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MAT 153: College Algebra and Trigonometry

Section 1.2 Linear Functions

In this course we will study many types of functions. In mathematics, functions are relations where each input produces exactly one output. Linear functions are used to describe patterns where the rate of change is constant, and will be the first type of function that we study.
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Goals:
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  • L: Be able to solve a linear equation.
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  • L: Be able to model a situation with appropriate linear equation(s) and interpret the solution.
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  • L: Be able to determine the slope or equation of a linear function given its graph or a table of values.
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  • F: Be able to determine inputs or outputs from a function table.
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  • F: Be able to determine inputs or outputs from a function graph.
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Definition 1.2.1.

Slope is a measure of the steepness of a line. For a linear function \(f(x)\text{,}\) the slope of a line can be calculated as the ratio
\begin{equation*} \text{slope}=\frac{\text{change in output}}{\text{change in input}}=\frac{f(x_2)-f(x_1)}{x_2-x_1} \end{equation*}
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Definition 1.2.2.

Equations of lines are often written in slope-intercept form,
\begin{equation*} y=mx+b \end{equation*}
where \(m\) is the slope of the line and \(b\) is the vertical intercept (the point where the line crosses the vertical axis). The point-slope form of a line is also often useful and is written
\begin{equation*} y-y_1=m(x-x_1) \end{equation*}
where \(m\) is the slope of the line and \((x_1,y_1)\) is any point on the line.
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Example 1.2.3.

Write an equation of each line below in slope-intercept form.
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  1. \(v\) as shown in the graph below.
    A graph of a linear function with negative slope. The line decreases from left to right and crosses the vertical axis at y equals 1.
    Figure 1.2.4.
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  2. Table 1.2.5.
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    \(t\) \(-1\) 0 1 2 3
    \(g(t)\) 2 1.5 1 0.5 0
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Definition 1.2.6.

The domain of a function is the set of inputs to the function. The range of a function is the set of outputs from the function.
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Problem 1.2.7.

Let \(f(x)=2x+3\text{.}\)
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  1. Graph the function on the domain \(-4 \leq x \leq 4\text{.}\) Be sure to label the \(x-\) and \(y-\)intercepts and the endpoints of the function on this domain.
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  2. Solve \(f(x)=-1\) for \(x\text{.}\)
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  3. What is \(f(\frac{1}{2})\text{?}\)
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Problem 1.2.8.

Table Table 1.2.9 shows a number pattern. Fill in the blanks in the table, and make a graph of the values in the table. Then, write a function equation for \(f\text{.}\)
Table 1.2.9. Patterns
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\(n\) \(f(n)\)
1 1
2 3
3 5
4 7
5
6
20
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Investigation 1.2.1.

Nancy is a landscape artist. She specializes in square ponds surrounded by hand-painted tiles. Customers can order a pond in any size square, starting with sides measuring 2 feet, and available in one-foot increments thereafter (sides of 3 feet, 4 feet, 5 feet, etc.). The tiles are 1-foot square, and are placed edge-to-edge along the entire outer perimeter of the pond. An example pond is shown below.
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A 4×4 grid with a centered 2×2 block of dark shaded squares.
Figure 1.2.10.
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  1. Make a table showing the number of border tiles needed for different pond sizes, from 2-foot sides up through 6-foot sides.
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  2. If the side length of the pond is increased by 1 foot, how many more border tiles are needed? Explain why the number of tiles is increasing according to this pattern.
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  3. How many border tiles are needed for a pond with sides of length 12 feet?
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  4. If Nancy orders 64 border tiles for an upcoming job, how large is the pond the customer wants?
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  5. Describe in words how to find the number of tiles needed for the border of a pond.
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  6. Write an equation for the number of tiles as a function of the length of one side of the pond.
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Problem 1.2.11.

A local doughnut shop charges $9 for the first dozen doughnuts purchased, and $0.50 for each additional individual doughnut purchased.
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  1. Write a linear function for \(C(d)\text{,}\) the cost of \(d\) doughnuts.
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  2. For what values of \(d\) does your equation from Item 1 make sense?
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  3. How much will 20 doughnuts cost?
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  4. Your club is having a fundraiser reselling doughnuts from this shop. You were given $20 to buy doughnuts to resell. How many doughnuts can you buy?
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  5. Using your answer from Item 4, what is the average cost of each doughnut?
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  6. Your club wants to sell the doughnuts for $1 and make a profit of at least $0.45 on each doughnut. What is the minimum number of doughnuts you need to buy (and sell) in order for this to happen?
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Problem 1.2.12.

Stephanie is planting her summer garden. She purchased a sunflower plant from a nursery and planted it when it was 6 cm tall. After she planted it, the sunflower plant grew at a constant rate of 3 cm per day.
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  1. Write a function, \(H(t)\) that gives the height of the sunflower plant as a function of \(t\text{,}\) the number of days since the plant was planted.
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  2. A worker at the nursery told Stephanie that her sunflower would reach its maximum height about 40 days after it was planted. What will the height be at that time?
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  3. How many days after planting would the sunflower be half its maximum height?
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  4. Consider your function from Item 1. What domain and range make sense in the context of this problem?
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Problem 1.2.13.

Isabella is considering renting a car for one day to drive to an event in another city. Company A charges $20 per day plus $0.25 per mile, and Company B charges $25 per day plus $0.15 per mile.
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  1. Write a linear function equation that models the one day cost of renting a car from Company A, \(C_A\text{,}\) as function of \(m\text{,}\) the number of miles driven.
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  2. Write a linear function equation that models the one day cost of renting a car from Company B, \(C_B\text{,}\) as function of \(m\text{,}\) the number of miles driven.
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  3. Graph \(C_A\) and \(C_B\) on the axes below. Use the graph to estimate the number of miles for which the cost for Company A is the same as the cost for Company B.
    A blank coordinate grid with the horizontal axis labeled m from 0 to 80 and the vertical axis labeled C(m) from 0 to 60.
    Figure 1.2.14.
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  4. Use algebra to find the number of miles for which the cost for Company A is the same as the cost for Company B.
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  5. If the event Isabella is going to is a 60 mile round trip drive, which rental company should she choose? How much will she save by choosing that company?
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