The Graphs of Quadratic EquationsLesson 9-1 and 9-2

Recall that a polynomial of degree 2, such as is called a quadratic polynomial.

You can use a quadratic polynomial to define a quadratic function

A quadratic function is a type of nonlinear function that models certain situations where the rate of change is not constant

All of these are quadratics:

y = x2

y = x2 + 2

y = x2 + x – 4

y = x2 + 2x – 3

So quadratic means that there must be

a x2 in what ever y equals

A quadratic equation is an equation that has a x2

value.

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

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

Quadratic Functions

The graph of a quadratic function is a parabola.

A parabola can open up or down.

NOTE: if the parabola opened left or right it would not be a function!

y

x

VertexMinimum

Vertexmaximum

Quadratic Functions

Standard form of a quadratic function

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.

y

x

The standard form of a quadratic function is

a > 0

a < 0

Line of symmetryy

x

Line of Symmetry

Parabolas have a symmetric property to them.

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

We call this line the line of symmetry.

The line of symmetry ALWAYS passes through the vertex.

Or, if we graphed one side of the parabola, we could “fold” (or REFLECT) it over, the line of symmetry to graph the other side.

Line of symmetry

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

When a quadratic function is in standard form

The equation of the line of symmetry is

y = ax2 + bx + c,

2ba

x

For example…

Using the formula…

This is best read as … the opposite of b divided by the quantity of 2 times a.

18

2 3x 18

6 3

Thus, the line of symmetry is

x = 3.

Line of symmetryy

x

Thus the line of symmetry is x = 3

Finding the vertex

To find the y – coordinate of the vertex, we need to plug the x – value into the original equation.

Thus, the line of symmetry gives us the x – coordinate of the vertex.

We know the line of symmetry always goes through the vertex.

Finding the vertex

STEP 1: Find the line of symmetry

STEP 2: Plug the x – value into the original equation to find the y value.

y = –2x2 + 8x –3

8 8 22 2( 2) 4ba

x

y = –2(2)2 + 8(2) –3

y = –2(4)+ 8(2) –3

y = –8+ 16 –3

Therefore, the vertex is (2 , 5)

The standard form of a quadratic function is given by

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

STEP 1: Find the line of symmetry using the equation

STEP 2: Find the coordinate of the vertex by substituting the x – value obtained on step 1 in the given equation to solve for y

STEP 3: Find others points using a table, and reflect them across the line of symmetry. Then connect the points with a smooth curve.

Graphing a quadratic equation

y = ax2 + bx + c

2bxa

-=

Graphing a quadratic function

STEP 1: Find the line of symmetry

( )4

12 2 2

bx

a

-= = =

y

x

Thus the line of symmetry is x = 1

y = 2x2 – 4x – 1Example:

Graphing a quadratic function

STEP 2: Find the vertex

y

x

Thus the vertex is (1,-3)

y = 2x2 – 4x – 1Example:

Since the x – value of the vertex is given by the line of symmetry, we need to plug in x = 1 to find the y – value of the vertex.

y = 2(1)2 – 4(1) – 1

y = -3

Graphing a quadratic function

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

y

x

x y = 2x2 – 4x – 1 y

2 y = 2(2)2 – 4(2) – 1 -1

3 y = 2(3)2 – 4(3) – 1 5

y = 2x2 – 4x – 1

Example

x

yGraph y = –2x2 + 4x + 5.

x y

1 7

2 5

0 5

3 –1

–1 –1

(3, –1)(–1, –1)

(2, 5)(0, 5)

(1, 7)Since a = –2 and b = 4, the graph opens down and the x-coordinate of the vertex is 1

)2(2

4