Vertex and Factor Form

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?