# A Math Primer

**Slope
**The slope of a line is a measure of how “tilted” the

line is . A highway sign might say something like

“6% grade ahead.” What does this mean, other than

that you hope your brakes work? What it means is

that the ratio of your drop in altitude to your

horizontal distance is 6%, or 6/100. In other words, if

you move 100 feet forward, you will drop 6 feet; if

you move 200 feet forward, you will drop 12 feet,

and so on.

We measure the slope of lines in much the same way,

although we do not convert the result to a percent.

Suppose we have a graph of an unknown straight

line. Pick any two different points on the line and

label them point 1 and point 2:

In moving from point 1 to point 2, we cover 4 steps

horizontally (the x direction) and 2 steps vertically

(the y direction):

Therefore, the ratio of the change in altitude to the

change in horizontal distance is 2 to 4. Expressing it

as a fraction and reducing , we say that the slope of

this line is

To formalize this procedure a bit, we need to think

about the two points in terms of their x and y

coordinates.

Now you should be able to see that the horizontal

displacement is the difference between the x

coordinates of the two points, or

4 = 5 – 1,

and the vertical displacement is the difference

between the y coordinates, or

2 = 4 – 2.

In general, if we say that the coordinates of point 1

are and the coordinates of point 2 are

,

then we can define the slope m as follows:

where and
are any two distinct

points on the line.

It is customary (in
the US) to use the letter m to

represent slope. No one knows why.

It makes no difference which two points are used

for point 1 and point 2. If they were switched,

both the numerator and the denominator of the

fraction would be changed to the opposite sign,

giving exactly the same result.

Many people find it useful to remember this

formula as “slope is rise over run.”

Another common notation is m = Δy/Δx, where

the Greek letter delta (Δ) means “the change in.”

The slope is a ratio of how much y changes per

change in x:

**Horizontal Lines**

A horizontal line has zero slope, because there is no

change in y as x increases. Thus, any two points will

have the same y coordinates, and since ,

**Vertical Lines**

A vertical line presents a different problem. If you

look at the formula

you see that there is a problem with the denominator.

It is not possible to get two different values for
and

, because if x changes then you are not on the

vertical line anymore. Any two points on a vertical

line will have the same x coordinates, and so

. Since the denominator of a fraction

cannot be zero, we have to say that **a vertical line
has undefined slope. **Do not confuse this with the

case of the horizontal line, which has a well-defined

slope that just happens to equal zero.

**Positive and Negative Slope**

The x coordinate increases to the right, so moving

from left to right is motion in the positive x direction.

Suppose that you are going uphill as you move in the

positive x direction. Then both your x and y

coordinates are increasing, so the ratio of rise over

run will be positive—you will have a positive

increase in y for a positive increase in x. On the other

hand, if you are going downhill as you move from

left to right, then the ratio of rise over run will be

negative because you lose height for a given positive

increase in x. The thing to remember is:

As you go from left to right,

Uphill = Positive Slope

Downhill = Negative Slope

And of course, no change in height means that the

line has zero slope.

Some Slopes

**Intercepts
**Two lines can have the same slope and be in

different places on the graph. This means that in

addition to describing the slope of a line we need

some way to specify exactly where the line is on the

graph. This can be accomplished by specifying one

particular point that the line passes through.

Although any point will do, it is conventional to

specify the point where the line crosses the y-axis.

This point is called the y-intercept, and is usually

denoted by the letter b. Note that every line except

vertical lines will cross the y-axis at some point, and

we have to handle vertical lines as a special case

anyway because we cannot define a slope for them.

Same Slopes, Different y-Intercepts

**Equations
**The equation of a line gives the mathematical

relationship between the x and y coordinates of any

point on the line.

Let’s return to the example we used in graphing

functions. The equation

y = 2x – 1

produces the following graph:

This line evidently has a slope of 2 and a y intercept

equal to –1. The numbers 2 and –1 also appear in the

equation—the coefficient of x is 2, and the additive

constant is –1. This is not a coincidence, but is due to

the standard form in which the equation was written.

**Standard Form (Slope- Intercept Form )
**If a linear equation in two unknowns is written in the

form

y = mx + b

where m and b are any two real numbers , then the

graph will be a straight line with a slope of m and a y

intercept equal to b.

**Point-Slope Form**

As mentioned earlier, a line is fully described by

giving its slope and one distinct point that the line

passes through. While this point is customarily the y

intercept, it does not need to be. If you want to

describe a line with a given slope m that passes

through a given point , the formula is

To help remember this formula, think of solving it

for m:

Since the point (x, y) is an arbitrary point on the line

and the point is another point on the line, this

is nothing more than the definition of slope for that

line.

**Two-Point Form
**Another way to completely specify a line is to give

two different points that the line passes through. If

you are given that the line passes through the points

and , the formula is

This formula is also easy to remember if you notice

that it is just the same as the point-slope form with

the slope m replaced by the definition of slope,

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