + ) + ( + + 9) = + + 9 Add ( b ) for each epression to both sides. ( - ) + ( + 3) = 5 Write the epressions as perfect squares; simplif. ( - ) + ( - (-3)) = 5 Write the equation in standard form. h =, k = -3, r = 5 Compare the equation to ( - h) + ( - k) = r. The center of the circle is (, -3). The radius of the circle is 5. Eercises Find the center and radius of each circle.. + - 0 = 0 (0, 5); 5. + = 5 (0, 0); 5 3. + + - = 5 (, 3); 5. + + + = -8 5. + + + = 3 (, );. + - 0 - = -0 (, 7); (5, ); 3 7. + + - 8 = -7 ( 8, ); 8. + - 8 + = -5 (, 3); 5 9. + - - = - (, 3); 3 0. + + 8 = 7 (, 0); 37 Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Circles in the Coordinate Plane In order to plan a circular ornamental garden, a landscaper uses grid paper in which the center of the garden is at (0, 8) and the diameter is 0 units. How can he model the circular garden with an equation? Step Identif the information that ou know. The center of the circle is (0, 8). The diameter of the circle is 0. Step Identif the information that ou want to find. You need to find h, k, and r. The diameter of the circle is 0 units, so the radius is 0 = 0 units. The center is (0, 8), so h = 0 and k = 8. Step 3 Substitute known values into the standard form of an equation for a circle. ( - h) + ( - k) = r Write the standard form. ( - 0) + ( - 8) = 0 Substitute 0 for h, 8 for k, and 0 for r. + ( - 8) = 00 Simplif. Eercises. An archaeologist is working on the ecavation of a circular shaped pit. If the center of the pit is (3, 7) and the radius is 30 ft, what is an equation for the circle? ( 3) + ( 7) = 900. A homeowner is planning a circular sandbo in the backard. She wants the diameter of the sandbo to be 5 ft. She uses graph paper and marks the center of the circle at (-, -5). What is an equation for the circle? ( + ) + ( + 5) = 5.5 3. A quilter is cutting pieces for a quilt with a circular design in the center. If the center of the circle is the origin and the radius of the circle is in., what is an equation for the circle? + =. The diameter of a plaground merr-go-round is ft. If it is placed on a grid with the center at (-9, 7), what equation models it? ( + 9) + ( 7) = 9 Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.