Bell Ringer 4215 Find the Axis of symmetry

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

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

Bell Ringer 4/6/15 #1 & 2 Find the axis of symmetry (Ao. S), and vertex of the following functions. 1. f(x) = X 2 – 4 2. f(x) = -2 X 2 – 8 x + 10 3. If a< 0 which way will the parabola open? 4. Graph the function f(x)= 2 x 2 – 4 x – 1 by solving for the Ao. S, vertex, and using x = 2 & x = 3.

7. 4 & 7. 5 Graphing Quadratic Functions • Definitions • 3 forms for a quad. function • Steps for graphing each form • Examples • 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. 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 negative, u opens down The x-coordinate of the vertex is at To find the y-coordinate of the vertex, plug the xcoordinate into the given eqn. The axis of symmetry is the vertical line x= Choose 2 x-values on either side of the vertex xcoordinate. Use the eqn to find the corresponding yvalues. Graph and label the 5 points and axis of symmetry on a coordinate plane. Connect the points with a smooth curve.

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

Standard Form: Transformations • g(x) f(x) = x 2 g(x) = x 2 + 4 z(x) = x 2 - 2 f(x) z(x)

Standard Form: Transformations 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 y 5 x -5 5

Transformations (Cont. ) •

Graph of Transformations (cont. )

Vertex Form Equation The Vertex form for the equation of a quadratic function is: f (x) = a(x – h)2 + k (a 0) • 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 = x 2 , h units to the right. If h<0 then parent function y = x 2 , h units to the left. K shift same as in standard form

Example: Graph f (x) = (x – 3)2 + 2 and find the vertex and axis. f (x) = (x – 3)2 + 2 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. y f (x) = (x – 3)2 + 2 g (x) = (x – 3)2 y = x 2 4 -4 (3, 2) vertex x 4

Example: Graph and find the vertex and x-intercepts of f (x) = – ( x – 3)2 + 16 y (3, 16) 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) 4 (– 1, 0) (7, 0) 4 x=3 x

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

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

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

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. 19

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 x corral 120 – 2 x Let x represent the width of the corral and 120 – 2 x the length. Area = A(x) = (120 – 2 x) x = – 2 x 2 + 120 x The graph is a parabola and opens downward. The maximum occurs at the vertex where a = – 2 and b = 120 – 2 x = 120 – 2(30) = 60 The maximum area occurs when the width is 30 feet and the length is 60 feet. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20 x