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Section 1.5 Linear Functions

Subsection 1.5.1 Slope-Intercept Form

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 \((0,b)\text{.}\)

Caution 1.118.

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 Figure 1.119 and Figure 1.120.
\(f (x) = 10 - 3x\)
\(\hphantom{00}x\hphantom{00}\) \(\hphantom{0}f(x)\hphantom{0}\)
\(0\) \(10\)
\(1\) \(7\)
\(2\) \(4\)
\(3\) \(1\)
\(4\) \(-2\)
described in detail following the image
deceasing line
Figure 1.119.
\(g(x) = -3+2x\)
\(\hphantom{00}x\hphantom{00}\) \(\hphantom{0}f(x)\hphantom{0}\)
\(0\) \(-3\)
\(1\) \(-1\)
\(2\) \(1\)
\(3\) \(3\)
\(4\) \(5\)
described in detail following the image
inceasing line
Figure 1.120.
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 \(f(a)=0\text{.}\)

Example 1.121.

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.
M atch the graphs in Figure 1.122 to Francine’s Internet cost with each company.
Solution.
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.
\begin{equation*} \text{Juno: } f(x) = 30 + 14.95x \end{equation*}
\begin{equation*} \text{ISP.com: } g(x) = 30 + 12.95x \end{equation*}
\begin{equation*} \text{peoplepc: } h(x) = 30 + 15.95x \end{equation*}
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.
described in detail following the image
internet provider costs
Figure 1.122.

Example 1.123.

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.
  1. Write a formula for Delbert’s Internet costs under each plan.
  2. Match Delbert’s Internet cost under each company with its graph in Figure 1.124.
described in detail following the image
dsl costs
Figure 1.124.
Solution.
  1. Earthlink: \(f (x) = 99 + 39.95x\text{;}\) DigitalRain: \(g(x) = 50 + 34.95x\text{;}\) FreeAmerica: \(h(x) = 149 + 34.95x\)
  2. DigitalRain: I; Earthlink: II; FreeAmerica: III

Note 1.125.

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 Figure 1.126). The collection of all linear functions \(f (x) = b + mx\) is called a two-parameter family of functions.
described in detail following the image
lines with common intercept or common slope
Figure 1.126.

Subsection 1.5.2 Slope-Intercept Method of Graphing

Look again at the lines in Figure 1.126: 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.

Example 1.127.

  1. Write the equation \(4x - 3y = 6\) in slope-intercept form.
  2. Graph the line by hand.
Solution.
  1. 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*}
  2. 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 Figure 1.128.)
described in detail following the image
slope-intercept method
Figure 1.128.

Note 1.129.

The slope of a line is a ratio and can be written in many equivalent ways. In Example 1.127, 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.

  1. Plot the point associated with the vertical intercept, \((0, b)\text{.}\)
  2. 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.
  3. Use an equivalent form of the slope to find a third point, and draw a line through the points.
described in detail following the image
slope-intercept method

Example 1.130.

Exercises 1.5.3 Exercises

1. Creating a Linear Function.

You biked 21 miles in 2 hours and 45 minutes. What was your average rate of change?
mi/hr
Hint.
Since we want our answer to have units miles per hour, we need to write 2 hours and 45 minutes in hours. So the time taken on the bike ride is 2.75 hours. The average rate of change will be given by the total number of miles traveled divided by the total number of hours.

2. Creating a Linear Function.

In 2001, a 256 MB USB flash drive cost $ 155. In 2017, a pack of ten 256 MB USB flash drives can be purchased for $ 23. Assuming the value of a flash drive has depreciated linearly with time, write a formula giving the cost of a single 256 MB USB flash drive \(C\) as a function of \(t\) the number of years since 2001. For this problem you can assume that you don’t get any special discount for purchasing ten instead of one.
C(t)=
Hint.
Since the input of the function is the number of years since 2001, we are given the initial value (vertical intercept). We know that 16 years later, the cost of a 10 pack is $ 23. Therefore, the cost of a single 256 MB USB flash drive costs $ 2.3. We need to use this information to calculate the slope. Recall the slope formula:
\begin{equation*} m=\frac{y_2-y_1}{t_2-t_1}. \end{equation*}

4. Interpreting the Components of a Linear Equation.

A child wants to make money by selling lemonade at their own lemonade stand. The cost, in dollars, of all the ingredients they need is $13.80. They decide to charge 0.6 dollars for a cup of lemonade, hence, their profit \(P(n)\) in dollars for selling \(n\) cups of lemonade is
\begin{equation*} P(n)=0.6 n-13.80. \end{equation*}
a) What is the \(n\) value of the \(n\)-intercept of the graph of \(P(n)\text{?}\) (That is, we are looking for the input-intercept.)
b) Which of the following accurately describes the \(n\)-intercept?
  • It is the most amount of cups of lemonade the child can sell.
  • It is the amount of money made by selling lemonade.
  • It is the number of cups of lemonade have to be sold for her to make a profit of zero dollars.
  • It is how much the child loses if they don’t sell any lemonade.
  • None of the above.

5. Finding Coefficients in a Linear Equation.

For the linear expression in \(x,\) give the constant term and coefficient of \(x.\)
\(-9-2 x - 4 x + 7.\)
Constant Term:
Coefficient of \(x\text{:}\)
Hint.
Hint
Simplify the expression to put it in the form \(mx+b\text{.}\)

6. Interpreting Linear Equations.

A report by the US Geological Survey indicates that glaciers in Glacier National Park, Montana, are shrinking. Recent estimates indicate the area covered by glaciers has decreased from over \(25.5 \ \mbox{km}^2\) in 1850 to about \(16.51 \ \mbox{km}^2\) in 1995. Let \(A = f(t)\) give the area (in square km) \(t\) years after 2000, and assume \(f(t) = 16.2 - 0.062t\text{.}\)
a) Find and explain the meaning of the slope. Which statement best explains its significance?
  • The area covered by glaciers is decreasing by \(16.2 \ \mbox{km}^2\) every year.
  • The total area covered by glaciers is increasing by \(0.062 \ \mbox{km}^2\) every year.
  • The area covered by glaciers is decreasing by \(62,000 \ \mbox{m}^2\) every year.
  • The total area covered by glaciers decreased by \(16.2 \ \mbox{km}^2\) from 1850 to 2000.
  • The area covered by glaciers is decreasing by \(62 \ \mbox{m}^2\) every year.
  • None of the above
b) Find and explain the meaning of the \(A\)-intercept. Which statement best explains its significance?
  • The area covered by glaciers is decreasing by \(16.2 \ \mbox{km}^2\) every year.
  • The total area covered by glaciers will decrease by \(0.062 \ \mbox{km}^2\) from 2000 to 2001.
  • The total area covered by glaciers decreased by \(16.2 \ \mbox{km}^2\) from 1850 to 2000.
  • The area covered by glaciers in 2000 was \(0.062 \ \mbox{km}^2\) .
  • The area covered by glaciers in 2000 was \(16.2 \ \mbox{km}^2\text{.}\)
  • None of the above
c) For both expressions listed below, enter the letter A-E of the statement which correctly completes the sentence. There are extra, unused statements.
  1. The number of years after 2000 that the total area covered by glaciers will be \(12 \ \mbox{km}^2\text{.}\)
  2. How much area will be covered by glaciers in 2012.
  3. The number of years after 2000 that the amount of glacier area that has disappeared is \(12 \ \mbox{km}^2\text{.}\)
  4. The amount of glacier area that disappears in 12 years.
  5. How much area will be covered by glaciers in 12 years.
  1. If \(f(t) = 12\text{,}\) then \(t\) is
  2. \(f( 12 )\) is
d) Evaluate \(f( 12 ) =\) .
e) Solve \(f(t) = 12\) .
\(t =\) (round your answer to one decimal place)
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