Linear relationships are relationships in which the rate of change is constant.
Linear Equation
A linear function is a function which has a constant rate of change.
Many phenomena can be modeled using linear functions \(y = f (x)\) where the equations have the form
\begin{equation*}
f (x) = (\text{starting value}) + (\text{rate of change}) \cdot x.
\end{equation*}
The rate of change is the slope of the graph, and the initial value is the value of \(f(0)\text{.}\) On the graph of \(f\text{,}\) the initial value corresponds to the point that lies on the vertical axis.
Vertical intercept / \(y\)-intercept
A vertical intercept is a point where a graph intersects the vertical axis. For a function \(f\text{,}\) the vertical intercept of the graph of \(f\) is the point \((0,f(0))\text{.}\) Because we often use the variable \(y\) on the vertical axis, the vertical intercept is often called the \(y\)-intercept.
For simplicity, we often use the variable \(b\) for \(f(0)\) so that the vertical intercept of a function \(f\) is at \((0,b)\text{.}\)
Caution119
We will often abuse notation and say that the \(y\)-intercept of a graph is \(b\text{.}\) What we really mean is that the \(y\)-intercept is the point \((0,b)\text{.}\)
We can write the equation for a linear function explicitly as
\begin{equation*}
f (x) = b + mx
\end{equation*}
where the constant term, \(b\text{,}\) is the \(y\)-intercept and \(m\text{,}\) the coefficient of \(x\text{,}\) is the slope of the line. This form for an equation of a line is called the slope-intercept form.
Slope-Intercept Form
If we write an equation of a linear function in the form,
\begin{equation*}
f (x) = b + mx
\end{equation*}
then \(m\) is the slope of the line, and \(b\) is the \(y\)-coordinate of the \(y\)-intercept of the line.
(You may have encountered the slope-intercept equation in the equivalent form \(y = mx + b\text{.}\))
For example, consider the two linear functions and their graphs shown in Figure120 and Figure121.
\(f (x) = 10 - 3x\)
\(g(x) = -3+2x\)
Figure120Figure121
Some observations:
We can see that the \(y\)-intercept of each line is given by the constant term, \(b\text{.}\)
By examining the table of values, we can also see why the coefficient of \(x\) gives the slope of the line:
For \(f (x)\text{,}\) each time \(x\) increases by \(1\) unit, \(y\) decreases by \(3\) units.
For \(g(x)\text{,}\) each time \(x\) increases by \(1\) unit, \(y\) increases by \(2\) units.
For each graph, the coefficient of \(x\) is a scale factor that tells us how many units \(y\) changes for \(1\) unit increase in \(x\text{.}\) But that is exactly what the slope tells us about a line.
Is is also useful to introduce the term \(x\)-intercept at this point.
\(x\)-intercept
An \(x\)-intercept is a point where a graph intersects the \(x\)-axis. It corresponds to the point \((a,0)\) where \(a\) is a value such that \(f(a)=0\text{.}\)
Example122
Francine is choosing an Internet service provider. She paid $30 for a modem, and she is considering three companies for service:
Juno charges $14.95 per month,
ISP.com charges $12.95 per month,
and peoplepc charges $15.95 per month.
Match the graphs in Figure123 to Francine's Internet cost with each company.
Francine pays the same initial amount, $30 for the modem, under each plan. The monthly fee is the rate of change of her total cost, in dollars per month. We can write a formula for her cost under each plan.
The graphs of these three functions all have the same \(y\)-intercept, but their slopes are determined by the monthly fees. The steepest graph, III, is the one with the largest monthly fee, peoplepc, and ISP.com, which has the lowest monthly fee, has the least steep graph, I.
Figure123
Example124
Delbert decides to use DSL for his Internet service.
Earthlink charges a $99 activation fee and $39.95 per month,
DigitalRain charges $50 for activation and $34.95 per month,
and FreeAmerica charges $149 for activation and $34.95 per month.
Write a formula for Delbert's Internet costs under each plan.
Match Delbert's Internet cost under each company with its graph in Figure125.
In the equation \(f (x) = b + mx\text{,}\) we call \(m\) and \(b\) parameters. Their values are fixed for any particular linear equation; for example, in the equation \(y = 2x + 3\text{,}\) \(m = 2\) and \(b = 3\text{,}\) and the variables are \(x\) and \(y\text{.}\) By changing the values of \(m\) and \(b\text{,}\) we can write the equation for any line except a vertical line (see Figure127). The collection of all linear functions \(f (x) = b + mx\) is called a two-parameter family of functions.
Figure127
SubsectionSlope-Intercept Method of Graphing
Look again at the lines in Figure127: There is only one line that has a given slope and passes through a particular point. That is, the values of \(m\) and \(b\) determine the particular line. The value of \(b\) gives us a starting point, and the value of \(m\) tells us which direction to go to plot a second point. Thus, we can graph a line given in slope-intercept form without having to make a table of values.
Example128
Write the equation \(4x - 3y = 6\) in slope-intercept form.
We solve the equation for \(y\) in terms of \(x\text{.}\) \begin{align*} -3y \amp =6 - 4x\\ y \amp = \frac{6 - 4x}{-3} =\frac{6}{-3}+\frac{-4x}{-3}\\ y \amp = -2+\frac{4}{3}x \end{align*}
We see that the slope of the line is \(m = \dfrac{4}{3}\text{,}\) and its vertical intercept is \(b = -2\text{.}\) We begin by plotting the \(y\)-intercept, \((0, -2)\text{.}\) We then use the slope to find another point on the line. We have
\begin{equation*}
m = \frac{\Delta y}{\Delta x}=\frac{4}{3}
\end{equation*}
so starting at \((0, -2)\text{,}\) we move \(4\) units in the \(y\)-direction and \(3\) units in the \(x\)-direction, to arrive at the point \((3, 2)\text{.}\) Finally, we draw the line through these two points. (See Figure129.)
Figure129
Note130
The slope of a line is a ratio and can be written in many equivalent ways. In Example128, the slope is equal to \(\dfrac{8}{6}\text{,}\) \(\dfrac{12}{9}\text{,}\) and \(\dfrac{-4}{-3}\text{.}\) We can use any of these fractions to locate a third point on the line as a check. If we use \(m = \dfrac{\Delta y}{\Delta x}= \dfrac{-4}{-3}\text{,}\) we move down \(4\) units and left \(3\) units from the \(y\)-intercept to find the point \((-3, -6)\) on the line.
Slope-Intercept Method for Graphing a Line
Plot the point associated with the vertical intercept, \((0, b)\text{.}\)
Use the definition of slope to find a second point on the line: Starting at the vertical intercept, move \(\Delta y\) units in the \(y\)-direction and \(\Delta x\) units in the \(x\)-direction. Plot a second point at this location.
Use an equivalent form of the slope to find a third point, and draw a line through the points.
Example131
Write the equation \(2y + 3x + 4 = 0\) in slope-intercept form.
