1 Linear functions

Any function of the form $y=f\left(x\right)=ax+b$ where $a$ and $b$ are constants is called a linear function. The constant $a$ is called the coefficient of $x$ , and $b$ is referred to as the constant term.

For example, $f\left(x\right)=3x+2$ , $g\left(x\right)=\frac{1}{2}x-7$ , $h\left(x\right)=-3x+\frac{2}{3}$ and $k\left(x\right)=2x$ are all linear functions.

The graph of a linear function is always a straight line. Such a graph can be plotted by finding just two distinct points and joining these with a straight line.

Example 10

Plot the graph of the linear function $y=f\left(x\right)=4x+3$ .

Solution

We start by finding two points. For example if we choose $x=0$ , then $y=f\left(0\right)=3$ , i.e. the first point has coordinates $\left(0,3\right)$ . Secondly, suppose we choose $x=5$ , then $y=f\left(5\right)=23$ . The second point is $\left(5,23\right)$ . These two points are then plotted and then joined by a straight line as shown in the following diagram.

Example 11

Plot graphs of the three linear functions $y=4x-3$ , $y=4x$ , and $y=4x+5$ , for $-2\le x\le 2$ .

Solution

For each function it is necessary to find two points on the line.

For $y=4x-3$ , suppose for the first point we choose $x=0$ , so that $y=-3$ . For the second point, let $x=2$ so that $y=5$ . So, the points $\left(0,-3\right)$ and $\left(2,5\right)$ can be plotted and joined. This is shown in the following diagram.

For $y=4x$ we find the points $\left(0,0\right)$ and $\left(2,8\right)$ . Similarly for $y=4x+5$ we find points $\left(0,5\right)$ and $\left(2,13\right)$ . The corresponding lines are also shown in the figure.

Task!

Refer to Example 11. Comment upon the effect of changing the value of the constant term of the linear function.

Answer

The value of the constant term is also known as the vertical or $y$ -axis intercept because this is the value of $y$ where the line cuts the $y$ axis.

Task!

State the vertical intercept of each of the following lines:

1. $y=3x+3$ ,
2. $y=\frac{1}{2}x-\frac{1}{3}$ ,
3. $y=1-3x$ ,
4. $y=-5x$ .

In each case you need to identify the constant term:

Answer

Example 12

Plot graphs of the lines $y=3x+3$ , $y=5x+3$ and $y=-2x+3$ .

Solution

Note that all three lines have the same constant term, that is 3. So all three lines pass through $\left(0,3\right)$ , the vertical intercept. A further point has been calculated for each of the lines and their graphs are shown in the following diagram.

Note from the graphs in Example 12 that as the coefficient of $x$ is changed the gradient of the graph changes. The coefficient of $x$ gives the gradient or slope of the line. In general, for the line $y=ax+b$ a positive value of $a$ produces a graph which slopes upwards from left to right. A negative value of $a$ produces a graph which slopes downwards from left to right. If $a$ is zero the line is horizontal, that is its gradient is zero. These properties are summarised in the next figure.

Figure 24 :

Task!

State the gradients of the following lines:

1. $y=7x+2$
2. $y=-\frac{1}{3}x+4$
3. $y=\frac{x+2}{3}$

In each case the coefficient of $x$ must be examined:

Answer

Task!

Which of the following lines has the steepest gradient ?

1. $y=\frac{17x+4}{5}$ ,
2. $y=9x-2$ ,
3. $y=\frac{1}{3}x+4$ .

Answer

Exercises

1. State the general form of the equation of a straight line explaining the role of each of the terms in your answer.
2. State which of the following functions will have straight line graphs.

   (a) $f\left(x\right)=3x-3$ , (b) $f\left(x\right)=x^{1∕2}$ , (c) $f\left(x\right)=\frac{1}{x}$ , (d) $f\left(x\right)=13$ , (e) $f\left(x\right)=-2-x$ .

3. For each of the following, identify the gradient and vertical intercept.
   1. $f\left(x\right)=2x+1$ ,
   2. $f\left(x\right)=3$ ,
   3. $f\left(x\right)=-2x$ ,
   4. $f\left(x\right)=-7-17x$ ,
   5. $f\left(x\right)=mx+c$ .

Answer