# 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 (x1, y1) and (x2, y2), then the vertical change is y2 - y1 and the horizontal change is x2 - x1.  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,

1. 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.
2. 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