Warm-Up: December 15, 2011 Divide and express the result in standard form

i

i

22

23

Homework Questions?

Quadratic Functions

Section 2.2

Quadratic Functions A quadratic function is any function that

can be written in the form

The graph of a quadratic function is a parabola.

Every parabola has a vertex at either its minimum or its maximum.

Every parabola has a vertical axis of symmetry that intersects the vertex.

cbxaxxf 2

Example Graphs

Vertex

Axis of Symmetry

Standard Form of a Quadratic Function

Vertex is at (h, k) Axis of symmetry is the line x=h If a>0, the parabola opens upward, U If a<0, the parabola opens downward,

khxaxf 2

Graphing Quadratics in Standard Form1. Determine the vertex, (h, k)2. Find any x-intercepts by replacing f(x) with 0

and solving for x3. Find the y-intercept by replacing x with 04. Plot the vertex, axis of symmetry, and y-

intercepts and connect the points. Draw a dashed vertical line for the axis of symmetry.

5. Check the sign of “a” to make sure your graph opens in the right direction.

Example 1 Graph the quadratic function. Give the equation of the parabola’s axis of

symmetry. Determine the graph’s domain and range. 12 2 xxf

You-Try #1 Graph the quadratic function. Give the equation of the parabola’s axis of

symmetry. Determine the graph’s domain and range 41 2 xxf

Graphing Quadratics in General Form General form is The vertex is at

x-intercepts can be found by quadratic formula (or sometimes by factoring and zero product property)

y-intercept is at (0, c) Graph the parabola using these points just

as we did before.

cbxaxxf 2

a

bf

a

b

2,

2

Example 3 Graph the quadratic function. Give the equation of the parabola’s axis of

symmetry. Determine the graph’s domain and range 342 2 xxxf

You-Try #3 Graph the quadratic function. Give the equation of the parabola’s axis of

symmetry. Determine the graph’s domain and range 462 xxxf

Minimum and Maximum Consider If a>0, then f has a minimum If a<0, then f has a maximum The maximum or minimum occurs at

The maximum or minimum value is

cbxaxxf 2

a

bx

2

a

bf

2

Example 4 (Page 266 #44) A football is thrown by a quarterback to a

receiver 40 yards away. The quadratic function

models the football’s height above the ground, s(t), in feet, when it is t yards from the quarterback. How many yards from the quarterback does the football reach its greatest height? What is that height?

5025.0)( 2 ttts

You-Try #4 (Page 266 #43) Fireworks are launched into the air. The

quadratic function

models the fireworks’ height, s(t), in feet, t seconds after they are launched. When should the fireworks explode so that they go off at the greatest height? What is that height?

420016)( 2 ttts

Assignment Page 264 #1-8 ALL (use your graphing

calculator for 5-8), #9-41 Every Other Odd