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Intermediate Algebra: Functions and Graphs

Katherine Yoshiwara

Section 1.6 Chapter Summary and Review

Subsection 1.6.1 Glossary

  • mathematical model
  • linear model
  • evaluate an expressions
  • solve an equation
  • increasing graph
  • decreasing graph
  • inequality
  • intercepts
  • linear equation
  • solution of an equation
  • graph of an equation
  • ordered pair
  • rate
  • rate of change
  • net change
  • slope
  • slope-intercept form
  • point-slope form

Subsection 1.6.2 Key Concepts

  1. A mathematical model is a simplified description of reality that helps us understand a system or process.
  2. We can describe a relationship between variables with a table of values, a graph, or an equation.
  3. Linear models have equations of the form:
    \begin{gather*} y = (\blert{\text{starting value}}) + (\blert{\text{rate of change}}) \cdot x \end{gather*}
  4. The general form for a linear equation is: \(Ax + By = C\text{.}\)
  5. We can use the intercepts to graph a line. The intercepts are also useful for interpreting a model.
  6. The graph of an equation in two variables is just a picture of all its solutions.
  7. Lines have constant slope.
  8. The slope of a line gives us the rate of change of one variable with respect to another.
  9. Formulas for Linear Models
    • slope: \(\qquad\qquad m=\dfrac{\Delta y}{\Delta x} = \dfrac{y_2-y_1}{x_2-x_1} \text{,}\) \(x_1 \ne x_2\)
    • slope-intercept form: \(\qquad\qquad y=b+mx \)
    • point-slopet form: \(\qquad\qquad y=y_1+m(x-x_1) \)
  10. The slope-intercept form is useful when we know the initial value and the rate of change.
  11. The point-slope form is useful when we know the rate of change and one point on the line.
.

Exercises 1.6.3 Chapter 1 Review Problems

1.

Last year, Pinwheel Industries introduced a new model calculator. It cost $2000 to develop the calculator and $20 to manufacture each one.
  1. Complete the table of values showing the total cost, \(C\text{,}\) of producing \(n\) calculators.
    \(n\) \(100\) \(500\) \(800\) \(1200\) \(1500\)
    \(C\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)
  2. Write an equation that expresses \(C\) in terms of \(n\text{.}\)
  3. Graph the equation by hand.
    grid
  4. What is the cost of producing 1000 calculators? Illustrate this as a point on your graph.
  5. How many calculators can be produced for $10,000? Illustrate this as a point on your graph.
Answer.
  1. \(n\) \(100\) \(500\) \(800\) \(1200\) \(1500\)
    \(C\) \(4000\) \(12,000\) \(18,000\) \(26,000\) \(32,000\)
  2. \(\displaystyle C = 20n + 2000\)
  3. cost of producing calculators
  4. $22,000
  5. 400

2.

The world’s oil reserves were 2100 billion barrels in 2005; total annual consumption is 28 billion barrels per year.
  1. Complete the table of values that shows the remaining oil reserves \(R\) in terms of time \(t\) (in years since 2005).
    \(t\) \(~ 5 ~\) \(~10~\) \(~15~\) \(~20~\) \(~25~\)
    \(R\) \(\) \(\) \(\) \(\) \(\)
  2. Write an equation that expresses \(R\) in terms of \(t\text{.}\)
  3. Find the intercepts and graph the equation by hand.
    grid
  4. What do the intercepts tell us about the world’s oil supply?
Answer.
  1. \(t\) \(~ 5 ~\) \(~10~\) \(~15~\) \(~20~\) \(~25~\)
    \(R\) \(1960\) \(1820\) \(1680\) \(1540\) \(1400\)
  2. \(\displaystyle R = 2100 - 28t\)
  3. \((75, 0)\text{,}\) \((0, 2100)\)
    grid
  4. \(t\)-intercept: The oil reserves will be gone in 2080; \(R\)-intercept: There were \(2100\) billion barrels of oil reserves in 2005.

3.

Alida plans to spend part of her vacation in Atlantic City and part in Saint-Tropez. She estimates that after airfare her vacation will cost $60 per day in Atlantic City and $100 per day in Saint-Tropez. She has $1200 to spend after airfare.
  1. Write an equation that relates the number of days, \(C\text{,}\) Alida can spend in Atlantic City and the number of days, \(T\text{,}\) in Saint-Tropez.
  2. Find the intercepts and graph the equation by hand.
    grid
  3. If Alida spends 10 days in Atlantic City, how long can she spend in Saint-Tropez?
  4. What do the intercepts tell us about Alida’s vacation?
Answer.
  1. \(\displaystyle 60C+100T=1200\)
  2. \((20,0) \text{,}\) \((0,12) \)
    grid
  3. 6 days
  4. She can spend 20 days in Atlantic City if she spends no time in Saint-Tropez, or 12 days in Saint-Tropez if she spends no time in Atlantic City.

Exercise Group.

For Problems 4–9, graph the equation on graph paper. Use the most convenient method for each problem.
4.
\(\dfrac{x}{6}-\dfrac{y}{12}=1 \)
Answer.
line with x-intercept 5, y-intercept -12
5.
\(50x=40y-20,000 \)
Answer.
line with x-intercept -400, y-intercept 500
6.
\(1.4x+2.1y=8.4 \)
Answer.
line with x-intercept 6, y-intercept 4
7.
\(3x-4y=0 \)
Answer.
line through origin with slope 3/4
8.
\(3x-4y=0 \)
Answer.
line through origin with slope -1/4
9.
\(4x=-12 \)
Answer.
verticalline x= -3

10.

The table shows the amount of oil, \(B\) (in thousands of barrels), left in a tanker \(t\) minutes after it hits an iceberg and springs a leak.
\(t\) \(0\) \(10\) \(20\) \(30\)
\(B\) \(800\) \(750\) \(700\) \(650\)
  1. Write a linear function for \(B\) in terms of \(t\text{.}\)
  2. Choose appropriate window settings on your calculator and graph your function.
  3. Give the slope of the graph, including units, and explain the meaning of the slope in terms of the oil leak.
Answer.
  1. \(\displaystyle B = 800 - 5t\)
  2. GC graph
  3. \(m = -5\) thousand barrels/minute: The amount of oil in the tanker is decreasing by 5000 barrels per minute.

11.

An interior decorator bases her fee on the cost of a remodeling job. The accompanying table shows her fee, \(F\text{,}\) for jobs of various costs, \(C\text{,}\) both given in dollars.
\(C\) \(5000\) \(10,000\) \(20,000\) \(50,000\)
\(F\) \(1000\) \(1500\) \(2500\) \(5500\)
  1. Write a linear function for \(F\) in terms of \(C\text{.}\)
  2. Choose appropriate window settings on your calculator and graph your function.
  3. Give the slope of the graph, including units, and explain the meaning of the slope in terms of the the decorator’s fee.
Answer.
  1. \(\displaystyle F = 500 + 0.10C\)
  2. GC graph
  3. \(m = 0.10\text{:}\) The fee increases by $0.10 for each dollar increase in the remodeling job.

Exercise Group.

For Problems 12–15, find the slope of the line segment joining the points.
12.
\((-1,4), ~(3,-2) \)
Answer.
\(\dfrac{-3}{2} \)
13.
\((5,0), ~(2,-6) \)
Answer.
\(2 \)
14.
\((6.2,1.4), ~(-2.3,4,8) \)
Answer.
\(-0.4 \)
15.
\((0,-6.4), ~(-5.6,3.4) \)
Answer.
\(-1.75\)

Exercise Group.

For Problems 16–17, which tables could describe variables related by a linear equation?
16.
  1. \(r\) \(E\)
    \(1\) \(5\)
    \(2\) \(\dfrac{5}{2} \)
    \(3\) \(\dfrac{5}{3} \)
    \(4\) \(\dfrac{5}{4} \)
    \(5\) \(1\)
  2. \(s\) \(t\)
    \(10\) \(6.2\)
    \(20\) \(9.7 \)
    \(30\) \(12.6 \)
    \(40\) \(15.8 \)
    \(50\) \(19.0\)
Answer.
neither
17.
  1. \(w\) \(A\)
    \(2\) \(-13\)
    \(4\) \(-23 \)
    \(6\) \(-33 \)
    \(8\) \(-43 \)
    \(10\) \(-53\)
  2. \(x\) \(G\)
    \(0\) \(0\)
    \(2\) \(5 \)
    \(4\) \(10 \)
    \(8\) \(20 \)
    \(16\) \(40\)
Answer.
both

Exercise Group.

For Problems 18–19, the table gives values for a linear equation in two variables. Fill in the missing values.
18.
\(d\) \(V\)
\(-5\) \(-4.8\)
\(-2\) \(-3 \)
\(\) \(-1.2\)
\(6\) \(1.8 \)
\(10\) \(\)
Answer.
\(d\) \(V\)
\(-5\) \(-4.8\)
\(-2\) \(-3 \)
\(1\) \(-1.2\)
\(6\) \(1.8 \)
\(10\) \(4.2\)
19.
\(q\) \(S\)
\(-8\) \(-8\)
\(-4\) \(36 \)
\(3\) \(\)
\(\) \(200 \)
\(9\) \(264\)
Answer.
\(q\) \(S\)
\(-8\) \(-8\)
\(-4\) \(36 \)
\(3\) \(168\)
\(5\) \(200 \)
\(9\) \(264\)

20.

The planners at AquaWorld want the small water slide to have a slope of 25%. If the slide is 20 feet tall, how far should the end of the slide be from the base of the ladder?
Answer.
80 ft

Exercise Group.

For Problems 21–24, find the slope and \(y\)-intercept of the line.
21.
\(2x-4y=5\)
Answer.
\(m=\dfrac{1}{2} \text{,}\) \(b= \dfrac{-5}{4} \)
22.
\(\dfrac{1}{2} x+\dfrac{2}{3} y=\dfrac{5}{6} \)
Answer.
\(m=\dfrac{3}{4} \text{,}\) \(b= \dfrac{5}{4} \)
23.
\(8.4x + 2.1y = 6.3\)
Answer.
\(m=-4 \text{,}\) \(b= 3 \)
24.
\(y - 3 = 0 \)
Answer.
\(m=0 \text{,}\) \(b= 3 \)

Exercise Group.

For Problems 25 and 26,
  1. Graph by hand the line that passes through the given point with the given slope.
  2. Find the equation of the line.
25.
\((-4,6)\text{;}\) \(~ m=\dfrac{-2}{3} \)
Answer.
  1. line through (-4,6), slope -2/3
  2. \(\displaystyle y=\dfrac{-2}{3}x +\dfrac{10}{3} \)
26.
\((2,-5)\text{;}\) \(~ m=\dfrac{3}{2} \)
Answer.
  1. line through (2,-5), slope 3/2
  2. \(\displaystyle y=\dfrac{3}{2}x -8 \)

27.

The rate at which air temperature decreases with altitude is called the lapse rate. In the troposphere, the layer of atmosphere that extends from the Earth’s surface to a height of about 7 miles, the lapse rate is about \(3.6\degree\text{F}\) for every 1000 feet. (Source: Ahrens, 1998)
  1. If the temperature on the ground is \(62\degree\text{F}\text{,}\) write an equation for the temperature, \(T\text{,}\) at an altitude of \(h\) feet.
  2. What is the temperature outside an aircraft flying at an altitude of 30,000 feet? How much colder is that than the ground temperature?
  3. What is the temperature at the top of the troposphere?
Answer.
  1. \(\displaystyle T = 62 - 0.0036h\)
  2. \(-46\degree\text{F}\text{;}\) \(108\degree\text{F}\)
  3. \(\displaystyle -71\degree\text{F}\)

Exercise Group.

For Problems 28 and 29, find an equation for the line passing through the two given points.
28.
\((3,-5) \text{,}\) \(~(-2,4) \)
Answer.
\(y=\dfrac{-9}{5}x+\dfrac{2}{5} \)
29.
\((0,8) \text{,}\) \(~(4,-2) \)
Answer.
\(y=\dfrac{-5}{2}x+8 \)

30.

The population of Maple Rapids was 4800 in 2005 and had grown to 6780 by 2020. Assume that the population increases at a constant rate.
  1. Make a table of values showing two data points.
  2. Find a linear equation that expresses the population, \(P\text{,}\) of Maple Rapids in terms of the number of years, \(t\text{,}\) since 2005.
  3. State the slope of the line, including units. What does it tell us about the population?
Answer.
  1. \(t\) \(0\) \(15\)
    \(P\) \(4800\) \(6780\)
  2. \(\displaystyle P = 4800 + 132t\)
  3. \(m = 132\) people/year: the population grew at a rate of \(132\) people per year.

Exercise Group.

For Problems 31 and 32,
  1. Find the slope and \(y\)-intercept of the line.
  2. Write an equation for the line.
31.
line
Answer.
  1. \(m=-2\text{;}\) \(b=3\)
  2. \(\displaystyle y=-2x+3\)
32.
line
Answer.
  1. \(m=\dfrac{3}{2} \text{;}\) \(b=-5\)
  2. \(\displaystyle y=\dfrac{3}{2}x-5\)

33.

What is the slope of the line whose intercepts are \((-5, 0)\) and \((0, 3)\text{?}\)
Answer.
\(\dfrac{3}{5} \)

34.

  1. Find the \(x\)- and \(y\)-intercepts of the line \(\dfrac{x}{4}-\dfrac{y}{6}=1 \text{.}\)
  2. What is the slope of the line in part (a)?
Answer.
  1. \((4,0) \) and \((0,-6) \)
  2. \(\displaystyle \dfrac{3}{2} \)

35.

  1. Find the \(x\)- and \(y\)-intercepts of the line \(y= 2+ \dfrac{3}{2}(x-4) \text{.}\)
  2. Find the point on the line whose \(x\)-coordinate is 4. Can there be more than one such point?
Answer.
  1. \(\left(\dfrac{8}{3},0 \right) \) and \((0,-4) \)
  2. \((4,3) \text{;}\) No

36.

Find an equation in slope-intercept form for the line of slope \(\dfrac{6}{5} \) that passes through \((-3, -4)\text{.}\)
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