11 4 Graphing Quadratic Functions 1 Graph quadratic

11. 4 Graphing Quadratic Functions 1. Graph quadratic functions of the form f(x) = ax 2. 2. Graph quadratic functions of the form f(x) = ax 2 + k. 3. Graph quadratic functions of the form f(x) = a(x – h)2. 4. Graph quadratic functions of the form f(x) = a(x – h)2 + k. 5. Graph quadratic functions of the form f(x) = ax 2 + bx + c.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Shape: Parabola Axis: x = 0 axis of symmetry x = 0 (y-axis) Vertex: (0, 0) vertex (0, 0) x & y-intercepts: Perfect Square Imaginary Solutions

Vertex: Axis: Vertex: x=0 Vertex: Axis: x=0 Vertical Shifts

Graph: Vertex: (0, -3) Axis: x=0 Axis is always x = x-coordinate of the vertex

Vertex: Axis: x=2 Vertex: Axis: x =-5 Horizontal Shifts Opposite of sign Vertex: Axis: x=3 Vertex: Axis: x =-4

ite s po Graph: Vertex: (2, 3) Axis: x=2 op e m sa

ite s po Graph: Vertex: (-1, 2) Axis: x = -1 op e m sa

What is the axis of symmetry for the function f(x) = (x + 3)2 + 5 ? a) x = 3 b) x = – 3 c) x = 5 d) y = – 3 11. 4 Copyright © 2011 Pearson Education, Inc. Slide 11 - 8

What is the axis of symmetry for the equation f(x) = (x + 3)2 + 5 ? a) x = 3 b) x = – 3 c) x = 5 d) y = – 3 11. 4 Copyright © 2011 Pearson Education, Inc. Slide 11 - 9

If a > 1, the graph is narrower. If 0 < a < 1, the graph is wider.

If a > 0, the graph opens upward. If a < 0, the graph opens downward.

Graph: Vertex & 4 other points is ax e g an r (-1, 4) Vertex: Direction: Down Shape: Same Axis: x = -1 y-intercept: (0, 3) Let x=0. Mirrored point: (-2, 3) x-intercepts: Too hard Pick a value for x: x = 1 (1, 0) Domain: (- ∞, ∞) Mirrored point: (-3, 0) Range: (- ∞, 4]

Vertex & 4 other points Graph: e is ng ax ra Vertex: (1, -3) Direction: Up Shape: Narrower Axis: x=1 y-intercept: (0, -1) Mirrored point: (2, -1) x-intercepts: Domain: (- ∞, ∞) Let x = 3 (3, 5) Mirrored point: (-1, 5) Range: [-3, ∞)

Graph: Vertex Formula 4 4+ ½ Vertex: a = 1, b = 4, c = -1 squared (-2, -5) Vertex: (-2, -5)

Vertex of a Quadratic Function in the Form f(x) = ax 2 + bx + c 1. The x-coordinate is . 2. Find the y-coordinate by evaluating Vertex: .

Find the vertex: f(x) = 3 x 2 – 12 x + 4 Vertex: (2, 8)

e sit o p op e m sa Read vertex from equation Vertex: (opposite, same) Use vertex formula

What are the coordinates of the vertex of the function f(x) = x 2 + 4 x + 5? a) ( 1, 2) b) (0, 4) c) ( 2, 1) d) (4, 0) 11. 4 Copyright © 2011 Pearson Education, Inc. Slide 11 - 18

What are the coordinates of the vertex of the function f(x) = x 2 + 4 x + 5? a) ( 1, 2) b) (0, 4) c) ( 2, 1) d) (4, 0) 11. 4 Copyright © 2011 Pearson Education, Inc. Slide 11 - 19

What is the vertex of y = – 2(x + 3)2 + 5? a) (– 3, 5) b) (3, – 5) c) (5, – 3) d) (2, – 3) 11. 4 Copyright © 2011 Pearson Education, Inc. Slide 11 - 20

What is the vertex of y = – 2(x + 3)2 + 5? a) (– 3, 5) b) (3, – 5) c) (5, – 3) d) (2, – 3) 11. 4 Copyright © 2011 Pearson Education, Inc. Slide 11 - 21

Graph: Vertex & 4 other points Vertex: (-3, -1) Direction: Up Shape: Same Axis: x = -3 y-intercept: (0, 8) Mirrored point: (-6, 8) x-intercepts: Domain: (- ∞, ∞) Let y=0. (-2, 0) (-4, 0) Range: [ -1, ∞)

Graph: Vertex & 4 other points Vertex: (-2, -1) Direction: Down Shape: Same Axis: x = -2 y-intercept: (0, -5) Mirrored point: (-4, -5) x-intercepts: None Let x = -1 (-1, -2) Mirrored point: (-3, -2) Domain: (- ∞, ∞) Range: (-∞, -1]