Bell Ringer 4/2/15 Find the Axis of symmetry, vertex, and solve the quadratic eqn. 1. f(x) = x 2 + 4x f(x) = x 2 + 2x - 3

Bell Ringer 4/3/15 Find the Axis of symmetry, vertex, and solve the quadratic eqn. 1. f(x) = x 2 + 2x - 3

Bell Ringer 4/6/15 #1 & 2 Find the axis of symmetry (AoS), and vertex of the following functions. 1.f(x) = X 2 – 4 2.f(x) = -2X 2 – 8x If a < 0 which way will the parabola open? 4.Graph the function f(x)= 2x 2 – 4x – 1 by solving for the AoS, vertex, and using x = 2 & x = 3.

7.4 & 7.5 Graphing Quadratic Functions Definitions Definitions 3 forms for a quad. function 3 forms for a quad. function Steps for graphing each form Steps for graphing each form Examples Examples Changing between eqn. forms Changing between eqn. forms

Quadratic Function A function of the form y=ax 2 +bx+c where a≠0 making a u-shaped graph called a parabola. A function of the form y=ax 2 +bx+c where a≠0 making a u-shaped graph called a parabola. Example quadratic equation:

Vertex- The lowest or highest point of a parabola. Vertex Axis of symmetry- The vertical line through the vertex of the parabola. Axis of Symmetry

Standard Form Equation y=ax 2 + bx + c If a is positive, u opens up If a is positive, u opens up If a is negative, u opens down The x-coordinate of the vertex is at The x-coordinate of the vertex is at To find the y-coordinate of the vertex, plug the x- coordinate into the given eqn. To find the y-coordinate of the vertex, plug the x- coordinate into the given eqn. The axis of symmetry is the vertical line x= The axis of symmetry is the vertical line x= Choose 2 x-values on either side of the vertex x- coordinate. Use the eqn to find the corresponding y- values. Choose 2 x-values on either side of the vertex x- coordinate. Use the eqn to find the corresponding y- values. Graph and label the 5 points and axis of symmetry on a coordinate plane. Connect the points with a smooth curve. Graph and label the 5 points and axis of symmetry on a coordinate plane. Connect the points with a smooth curve.

Graph y = 2x 2 - 8x + 6 Bell Ringer 4/7/15

Standard Form: Transformations f(x) = x 2 g(x) = x z(x) = x z(x) g(x) f(x)

5 y x Simple Quadratic Functions The simplest quadratic functions are of the form f (x) = ax 2 (a  0) These are most easily graphed by comparing them with the graph of y = x 2. Example: Compare the graphs of, and Standard Form: Transformations

Transformations (Cont.)

Graph of Transformations (cont.)

Vertex Form Equation If a > 0, parabola opens up If a < 0, parabola opens down. The vertex is the point (h, k). The axis of symmetry is the vertical line x = h. If h>0 then parent function y = x2, h units to the right. If h<0 then parent function y = x2, h units to the left. K shift same as in standard form The Vertex form for the equation of a quadratic function is: f (x) = a(x – h) 2 + k (a  0)

Example: f(x) = (x –3) Example: Graph f (x) = (x – 3) and find the vertex and axis. f (x) = (x – 3) is the same shape as the graph of g (x) = (x – 3) 2 shifted upwards two units. g (x) = (x – 3) 2 is the same shape as y = x 2 shifted to the right three units. f (x) = (x – 3) g (x) = (x – 3) 2 y = x x y 4 4 vertex (3, 2)

x y 4 4 Vertex and x-Intercepts Example: Graph and find the vertex and x-intercepts of f (x) = – ( x – 3) a < 0  parabola opens downward. h = 3, k = 16  axis x = 3, vertex (3, 16). Find the x-intercepts by solving x = 7, x = –1 x-intercepts (7, 0), (–1, 0) x = 3 (7, 0)(–1, 0) (3, 16)

Example 3: Graph y=-.5(x+3) 2 +4 a is negative (a = -.5), so parabola opens down. a is negative (a = -.5), so parabola opens down. Vertex is (h,k) or (-3,4) Vertex is (h,k) or (-3,4) Axis of symmetry is the vertical line x = -3 Axis of symmetry is the vertical line x = -3 Table of values x y Table of values x y Vertex (-3,4) (-4,3.5) (-5,2) (-2,3.5) (-1,2) x=-3

Now you try one! Ex. 4 y=2(x-1) 2 +3 Open up or down? Open up or down? Vertex? Vertex? Axis of symmetry? Axis of symmetry? Table of values with 5 points? Table of values with 5 points?

(-1, 11) (0,5) (1,3) (2,5) (3,11) X = 1

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19 Example: Basketball Example: A basketball is thrown from the free throw line from a height of six feet. What is the maximum height of the ball if the path of the ball is: The path is a parabola opening downward. The maximum height occurs at the vertex. At the vertex, So, the vertex is (9, 15). The maximum height of the ball is 15 feet.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20 Example: Maximum Area Example: A fence is to be built to form a rectangular corral along the side of a barn 65 feet long. If 120 feet of fencing are available, what are the dimensions of the corral of maximum area? barn corral x x 120 – 2x Let x represent the width of the corral and 120 – 2x the length. Area = A(x) = (120 – 2x) x = –2x x The graph is a parabola and opens downward. The maximum occurs at the vertex where a = –2 and b = – 2x = 120 – 2(30) = 60 The maximum area occurs when the width is 30 feet and the length is 60 feet.