1. Warm Up for Lesson 3.5 1)Solve: x 2 – 8x – 20 = 0 2) Sketch the graph of the equation y = 2x – 4

2. Graphing Quadratic Functions Vertex Form Section 3.5 Essential Question: How do I analyze and graph quadratic functions in vertex form?

3. The graph of a quadratic is called a parabola. We can use a table of values to graph any quadratic function. xy -2 0 1 2 4 4 1 0 1

4. -8 -6 -4 -2 2 4 6 8 86428642 -2 -4 -6 -8 (1). y = x 2

5. Characteristic of: y = x 2 (a). Domain: (b). Range: (c). Vertex: (d). Axis of symmetry: (e). Opens: (f). Max or Min: (g). x-intercept: (h). y-intercept: (j). Increasing: (k). Decreasing: All reals or -∞ < x < ∞ y  0 or 0 ≤ y < ∞ (0, 0) x = 0 upward Min (0, 0) x < 0 or -∞ < x < 0 x > 0 or 0 < x < ∞

6. We can shift the graph of y = x 2 both horizontally and vertically. 2. y = (x – 2) 2 + 3 is created by shifting the parent graph (y = x 2 ) 2 units _____ and 3 units _____. 3. y = (x + 4) 2 – 5 is created by shifting the parent graph (y = x 2 ) 4 units _____ and 5 units _____. 4. y = -x 2 is created by reflecting the parent graph (y = x 2 ) about the ____ axis. right up left down x

7. -8 -6 -4 -2 2 4 6 8 86428642 -2 -4 -6 -8 (5). Graph: y = (x + 2) 2 – 1 V = (-2, -1) xy -2 0 3 3 0 0 -4 -3

8. Identify each characteristic of: y = (x + 2) 2 – 1 (a). Domain: (b). Range: (c). Vertex: (d). Axis of symmetry: (e). Opens: (f). Max or Min: (g). x-intercept: (h). y-intercept: *(i). Extrema: (j). Increasing: (k). Decreasing: *(l). Rate of change (0 ≤ x ≤ 4): All reals or -∞ < x < ∞ y  -1 or -1 ≤ y < ∞ (-2, -1) x = -2 upward Min (-3, 0) and (-1, 0) (0, 3) Min value = -1 x > -2 or -2 < x < ∞ x < -2 or -∞ < x < -2

9. Estimated Rate of change over the interval (0 ≤ x ≤ 4): y = (x + 2) 2 – 1 y = (0 + 2) 2 – 1 = 4 – 1 = 3 y = (4 + 2) 2 – 1 = 36 – 1 = 35 (0, 3) (4, 35)

10. -8 -6 -4 -2 2 4 6 8 86428642 -2 -4 -6 -8 (6). Graph: y = -(x – 4) 2 V = (4, 0) xy 40 3 2 5 6 -4 -4

11. Identify each characteristic of: y = -(x – 4) 2 (a). Domain: (b). Range: (c). Vertex: (d). Axis of symmetry: (e). Opens: (f). Max or Min: (g). x-intercept: (h). y-intercept: *(i). Extrema: (j). Increasing: (k). Decreasing: *(l). Rate of change (4≤ x ≤ 5): All reals or -∞ < x < ∞ y ≤ 0 or -∞ < y ≤ 0 (4, 0) x = 4 downward Max (4, 0) (0, -16) Max value = 0 x < 4 or -∞ < x < 4 x > 4 or 4 < x < ∞

12. Estimated Rate of change over the interval (4 ≤ x ≤ 5): y = -(x – 4) 2 y = -(4 – 4 ) 2 = -(0) 2 = 0 y = -(5 – 4 ) 2 = -(1) 2 = -1 (4, 0) (5, -1)

13. Vertex Form for Quadratic Functions: y = a(x – h) 2 + k Vertex has coordinates (h, k). If a > 0, the parabola opens up (vertex min) If a < 0, the parabola opens down (vertex max) To find x-intercept (zeros), let y = 0. To find y-intercept, let x = 0.

14. Axis of symmetry: x = h Extrema: max or min value is y-coordinate of vertex. Interval of increasing and decreasing: look at x- coordinate of vertex. Rate of change: Note: since the graph is not linear, the rate of change will vary and will NOT have a constant value.

15. -8 -6 -4 -2 2 4 6 8 86428642 -2 -4 -6 -8 (7). Graph: y = ½(x + 3) 2 – 4 V = (-3, -4) xy -3-4 -5 -7 1 -2 4 4

16. Identify each characteristic of: y = ½(x + 3) 2 – 4 (a). Domain: (b). Range: (c). Vertex: (d). Axis of symmetry: (e). Opens: (f). Max or Min: (g). x-intercept: (h). y-intercept: (i). Extrema: (j). Increasing: (k). Decreasing: (l). Rate of change (-7 ≤ x ≤ -5): All reals or -∞ < x < ∞ y  -4 or -4 ≤ y < ∞ (-3, -4) x = -3 upward Min Min value = -4 x > -3 or -3 < x < ∞ x < -3 or -∞ < x <-3 (-0.17, 0) and (-5.82, 0) (0, 0.5)

17. **x-intercepts: y = ½(x + 3) 2 – 4 0 = ½(x + 3) 2 – 4 4 = ½(x + 3) 2 8 = (x + 3) 2 **y-intercepts: y = ½(x + 3) 2 – 4 y = ½(0 + 3) 2 – 4 y = ½(3) 2 – 4 y = 4.5 – 4 y = 0.5

18. Estimated Rate of change over the interval (-7 ≤ x ≤ -5): y = ½(x + 3) 2 – 4 y = ½(-7 + 3) 2 – 4 = ½(16) – 4 = 4 y = ½(-5 + 3) 2 – 4 = ½(4) – 4 = -2 (-7, 4) (-5, -2)

19. Without graphing, find the x and y intercepts for the graph of: (8). y = 7(x – 6) 2 – 14 x-int: (7.41, 0) and (4.59, 0)

20. Without graphing, find the x and y intercepts for the graph of: (8). y = 7(x – 6) 2 – 14 y-int: (0, 238)