Name Class Date. Quadratic Functions and Transformations

Transcription

1 4-1 Reteaching Parent Quadratic Function The parent quadratic function is y = x. Sustitute 0 for x in the function to get y = 0. The vertex of the parent quadratic function is (0, 0). A few points near the vertex are: Quadratic Functions and Transformations x y The graph is symmetrical aout the line x = 0. This line is the axis of symmetry. Vertex Form of a Quadratic Function The vertex form of a quadratic function is y = a(x h) + k. The graph of this function is a transformation of the graph of the parent quadratic function y = x. The vertex of the graph is (h, k). If a = 1, you can graph the function y sliding the graph of the parent function h units along the x-axis and k units along the y-axis. What is the graph of y = (x + 3) +? What are the vertex and axis of symmetry of the function? Step 1 Write the function in vertex form: y = 1[x (3)] + Step Find the vertex: h = 3, k =. The vertex is (3, ). Step 3 Find the axis of symmetry. Since the vertex is (3, ), the graph is symmetrical aout the line x = 3. The axis of symmetry is x = 3. Step 4 Because a = 1, you can graph this function y sliding the graph of the parent function 3 units along the x-axis and units along the y-axis. Plot a few points near the vertex to help you sketch the graph. x y Prentice Hall Algera Teaching Resources 9

2 4-1 Reteaching (continued) Quadratic Functions and Transformations If a 1, the graph is a stretch or compression of the parent function y a factor of a. 0 < a < 1 a > 1 The graph is a vertical compression The graph is a vertical stretch of the parent function. of the parent function What is the graph of y = (x + 3) +? Step 1 Write the function in vertex form: y = [x (3)] + Step The vertex is (3, ). Step 3 The axis of symmetry is x = 3. Step 4 Because a =, the graph of this function is a vertical stretch y of the parent function. In addition to sliding the graph of the parent function 3 units left and units up, you must change the shape of the graph. Plot a few points near the vertex to help you sketch the graph. x y Graph each function. Identify the vertex and axis of symmetry. 1. y = (x 1) + 3. y = (x + 4) 3. y = (x + ) y = (x 1) y = 1 (x + 4) 6. y = 0.9(x + ) + 1 Prentice Hall Algera Teaching Resources 10

3 4- Reteaching Standard Form of a Quadratic Function The graph of a quadratic function, y = ax + x + c, where a 0, is a paraola. The axis of symmetry is the line x. a The x-coordinate of the vertex is. The y-coordinate of the vertex is a y f a, or the y-value when x. a The y-intercept is (0, c). What is the graph of y = x 8x + 5? 8 8 x = = = Find the equation of the axis of symmetry. a 4 x-coordinate of vertex: f a f = () 8() + 5 = = 3 a Find the y-value when x =. y-coordinate of vertex: 3 The vertex is (, 3). y-intercept: (0, 5) The y-intercept is at (0, c) = (0, 5). Because a is positive, the graph opens upward, and the vertex is at the ottom of the graph. Plot the vertex and draw the axis of symmetry. Plot (0, 5) and its corresponding point on the other side of the axis of symmetry. Graph each paraola. Lael the vertex and the axis of symmetry. 1. y = 3x + 6x 9. y = x 8x y = x 8x y = x 1x 7 Prentice Hall Algera Teaching Resources 19

4 4- Reteaching (continued) Standard Form of a Quadratic Function Standard form of a quadratic function is y = ax + x + c. Vertex form of a quadratic function is y = a(x h) + k. For a paraola in vertex form, the coordinates of the vertex are (h, k). What is the vertex form of y = 3x 4x + 50? y = ax + x + c y = 3x 4x + 50 Verify that the equation is in standard form. = 4, a = 3 Find and a. x -coordinate = a For an equation in standard form, the x-coordinate of the vertex can e found y using x. a 4 (3) Sustitute. = 4 Simplify. y-coordinate = 3(4) 4(4) + 50 y = 3(x 4) + = Simplify. Sustitute 4 into the standard form to find the y-coordinate. Sustitute 4 for h and for k into the vertex form. Once the conversion to vertex form is complete, check y multiplying. y = 3(x 8x + 16) + y = 3x 4x + 50 The result is the standard form of the equation. Write each function in vertex form. Check your answers. 5. y = x x 3 6. y = x + 4x y = x + 3x y = x 9x 9. y = x + x 10. y = x + 5x y = 4x + 8x 3 1. y = 3 4 x + 9x 13. y = x + x + 1 Write each function in standard form. 14. y = (x 3) y = (x 1) y = 3(x + 4) + 1 Prentice Hall Algera Teaching Resources 0

5 4-4 Reteaching Factoring Quadratic Expressions What is 6x 5x 4 in factored form? a = 6, = 5, and c = 4 Find a,, and c; they are the coefficients of each term. ac = 4 and = 5 We are looking for factors with product ac and sum. Factors of 4 1, 4 1, 4, 1,1 3, 8 3,8 4, 6 4, 6 Sum of factors The factors 3 and 8 are the comination whose sum is 5. Rewrite the middle term using the factors you found. Find common factors y grouping the terms in pairs. Rewrite using the Distriutive Property. Check (3x 4)(x + 1) You can check your answer y multiplying the factors together. 6x + 3x 8x 4 6x 5x 4 Rememer that not all quadratic expressions are factorale. Factor each expression. 1. x + 6x + 8. x 4x x 6x x 11x x 7x x + 16x x 5x x 19x 6 9. x x x + 9x x + 1x x 8x x 3x x + 9x x 6x x 10x x + 4x x + 7x x 8x x 5x 3 1. x 6x x + x 8 Prentice Hall Algera Teaching Resources 39

6 4-4 a + a + = (a + ) a a + = (a ) a = (a + )(a ) Reteaching (continued) Factoring Quadratic Expressions Factoring perfect square trinomials Factoring a difference of two squares What is 5x 0x + 4 in factored form? There are three terms. Therefore, the expression may e a perfect square trinomial. a = 5x and = 4 Find a and. a = 5x and = Take square roots to find a and. Check that the choice of a and gives the correct middle term. a = 5x = 0x Write the factored form. a a + = (a ) 5x 0x + 4 = (5x ) Check (5x ) You can check your answer y multiplying the factors together. (5x )(5x ) Rewrite the square in expanded form. 5x 10x 10x + 4 Distriute. 5x 0x + 4 Simplify. Factor each expression. 3. x 1x x + 30x x 1x x x 4x x x x + 4x x 3x x x x + 16x x 100x x x 4x 9 Prentice Hall Algera Teaching Resources 40

7 4-5 Reteaching Quadratic Equations There are several ways to solve quadratic equations. If you can factor the quadratic expression in a quadratic equation written in standard form, you can use the Zero-Product Property. If a = 0 then a = 0 or = 0. What are the solutions of the quadratic equation x + x = 15? x + x = 15 Write the equation. x + x 15 = 0 Rewrite in standard form, ax + x + c = 0. (x 5)(x + 3) = 0 Factor the quadratic expression (the nonzero side). x 5 = 0 or x + 3 = 0 Use the Zero-Product Property. x = 5 or x = 3 Solve for x. 5 x or x = 3 Check the solutions: x = : x = 3: (3) + (3) = = 15 Both solutions check. T e solutions are x = 5 and x = 3. Solve each equation y factoring. Check your answers. 1. x 10x + 16 = 0. x + x = x + 9x = 4. x 4x = 0 5. x = 7x x = 5x x 7x = 1 8. x + 10x = 0 9. x + x = 10. 3x 5x + = x = 5x 6 1. x + x = 0 Prentice Hall Algera Teaching Resources 49

8 4-5 Reteaching (continued) Quadratic Equations Some quadratic equations are dif cult or impossile to solve y factoring. You can use a graphing calculator to find the points where the graph of a function intersects the x-axis. At these points f(x) = 0, so x is a zero of the function. The values r 1 and r are the zeros of the function y = (x r 1)(x r ). The graph of the function intersects the x-axis at x = r 1, or (r 1, 0), and x = r, or (r, 0). What are the solutions of the quadratic equation 3x = x + 7? Step 1 Rewrite the equation in standard form, ax + x + c = 0. 3x x 7 = 0 Step Enter the equation as Y1 in your calculator. Step 3 Graph Y1. Choose the standard window and see if the zeros of the function Y1 are visile on the screen. If they are not visile, zoom out and determine a etter viewing window. In this case, the zeros are visile in the standard window. Step 4 Use the ZERO option in the CALC feature. For the first zero, choose ounds of and 1 and a guess of 1.5. The screen display gives the first zero as x = Similarly, the screen display gives the second zero as x = The solutions to two decimal places are x = 1.3 and x = Solve the equation y graphing. Give each answer to at most two decimal places. 13. x = x = 5x x + 7x = x + x = x + 3x + 1 = x = x x 5x + 9 = = x + 3x 1. x 6x = 7. x = 8x + 8 Prentice Hall Algera Teaching Resources 50

9 4-7 Reteaching The Quadratic Formula You can solve some quadratic equations y factoring or completing the square. You can solve any quadratic equation ax + x + c = 0 y using the Quadratic Formula: x a 4ac Notice the symol in the formula. Whenever 4ac is not zero, the Quadratic Formula will result in two solutions. What are the solutions for x + 3x = 4? Use the Quadratic Formula. x x + 3x 4 = 0 Write the equation in standard form: ax + x + c = 0 a = ; = 3; c = 4 4 a ac or 4 4 Check your results on your calculator. Replace x in the original equation with and. Both values 4 4 for x give a result of 4. The solutions check. a is the coefficient of x, is the coefficient of x, c is the constant term. Write the Quadratic Formula. Sustitute for a, 3 for, and 4 for c. Simplify. Write the solutions separately. What are the solutions for each equation? Use the Quadratic Formula. 1. x + 7x 3 = 0. x + 6x = x = 4x x + 81 = 36x 5. x + 1 = 5 7x 6. 6x 10x + 3 = 0 Prentice Hall Algera Teaching Resources 69

10 4-7 Reteaching (continued) The Quadratic Formula There are three possile outcomes when you take the square root of a real numer n: n > 0 = 0 < 0 two real values (one positive and one negative) one real value (0) no real values 4ac Now consider the quadratic formula: x. The value under a the radical symol determines the numer of real solutions that exist for the equation ax + x + c = 0: 0 4 ac 0 0 two real solutions one real solution no real solutions The value under the radical, 4ac, is called the discriminant. What is the numer of real solutions of 3x + 7x =? 3x + 7x = 3x + 7x = 0 Write in standard form. a = 3, = 7, c = Find the values of a,, and c. 4ac Write the discriminant. (7) 4( 3)( ) Sustitute for a,, and c Simplify. The discriminant, 5, is positive. The equation has two real roots. 5 What is the value of the discriminant and what is the numer of real solutions for each equation? 7. x + x 4 = 0 8. x + 13x 40 = 0 9. x + x + 5 = x = 18x x + 7x + 44 = x x 13. x + 7 = 5x 14. 4x + 5x = x + 5 = 3x x x x x 9 x x Prentice Hall Algera Teaching Resources 70

11 4-8 Reteaching A complex numer consists of a real part and an imaginary part. It is written in the form a + i, where a and are real numers. 1 Complex Numers i i and When adding or sutracting complex numers, comine the real parts and then comine the imaginary parts. When multiplying complex numers, use the Distriutive Property or FOIL. What is (3 i) + ( + 3i)? (3 i) + ( + 3i) = Circle real parts. Put a square around imaginary parts. = (3 + ) + ( 1 + 3)i Comine. = 5 + i Simplify. What is the product (7 3i)( 4 + 9i)? Use FOIL to multiply: (7 3i)( 4 + 9i) = 7( 4) + 7(9i) + ( 3i)( 4) + ( 3i)(9i) (7 3i)( 4 + 9i) = i + 1i 7i First = 7( 4) = i 7i Outer = 7(9i) You can simplify the expression y sustituting 1 for i. Inner = ( 3i)( 4) (7 3i)( 4 + 9i) = i 7( 1) Last = ( 3i)(9i) = i Simplify each expression. 1. i + ( 4 i). (3 + i)( + i) 3. (4 + 3i)(1 + i) 4. 3i(1 i) 5. 3i(4 i) 6. 3 ( + 3i) + ( 5 + i) 7. 4i(6 i) 8. (5 + 6i) + ( + 4i) 9. 9(11 + 5i) Prentice Hall Algera Teaching Resources 79

12 4-8 Reteaching (continued) Complex Numers The complex conjugate of a complex numer a + i is the complex numer a i. (a + i)(a i) = a + To divide complex numers, use complex conjugates to simplify the denominator. What is the quotient 4 5 i? i 4 5i i 4 5i i i i 8 4i 10i 5i i i 8 4i 10i 5i i i i 5 5 The complex conjugate of i is + i. Multiply oth numerator and denominator + i. Use FOIL to multiply the numerators. Simplify the denominator. (a + i)(a i) = a + Sustitute 1 for i. Simplify. Write as a complex numer a + i. Find the complex conjugate of each complex numer i i 1. i i i i Write each quotient as a complex numer i 1 i i 18. i i i 3 i 0. 4 i 3i 1. 6 i 5i Prentice Hall Algera Teaching Resources 80

13 4-6 Reteaching Completing the Square Completing a perfect square trinomial allows you to factor the completed trinomial as the square of a inomial. Start with the expression x + x. Add. Now the expression is x x, x x x which can e factored into the square of a inomial:. To complete the square for an expression ax + ax, first factor out a. Then find the value that completes the square for the factored expression. What value completes the square for x + 10x? Think Write the expression in the form a(x + x). Find. Add to the inner expression to complete the square. Factor the perfect square trinomial. Find the value that completes the square. Write x + 10x = (x 5x) x 5x x 5x 4 5 x What value completes the square for each expression? 1. x + x. x 4x 3. x + 1x 4. x 0x 5. x + 5x 6. x 9x 7. x 4x 8. 3x + 1x 9. x + 6x 10. 5x + 80x 11. 7x + 14x 1. 3x 15x Prentice Hall Algera Teaching Resources 59

14 4-6 Reteaching (continued) Completing the Square You can easily graph a quadratic function if you first write it in vertex form. Complete the square to change a function in standard form into a function in vertex form. What is y = x 6x + 14 in vertex form? Think Write Write an expression using the terms that contain x. Find. Add to the expression to Complete the square. Sutract 9 from the expression so that the equation is unchanged. x 6x 6 3 x 6x + ( 3) = x 6x + 9 y = x 6x Factor the perfect square trinomial. y = (x 3) Add the remaining constant terms. y = (x 3) + 5 Rewrite each equation in vertex form. 13. y = x + 4x y = x 6x y = x + 4x y = x x y = x + 8x y = x 6x y = x + 10x y = x + x 8 1. y = x + 4x 3. y = 3x 1x + 8 Prentice Hall Algera Teaching Resources 60

15 4-3 Reteaching Modeling With Quadratic Functions Three non-collinear points, no two of which are in line vertically, are on the graph of exactly one quadratic function. A paraola contains the points (0, ), (1, 5), and (, ). What is the equation of this paraola in standard form? If the paraola y = ax + x + c passes through the point (x, y), the coordinates of the point must satisfy the equation of the paraola. Sustitute the (x, y) values into y = ax + x + c to write a system of equations. First, use the point (0, ). y = ax + x + c Write the standard form. = a(0) + (0) + c = c Sustitute. Simplify. Use the point (1, 5) next. 5 = a(1) + (1) + c Sustitute. 5 = a + c Simplify. Finally, use the point (, ). = a() + () + c Sustitute. = 4a + + c Simplify. Because c =, the resulting system has two variales. Simplify the equations aove. a = 7 4a + = 4 Use elimination to solve the system and otain a = 3, = 4, and c =. Sustitute these values into the standard form y = ax + x + c. The equation of the paraola that contains the given points is y = 3x 4x. Find an equation in standard form of the paraola passing through the given points. 1. (0, 1), (1, 5), (1, 5). (0, 4), (1, 9), (, 0) 3. (0, 1), (1, 4), (3, ) 4. (1, 1), (, 0), (, 0) 5. (1, 5), (0, 1), (, 1) 6. (1, 3), (, 3), (1, 3) Prentice Hall Algera Teaching Resources 9

16 4-3 Reteaching (continued) Modeling With Quadratic Functions A soccer player kicks a all of the top of a uilding. His friend records the height of the all at each second. Some of her data appears in the tale. a. What is a quadratic model for these data?. Use the model to complete the tale. Use the points (0, 11), (1, 19), and (5, 19) to find the quadratic model. Sustitute the (t, h) values into h = at + t + c to write a system of equations. (0, 11): 11 = a(0) + (0) + c c = 11 (1, 19): 19 = a(1) + (1) + c a + + c = 19 Time (s) Height (ft) (5, 19): 19 = a(5) + (5) + c 5a c = 19 Use c = 11 and simplify the equations to otain a system with just two variales. a + = 80 5a + 5 = 80 Use elimination to solve the system. The quadratic model for the data is h = 16t + 96t + 11 Now use this equation to complete the tale for the t-values, 3, 4, 6, and 7. t = : h = 16() + 96() + 11 = = 40 t = 3: h = 16(3) + 96(3) + 11 = = 56 t = 4: h = 16(4) + 96(4) + 11 = = 40 t = 6: h = 16(6) + 96(6) + 11 = = 11 t = 7: h = 16(7) + 96(7) + 11 = = 0 Exercise Time (s) 7. The numer n of Brand X shoes in stock at the eginning of month t in a store follows a quadratic model. In January (t = 1), there are 36 pairs of shoes; in March (t = 3), there are 5 pairs; and in Septemer, there are also 5 pairs. a. What is the quadratic model for the numer n of pairs of shoes at the eginning of month t?. How many pairs are in stock in June? Height (ft) Prentice Hall Algera Teaching Resources 30