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### Does the Graph of the Equation Rise or Fall?

```Date: 10/22/2002 at 07:31:39
From: Becky Della Noce
Subject: Linear relations and function (slopes and intercepts)

Hi,

We are learning about graphs and how to graph the x and y intercepts,
whether it is rise or fall. The question I don't understand is:

Determine whether the graph of each equation rises to the right or
falls to the right, is horizontal, or is vertical.

The example question to figure out is 2x + 12 = 0.

Thanks a lot,
Becca
```

```
Date: 10/22/2002 at 11:47:13
From: Doctor Ian
Subject: Re: Linear relations and function (slopes and intercepts)

Hi Becca,

y = 2x + 12

Here's one way to think about it. Suppose x is zero. Then

y = 2(0) + 12

= 12

So the line crosses the y-axis at (0,12). We can label that point p on
the graph below:

|
p
|
|
|
--------- ---------
|

Now, let's consider what happens if we increase x. The 12 stays the
same, no matter what value of x we pick. But we're going to add 2x to
that, and 2x gets bigger as x gets bigger, right?

So as we move to the right - that is, as we choose bigger values of x
- the value of y has to increase. So the graph must slope upward as we
go to the right:

|          .  more x means more y
|          .
p...........
|
|
|
--------- ---------
|

Does that make sense?  Now, what if the equation were

y = -2x + 12

We'd still have our point p in the same place, but now every time we
make x bigger, we make y _smaller_, because we're subtracting instead

|
|
p...........
|          .
|          .  more x means less y
|
--------- ---------
|

The only difference between the two equations is the sign of the x
term. In the case where the sign is positive (more x means more y),
the line slopes upward. In the case where the sign is negative (more
x means less y), the line slopes downward.

What if the slope is zero?  Then we have

y = 0x + 12

or just

y = 12

which is to say, the value of y is _always_ 12, regardless of what x
is. This is a horizontal line - a line that has the same value for y
everywhere.

And what about a vertical line? That's like a horizontal line, but
instead of keeping y constant, we keep x constant, e.g.,

x = 3

Now, the equation you started with can be rearranged to look like
this:

2x + 12 = 0

2x = -12

x = -6

So it's the equation of a vertical line. The value of x is -6, no
matter _what_ value you choose for y.

Does this help?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Linear Equations

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