# 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

ADDING 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:

When we ADD two lines, we can say that
we add
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

So to answer the question:
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?

Think again about the
relationship

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?

Need help? Click here

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?