Assignment #1

Explorering Linear Functions

By: Elizabeth Gore

In the following pages we will investigate graphically what happens to linear functions when they are added, multiplied, divided, and composed.

We will be working with randomly selected functions for eachof these explorations.

In the first case we will look at the sum of the two linear functions.

Here is a bit of background information about algebraically adding linear functions.

We will take the general case for each function.

Where a,b,c, and d are real numbers.

So, when we want to form our new function, h(x), we will add these two function together like so:

Since addition of real number gives a new real number, we can substitute new letters in to stand for the addition of the real numbers.

We get a new linear function

Let take a look graphically at these linear functions.

MULTIPLICATION

In this part of our investigation, we will see what happens when we multiply linear functions.

Again, it is helpful to look at what happens algebraically before we look at graphs.

Where a,b,c, and d are real numbers.

So, to form h(x), we are going to multiply the functions:

Since a,b,c, and d are all real numbers, we know that musliplying them will give us real numbers as well.

So, we could call the product of these numbers seperate letters themselves. Doing so will reveal something important.

So now when we substitute these letters for the numbers we would get when multiplying the first real numbers we have:

Hopefully this brings back memories of the quadratic equation.

So you remember what the graph of a quadratic euation looks like?

DIVISION

In this part of our investigation, we will obviously be dividing our linear functions.

Where a,b,c, and d are real numbers.

Looking at the general case of h(x) does not reveal too much to begin with.

What we do know is that cx+d can never equal zero, because division by zero is not defined.

COMPOSITION

Finally, we have composition of our two linear functions

Where a,b,c, and d are real numbers.

H(x) will show how these general functions are conposed.

Since we have multiplied and added real numbers, we can do as we did in the multiplication section and gives those terms new names.

So we now have a function that looks like

We get a new linear function.