Quadratic Functions

quadratic function

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 Icon 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

Change the function to f(x)=sin(x) and f(x)=1/x

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 \(\vec{v} \).

Image

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 Image Vector tool), you can use the Image Translate tool to make a translated graph.

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

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