Graphing Quadratic Functions

MA.912.A.2.6 Identify and graph functions

MA.912.A.7.6 Identify the axis of symmetry, vertex, domain, range, and intercept(s) for a given parabola

Quadratic Function

y = ax2 + bx + c

Quadratic Term Linear Term Constant Term

What is the linear term of y = 4x2 – 3? 0x What is the linear term of y = x2- 5x ? -5x What is the constant term of y = x2 – 5x? 0 Can the quadratic term be zero? No!

Quadratic FunctionsThe graph of a quadratic function is a:

A parabola can open up or down.

If the parabola opens up, the lowest point is called the vertex (minimum).

If the parabola opens down, the vertex is the highest point (maximum).

NOTE: if the parabola opens left or right it is not a function!

y

x

Vertex

Vertex

parabola

y = ax2 + bx + c

The parabola will open down when the a value is negative.

The parabola will open up when the a value is positive.

Standard Form

y

x

The standard form of a quadratic function is:

a > 0

a < 0

a 0

y

x

Axis of Symmetry

Axis of Symmetry

Parabolas are symmetric.

If we drew a line down the middle of the parabola, we could fold the parabola in half.

We call this line the Axis of symmetry.

The Axis of symmetry ALWAYS passes through the vertex.

If we graph one side of the parabola, we could REFLECT it over the Axis of symmetry to graph the other side.

Find the Axis of symmetry for y = 3x2 – 18x + 7

Finding the Axis of SymmetryWhen a quadratic function is in standard form

the equation of the Axis of symmetry is

y = ax2 + bx + c,

2ba

x

This is best read as …

‘the opposite of b divided by the quantity of 2 times a.’

18

2 3x 18

6 3

The Axis of symmetry is x = 3.

a = 3 b = -18

Finding the VertexThe Axis of symmetry always goes through the _______. Thus, the Axis of symmetry gives us the ____________ of the vertex.

STEP 1: Find the Axis of symmetry

Vertex

Find the vertex of y = -2x2 + 8x - 3

2ba

x a = -2 b = 8

x 8

2( 2)

8

4 2

X-coordinate

The x-coordinate of the vertex is 2

Finding the Vertex

STEP 1: Find the Axis of symmetry

STEP 2: Substitute the x – value into the original equation to find the y –coordinate of the vertex.

8 8 22 2( 2) 4

ba

x

The vertex is (2 , 5)

Find the vertex of y = -2x2 + 8x - 3

y 2 2 2 8 2 3

2 4 16 3

8 16 3

5

Graphing a Quadratic Function

There are 3 steps to graphing a parabola in standard form.

STEP 1: Find the Axis of symmetry using:

STEP 2: Find the vertex

STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve.

MAKE A TABLE

using x – values close to the Axis of symmetry.

2ba

x

STEP 1: Find the Axis of symmetry

( )4

12 2 2

bx

a

-= = =

y

x

Graph : y 2x 2 4x 1

Graphing a Quadratic Function

STEP 2: Find the vertex

Substitute in x = 1 to find the y – value of the vertex.

( ) ( )22 1 4 1 1 3y = - - =-

Vertex : 1, 3

x 1

5

–1

( ) ( )22 3 4 3 1 5y = - - =

STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve.

y

x

( ) ( )22 2 4 2 1 1y = - - =-

3

2

yx

Graphing a Quadratic Function

Graph : y 2x 2 4x 1

y

x

Y-intercept of a Quadratic Function

y 2x 2 4 x 1 Y-axis

The y-intercept of aQuadratic function canBe found when x = 0.

y 2x 2 4x 1

2 0 2 4(0) 1

0 0 1

1

The constant term is always the y- intercept

The number of real solutions is at most two.

Solving a Quadratic

No solutions

6

4

2

-2

5

f x = x2-2 x +56

4

2

-2

5

2

-2

-4

-5 5

One solution

X = 3

Two solutions

X= -2 or X = 2

The x-intercepts (when y = 0) of a quadratic function are the solutions to the related quadratic equation.

Identifying Solutions

4

2

-2

-4

5

X = 0 or X = 2

Find the solutions of 2x - x2 = 0

The solutions of this quadratic equation can be found by looking at the graph of f(x) = 2x – x2

The x-intercepts(or Zero’s) of f(x)= 2x – x2

are the solutions to 2x - x2 = 0