Characteristics of Quadratic Functions

Characteristics of

quadratic functions

Click next.

For now, use your calculators' "calculate zeros"

(2nd calc, zeros).

There are other methods but we won't go into

them here.

to solve for zeros

For a function rule, they're called

For a graph, they're called

zeros

Click next.

important characteristics

if you've already found the vertex

Finding the line of symmetry is easy...

If the vertex is at (2, 10) then the line of symmetry is x = 2.

It's "x equals" whatever the x-coordinate is of the vertex.

Click the circle for another example.

If the vertex is at (-3, 4)

then the line of symmetry is

x = -3

All quadratics have these characteristics.

These can all be figured out using numbers from the function rule.

or a

Vertices can either be a

if you have a function in

The vertex is at (1, -4)

y = x - 2x - 3

2

h is 1 (remember it's the opposite) and k is -4

y = (x - 1) - 4

2

Finding the y-intercept

If x = 0, then you're sitting on the y-axis.

So, in order to find the y-intercept:

Plug a zero into the x's and solve for y

In this example, the y-intercept is -3

y = ax + bx + c

2

y = x - 2x - 3

general form

2

The constants are a, b, and c.

Once again, a cannot be zero because that would

cancel out the x term and make the rule a linear function.

General form is also called "Standard Form."

if you have a function in

Then use that number to plug into x to find the y-coordinate.

Click this circle for an example.

to find the x-coordinate.

2

y = x - 2x - 3

Here's our example.

Use the formula

b is -2

a is 1.

To find y, we plug the 1 back into the x's in the original function rule.

To find x we use

The vertex is at

(1, -4)

Click next.

2

The vertex is where the graph changes directions.

In this case,

it's changing from decreasing to increasing.

It's going down, down, down and then

right here

it starts going up, up, up.

This is y = x + 2x - 4

There are two forms that a quadratic can be written in.

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That would make the function a linear.

oh, and a is not allowed to be zero

function rule forms

2

y = a( x - h) + k

vertex form

The constants h and k make up the coordinates of

the graph's vertex, which we'll discuss in a little bit.

The vertex point would be (h, k).

Notice that the h is being subtracted in the form.

That means...

A positive h in the form means a negative x coordinate in the vertex.

A negative h in the form means a positive x coordinate in the vertex.

Click on either form or click next.

Click on a term or click next.

2

the parent function

When x is 0, y is 0

When x is 1, y is 1

When x is 2, y is 4

When x is 3, y is 9

and so on...

The parent function of a quadratic

(the function upon which all other quadratics are built) is y = x

2

By Rath Loeung

Quadratics are easy to spot.

Look for an x being sqared.

If there is no exponent larger

than a 2, then it's a

quadratic function.

2

y = x + 2x - 4

Here's what it looks like graphed

Notice its shape.

This U-shape is called a