UNIT 8: SOLVING AND GRAPHING QUADRATICS. 8-1 Factoring to Solve Quadratic Equations. Solve each equation:

Transcription

1 UNIT 8: SOLVING AND GRAPHING QUADRATICS 8-1 Factoring to Solve Quadratic Equations Zero Product Property For all numbers a & b Solve each equation: If: ab 0, 1. (x + 3)(x 5) = 0 Then one of these is true: 1. a = 0,. b = 0, 3. Both a and b = 0 Steps 1. Make the equation equal zero. Factor 3. Set each factor equal to zero. x 36 = 5x 4. Solve 5. Check 3. a 4a = 144 1

2 8- Solving Quadratic Equations (By Finding Square Roots) To Solve Quadratic Equations by Finding Square Roots: Solve. 1. x = Get x term alone on one side of equal sign.. Square root both sides of the equation. 3. Make sure to have BOTH the positive and negative solution! 4. Always write answer in simplest radical form!. y = x + 1 = x + 11 = x 1 = 5 6. (x ) = 5

3 8-3 Completing the Square You can solve some quadratic equations by taking the square root of each side: Solve each equation by completing the square: 1. x + 1x + 36 = 5 But, you won t always have a perfect square.so, you will have to complete the Square! Steps to completing the square 1. Get the constant alone on one side & make sure the coefficient of x is one.. x 6x + 8 = 0. Find ½ of b (the coefficient of x) 3. Square your result from step 4. Add the result to both sides of the equation 5. Then solve by taking the square root. 3. n + 4n 9 = x + 1x = 4 3

4 5. x² 5x 5 = 0 6. n² - 7n - = 0 7. x² +3x -3 = 0 4

5 8-4 Using the Quadratic Formula (QF) and The Discriminant Derive the Quadratic Formula ax + bx + c = 0 b 4ac The discriminant, can show how many solutions a quadratic equation has. If b 4ac >0 then there are two real roots If b 4ac <0 then there are no real roots If =0 then there is one real root b 4ac Without solving tell how many real solutions the following equations have. 1. x + 6x + 8 = 0. 3x x + 3 = n 8n = 0 4. n + 4n + 4 = 0 5

6 8-5 Using the Quadratic Formula If Quadratic Formula ax bx c a 0, 0, and b 4ac 0, then Use the quadratic formula to solve: 1. x + 6x + 8 = 0 x b b 4ac a. x + 7x 15 = n 8n = 0 6

7 4. 3x x + 3 = 0 5. n 4n x 3x 8 = 0 7

8 Graphing Quadratic Functions Vocabulary Quadratic function: A function that can be written in the standard form y = ax + bx + c where a 0 Parabola: The graph of a quadratic function y = ax + bx + c Vertex: The lowest of highest point on a parabola. Axis of Symmetry: The line that divides a parabola into mirror images and passes through the vertex. Vertex form: The form y = a(x h) + k, where the vertex of the graph is (h, K) and the axis of symmetry is x = h. Intercept form: The form y = a(x p)(x q), where the x intercepts of the graph are p and q. Minimum value: The y coordinate of the vertex for y = ax + bx + c when a > 0. Maximum value: The y coordinate of the vertex for y = ax + bx + c when a < Graphing Standard Form y = ax + c Graphing Standard Form y = ax + c Steps To graph any equation you can: 1. Choose 5 simple x - values. Substitute them in the function to find the matching 5 y - values 3. Graph the ordered pairs found and draw a parabola through the points. 4. Remember that if a > 0, then parabola opens up and if a < 0, then parabola opens down. Practice: 1. y = x. y = -x - 1 8

9 8-7 Graphing Standard Form y = ax +bx+ c Steps for graphing y = ax + bx + c 1. Draw the axis of symmetry. It is the line x = b a.. Find and plot the vertex. The x coordinate of the vertex is b a. Substitute this value for x in the function and evaluate to find the y coordinate of the vertex. 3. Plot two points on one side of the axis of symmetry. Use symmetry to plot two more points on the opposite side. 4. Graph or draw a parabola through the points. 1. Graph y = 3x + 6x 1 The function is in standard form y = ax + bx + c where a = 3, b = 6, and c = -1. Because a > 0, the parabola opens up. 1. Draw the axis of symmetry x = b = 6 = 1 a (3). Find and plot the vertex. The x coordinate of the vertex is -1. Find the y coordinate. y = 3x + 6x 1 y = 3( 1) + 6( 1) 1 = 4 The vertex is (-1, -4) 3. Plot two points to the left of the axis of symmetry. Evaluate the function for two x values that are less than -1, such as - and -3. y = 3( ) + 6( ) 1 = 1 y = 3( 3) + 6( 3) 1 = 8 Plot the points (-, -1) and (-3, 8). Plot their mirror images by counting the distance to the axis of symmetry and then counting the same distance beyond the axis of symmetry. 4. Draw a parabola through the points.. Graph y = x 6x + 5 Try This 9

10 a. Graph y = x + 4x 3 b. Graph y = x + x

11 8-8 Vertex Form of a Quadratic Function Steps to graphing y = a(x h) + k 1. Draw the axis of symmetry. It is the line x = h.. Plot the vertex, (h, k). 3. Plot two points on one side of the axis of symmetry. Use symmetry to plot two more points on the opposite side. 4. Graph or draw a parabola through the points. 1. Graph y = (x 3) + The function is in vertex form y = a(x h) + k where a =, h = 3 and k =. Because a > 0 the parabola opens up. 1. Draw the axis of symmetry, x = h = 3. Plot the vertex (h, k) = (3, ) 3. Plot points. The x values 1 and are to the right of the axis of symmetry. x = 1: y = (1 3) + = 10 x = : y = ( 3) + = 4 Plot the points (1, 10) and (, 4). Then plot their mirror images across the axis of symmetry. 4. Graph or draw a parabola through the points.. Graph y = (x 3) + 4 The function is in vertex form y = a(x h) + k where a = -, h = 3 and k = 4. Because a < 0 the parabola opens down. 1. Draw the axis of symmetry, x = h = 3. Plot the vertex (h, k) = (3, 4) 3. Plot points. The x values 1 and are to the right of the axis of symmetry. x = 1: y = (1 3) + 4 = 4 x = : y = ( 3) + 4 = Plot the points (1, -4) and (, ). Then plot their mirror images across the axis of symmetry. 4. Graph or draw a parabola through the points. 11

12 Try This a. y = (x 4) b. y = (x 3) + 4 1

13 8-9 Intercept Form of a Quadratic Function Steps for graphing y = a(x p)(x q) 1. Draw the axis of symmetry. It is the line x = p+q. Find and plot the vertex. The x coordinate of the vertex is p+q to find the y coordinate of the vertex. 3. Plot the points where the x intercepts, p and q, occur. 4. Graph. Substitute this value for x in the function 1. Graph y = (x )(x 4) The function is in intercept form y = a(x p)(x q) where a = -1, p =, and q = 4. Because a < 0, the parabola opens down. 1. Draw the axis of symmetry. x = p+q = +4 = 3. Find and plot the vertex. The x coordinate of the vertex is x = 3. Calculate the y coordinate of the vertex. y = (x )(x 4) y = (3 )(3 4) = 1 Plot the vertex (3, 1) 3. Plot the points where the x intercepts occur. The x intercepts are and 4. Plot the points (, 0) & (4, 0) 4. Graph Try This a. y = (x 3)(x + 1) 13

14 b. y = (x + 4)(x ) Find the Minimum or Maximum Value Tell whether the function y = (x + 4)(x ) has a minimum value or maximum value. Then find the min or max values. The function is in intercept form y = a(x p)(x q) where a =, p = -4, and q =. Because a > 0, the function has a minimum value. Find the y coordinate of the vertex. p + q x = = 4 + = 1 y = ( 1 + 4)( 1 ) = 18 The minimum value of the function is -18. Try This a. y = (x + 6)(x ) b. y = 3(x 1)(x 5) c. y = 4(x 3)