FURTHER GRAPHING OF QUADRATIC FUNCTIONS Section 11.6.

FURTHER GRAPHING OF QUADRATIC FUNCTIONS

Section 11.6

Further Graphing of Quadratic Functions

Section 11.6

Graph a quadratic equation by plotting points.

Identify the vertex of a parabola.

Quadratic Functions and Their Graphs

Graph by plotting points.

X Y

-2

-1

0

1

2

Section 11.6

2 2 3y x x

22 2 2 3y

21 2 1 3y

20 2 0 3y

21 2 1 3y

22 2 2 3y

X Y

-2 -3

-1 -4

0 -3

1 0

2 7

Quadratic Functions and Their Graphs

Graph by plotting points.

Quadratic Function A function that can be

written in the form y = ax2 + bx + c, where a, b, and c are real numbers and a ≠ 0. The shape of the graph of

a quadratic function is called a parabola.

The maximum or minimum value is called the vertex and has ordered pair (h, k).

All parabolas have an axis of symmetry, which is a vertical line running through the vertex, equation x = h.

Section 11.6

2 2 3y x x

Vertex(-1, -4)

Axis of symmetryx = -1

Quadratic Functions and Their Graphs

Solve . Quadratic Function A function that can be

written in the form y = ax2 + bx + c, where a, b, and c are real numbers and a ≠ 0. Solving the equation

equal to zero is the same as saying y=0, or finding the x-intercepts.

Because of symmetry, the x-intercepts will be equidistant from the vertex.

Section 11.6

2 2 3 0x x

Vertex(-1, -4)

Axis of symmetryx = -1

Quadratic Functions and Their Graphs

Section 11.6

Given an equation of the form y = ax2 + bx + c, the equation of the axis of symmetry can be found using the formula:

Since the axis of symmetry runs through the vertex, this formula also finds the x-coordinate of the vertex. To get the y-coordinate, substitute the found x-coordinate

back into the quadratic equation.

2

bx

a

Deriving a Formula for Finding the Vertex

Section 11.6

To find the vertex of a parabola in standard form: Calculate the x-

coordinate using the formula

Substitute this value into the original function to calculate the y-coordinate

Determine the value of the vertex and graph using the calculator.1.

2.

2

b

a

2 6 8y x x

23 2y x x

3, 1

2316 12,

Quadratic Functions and Their Graphs

An object is thrown upward from the top of a 100-foot cliff. Its height in feet above ground after t seconds is given by the function f(t) = -16t2 +10t +100. Find the maximum height of the object and the number of seconds it took for the object to reach its maximum height. Minimum/Maximum is the VERTEX

After 5/16ths of a second, the object reaches its maximum height of 101 and 9/16 feet.

Section 11.6