# Graphing Quadratic Equations

A Quadratic Equation in Standard Form

(**a**, **b**, and **c** can have any value, except that **a** can't be 0.)

Here is an example:

## Graphing

You can graph a Quadratic Equation using the Function Grapher, but to **really understand** what is going on, you can make the graph yourself. Read On!

## The Simplest Quadratic

The simplest Quadratic Equation is:

f(x) = x^{2}

And its graph is simple too:

This is the curve f(x) = x^{2}

It is a parabola.

Now let us see what happens when we introduce the "a" value:

f(x) = ax^{2}

- Larger values of
**a**squash the curve inwards - Smaller values of
**a**expand it outwards - And negative values of
**a**flip it upside down

## Play With ItNow is a good time to play with the |

## The "General" Quadratic

Before graphing we **rearrange** the equation, from this:

f(x) = ax^{2} + bx + c

To this:

f(x) = a(x-h)^{2} + k

Where:

- h = −b/2a
- k =
h**f(****)**

In other words, calculate **h** (= −b/2a), then find **k** by calculating the whole equation for **x=h**

## But Why?

The wonderful thing about this new form is that h and k show us the very lowest (or very highest) point, called the **vertex**:

And also the curve is symmetrical (mirror image) about the **axis** that passes through **x=h**, making it easy to graph

### So ...

**h**shows us how far left (or right) the curve has been shifted from x=0**k**shows us how far up (or down) the curve has been shifted from y=0

Lets see an example of how to do this:

### Example: Plot f(x) = 2x^{2} − 12x + 16

First, let's note down:

**a = 2,****b = −12,**and**c = 16**

Now, what do we know?

- a is positive, so it is an "upwards" graph ("U" shaped)
- a is 2, so it is a little "squashed" compared to the
**x**graph^{2 }

Next, let's calculate h:

**3**

And next we can calculate k (using h=3):

*3*

**f(***= 2(3)*

**)**^{2}− 12·3 + 16 = 18−36+16 =

**−2**

So now we can plot the graph (with real understanding!):

We also know: the **vertex** is (3,−2), and the **axis** is x=3

## From A Graph to The Equation

What if we have a graph, and want to find an equation?

### Example: you have just plotted some interesting data, and it looks Quadratic:

Just knowing those two points we can come up with an equation.

Firstly, we know **h** and **k** (at the vertex):

(h, k) = (1, 1)

So let's put that into this form of the equation:

f(x) = a(x-h)^{2} + k

f(x) = a(x−1)^{2} + 1

Then we calculate "a":

**(0, 1.5)**so:f(0) = 1.5

**a(x−1)**at x=0 is:f(0) = a(0−1)

^{2}+ 1^{2}+ 1

**f(0)**so make them equal: a(0−1)

^{2}+ 1 = 1.5

And here is the resulting Quadratic Equation:

f(x) = 0.5(x−1)^{2} + 1

Note: This may not be the **correct** equation for the data, but it’s a good model and the best we can come up with.