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 $$x+3=0 \Leftrightarrow x=-3$$.