Warm-UpFind a linear function that describes the situation, and solve the problem.

4 minutes

1) A tractor rents for $50, plus $5 per engine hour. How much does it cost to rent and run the tractor for 6 hours?

12.4 Quadratic Functions12.4 Quadratic Functions12.4 Quadratic Functions12.4 Quadratic FunctionsObjectives:

To graph quadratic functions

Graphs of Quadratic Functions

Using a graphing calculator, graph the equation y = x2.

What is the lowest point on the graph?

Which axis divides the graph in half?

(0,0)

y-axis

Using a graphing calculator, graph each of the following equations.

y = 2x2 y = 2x2 + 1 y = 2x2 – 4x

Quadratic FunctionA function f defined by an equation of the form y = ax2 + bx + c, where a,b, and c are real numbers and a = 0, is a quadratic function and can be written f(x) = ax2 + bx + c.

The graph of a quadratic function is called a parabola.

Example 1Graph the quadratic function f(x) = -x2.

f(x) = -x2

x f(x)

-2

-1

0

1

2

-4-10

-1-4

x

y

Example 1Graph the quadratic function f(x) = -x2.

x

yThe vertex is the maximum or minimum point of a parabola.

vertex

If the graph of a parabola is folded so that the two sides of the parabola coincide, then the fold line is the axis of symmetry.

axis of symmetry

Vertex and Axis of Symmetry

For a parabola defined by the equation y = ax2 + bx + c:

1) the x-coordinate of the vertex isb2a

2) the axis of symmetry is the lineb

x2a

Example 2Find the vertex and axis of symmetry, then graph the quadratic function f(x) = 2x2 - 8x + 4 .

x

yb

x coordinate of vertex :2a

( 8)2(2)

y-coordinate of vertex:

y = 2x2 – 8x + 4 = 2(2)2 – 8(2) + 4 = 8 – 16 + 4 = -4vertex: (2,-4)

84

2

Example 2Find the vertex and axis of symmetry, then graph the quadratic function f(x) = 2x2 - 8x + 4 .

x

y

axis of symmetry:

x = 2

bx

2af(x) = 2x2 – 8x + 4

x f(x)

0

1

2

3

4

4-2-4

-24

PracticeFind the vertex and axis of symmetry, then graph the function.

1) f(x) = -2x2 + 4x + 1

2) g(x) = x2 -3x + 1

Homework

p.554-555 #1-19 odds