Graphing Quadratic Functions 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. 6.Solve applications involving parabolas.

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

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

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

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

Graph: Vertex: Axis: (2, 3) x = 2 opposite same

Graph: Vertex: Axis: (-1, 2) x = -1 opposite same

Slide Copyright © 2011 Pearson Education, Inc. What is the axis of symmetry for the function f(x) = (x + 3) ? a) x = 3 b) x = –3 c) x = 5 d) y = –3 11.4

Slide Copyright © 2011 Pearson Education, Inc. What is the axis of symmetry for the equation f(x) = (x + 3) ? a) x = 3 b) x = –3 c) x = 5 d) y = –3 11.4

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

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

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

Graph: Vertex: (1, -3) Direction: Up Shape:Narrower Axis: x = 1 axis r a n g e Vertex & 4 other points y-intercept: (0, -1) Mirrored point: (2, -1) x-intercepts: Let x = 3 (3, 5) Mirrored point: (-1, 5) Range: Domain: (- ∞, ∞) [-3, ∞)

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

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:

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

Read vertex from equation Vertex: (opposite, same) opposite same Use vertex formula

Slide Copyright © 2011 Pearson Education, Inc. What are the coordinates of the vertex of the function f(x) = x 2 + 4x + 5? a) (  1, 2) b) (0, 4) c) (  2, 1) d) (4, 0) 11.4

Slide Copyright © 2011 Pearson Education, Inc. What are the coordinates of the vertex of the function f(x) = x 2 + 4x + 5? a) (  1, 2) b) (0, 4) c) (  2, 1) d) (4, 0) 11.4

Slide Copyright © 2011 Pearson Education, Inc. 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

Slide Copyright © 2011 Pearson Education, Inc. 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

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

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