# Quadratic Functions

The most common way to write a quadratic function is to use **general form**:

\[y=ax^2+bx+c\]

When analyzing
the graph of a quadratic function, or the correspondence between the graph and solutions to quadratic
equations, two other forms are more suitable: **vertex form**
and **factor form**.

## Vertex Form

### Exercise 1

Input a slider
called `k` into a GeoGebra window.

Enter the function \(y=x^2+k\) in the Input bar.

Move the slider `k`!

What effect does `k` have on the shape of the graph? What effect does `k` have on
the position of the graph?

### Exercise 2

Input another slider `h` into the GeoGebra window.

Change the function `y` to \(y=(x-h)^2\)

Move the slider `h`!

What effect does `h` have on the shape of the graph? What effect does `h` have on
the position of the graph?

### Exercise 3

Sketch the graph of \(y=(x+1)^2+2\) **on
a paper without using technology!**

Change the GeoGebra function to \(y=(x-h)^2+k\). Graph \(y=(x+1)^2+2\) using GeoGebra.

### Exercise 4

Input another slider `a` and adjust the function to \(y=a(x-h)^2+k\).

Move the slider `a`. Try putting a trace on the graph before moving
the slider.

What effect does `a` have on the shape of the graph? What effect does `a` have on
the position of the graph?

### Exercise 5

Sketch the graphs of following functions **on a paper without
using technology!**

**a)** \(y=2(x-1)^2-2\)

**b)** \(y=-(x+1)^2+2\)

**c)** \(y=0.5(x-1)^2\)

## Factor Form

### Exercise 6

Input two sliders `α` and `β` into a GeoGebra window.

Enter the function \(y=(x-\alpha )(x-\beta )\) in the Input bar.

Move the sliders!

From looking at the graph, how could you find the valuse of `α`
and `β`?

### Exercise 7

Input another slider `a` into the GeoGebra window.

Change the function to \(y=a(x-\alpha )(x-\beta )\).

Move the slider `a`!

What effect does the value of `a` have on the intercepts of the graph?

### Exercise 8

If you have a quadratic function in general form, how do you rewrite it to

vertex form? factor form?

Can you always rewrite it to

vertex form? factor form?

If not, why not?

## Translation using vector notation

In the applet above, the graph of \(g(x)\) is the graph of \(f(x)\) translated 2 units along the
`x`-axis and -1 unit along the `y`-axis. Each point on the graph of \(f(x)\) is translated
by a vector \(\vec{v} \).

The graph of the function \(g(x)=f(x-h)+k\), is the graph of \(f(x)\) translated \(h\) units along the
`x`-axis and \(k\) unit along the `y`-axis. Note the negative sign in front of \(h\).

Change the function to \(f(x)=\sin (x)\) in the input box in the applet above and observe the equation of \(g(x)\). Change the function to \(f(x)=1/x\).

### Using the tool Translate Object by Vector

If you insert a function and a vector (using the Vector tool), you can use the Translate tool to make a translated graph.

### Explanation

The graph of \(g(x)=x^2+3\), is the graph of \(f(x)=x^2\) translated **3 units** along the `y`-axis.

The graph of \(g(x)=(x+3)^2\), is the graph of \(f(x)=x^2\) translated **-3 units** along the
`x`-axis.

In order to understand the negative sign when translating along the `x`-axis, write the new function as
\(f(x+3)=(x+3)^2\). We know that this function has its vertex when the expression within the brackets is 0, i.e.
when \(x+3=0 \Leftrightarrow x=-3 \).

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License