**Home**
/ **Review
Topics**

# Linear
Functions

### Definitions

A linear function has the
form *f(x) = mx + b* for some constants *m* and *b*.
The graph is a line with slope *m* and *y*-intercept *(0,b).
*Its domain is all real numbers since any real number can be substituted
for *x*. For any non-horizontal line, the range is also
all real numbers.

Any linear
function has a constant rate of increase or decrease. This constant
rate of change is the slope of the line and is represented by *m*
in the equation. Use the following interactive example to explore
how the graph of the line _{} changes as the values of the coefficients
m and b change. Note any useful observations in your journal.

__ __

### Slope of a Line

The slope is the constant
rate of change of a linear function. It can be though of as the ratio
of the vertical change to the horizontal change between two points
on the graph of a line. If the two points are (x_{1},
y_{1}) and (x_{2}, y_{2}), then the vertical
change is y_{2} - y_{1} and the horizontal change
is x_{2} - x_{1}. Hence we can use the formula

_{}

to determine the slope
of a line if we know two points on that line. See Figure 1 below.

**
Figure 1**

Explore
the following interactive example to see how the slope of a line is
computed when two points are known. What observations can you make?

In general, when the slope
of a linear function is positive, the function will be increasing.
That is, the graph will be a line that rises from left to right. When
the slope is negative, the function will be decreasing and the graph
will be a line that falls from left to right. If the slope of a linear
function is 0, then the function is neither increasing nor decreasing,
but is constant. The graph of a constant function is a horizontal
line.

Use
the next interactive example to see how different values of the slope,
m, affect the graph of the linear function y = mx + b.

### Horizontal and Vertical
Intercepts

The points where a line
crosses the vertical and horizontal axes are known as the vertical
and horizontal intercepts. (These points are often referred to as
the x-intercept and the y-intercept.) Given a linear function *f(x)
= mx + b,*

- The
__vertical intercept__
(y-intercept) is found by evaluating the function when the input
variable, *x*, is 0 and is always the same as the constant
*b.* It can be thought of as the original value of the function.
- The
__horizontal intercept__
(x-intercept) is the value of the variable *x *when the function
value is 0. It is found by solving the equation *0= mx +
b.*

#### Interactive
Example

Now
explore how the values of the y-intercept, b, affect the graph of
the linear function y = mx + b.

#### Algebraic
Example

Find the vertical and horizontal
intercept of the linear function _{}.

Solution

Since f*(0) = -7.2(0)
+ 250 = 250*, the vertical intercept is 250. This means that the
graph of the linear function crosses the horizontal axis at the point
(0, 250). Also notice that this is the value of *b *in the linear
function *f(x) = mx + b.*

To find the horizontal
intercept we can replace *f(x)* with 0 and solve the linear equation
_{} The solution is given below.

_{}

The horizontal intercept
is 34.7. This is the point (34.7, 0) on the graph of the linear function.
A graph of the linear function is shown in Figure 2.

Figure 2