Assignment 1: Graphs

Exploring Linear Combinations

Here is a graph of the line y = 1

And this is a graph of the line y = 3

In arithmetic, we are accustomed to ADDING and SUBTRACTING.
We can also add and subtract (so to speak) lines and other functions.

So, we can ADD y = 1 and y = 3.

What might ADDING mean with reference to LINES?

The red line shown in the graph below can be viewed as the result of
y = 1 and y = 3.

Can you determine how y = 1 and y = 3 were added to "create" the red line?

Does the following graph help make sense of adding the lines?

To get an idea of what is going on in a more formal sense, let's rename the two equations we started with.

Rather than y = 1 and y = 3,
let's change the notation so we can be sure that we are always talking about two different lines, like this:

So, we have

and, of course,

But, what does that "4" mean?

When we add two lines, what do we get?
In other words,

Remember we said that this graph represents the addition of y = 1 and y = 3

a

So when we add two lines,
we get another line

a

In this case, we "created" a line
y = 4

We can respond with:

But, is it clear what means?

Let's try these two linear equations:

y= x + 3
and
y = x + 2

What do you think we will get by adding these two linear equations?

Did you come up with something like this?

Is this a "good" solution?

How can we tell?

If
, we need:

(x+ 2) + ( x+ 3)

What do you get now?

Did you get something like this?

What is the equation of the red line?

y = 2x + 5

Do you understand why?

Do you have an interpretation of how these line all relate to each other?

Let's go back to the question of what tells us.

Remember that we identify points on a line with coordinate pairs and that this is a fancy term meaning that any point has an x-component and a y=component.

So the equation tells us to add the y=components.

Let's take another look at the equations y= x + 3 and y = x + 2 when x = 0 and x = -2.

When x = 0, y= x + 3 becomes y= 0 + 3 = 3.

Similarly, at x = 0, y = x + 2 becomes y = 0 + 2 = 2.

In this case, gives us 3 + 2 = 5.

Look at the graph above again: What is the y-value of the red line at x = 0?

At x = 0, y = 2x + 5 = 5

When x = -2, y= x + 3 becomes y= -2 + 3 = 1.

Similarly, at x = -2, y = x + 2 becomes y = -2 + 2 = 0.

And y = 2x + 5 becomes y = 2(-2) + 5 = -4 + 5 = 1.

Look again at the graph: Does the result y = 1 for y = 2x + 5 seem correct?

Here is another look at the same three lines

As you move a point along the horizontal axis,
how do the y-values for the three lines relate to each other?