1–4 For the following lines, find the slopes between the two given
points by finding the change in output divided by the change in input.
What is the ratio of the output to the input at each of the points?
Which are proportional relations? Which are increasing and which are
decreasing? Sketch a graph.
1. y = 2 x + 3, using points with x = 1 and x = 3.
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2. z = -5 w , using points with w = 1 and w = 3.
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3. z = 5( w - 2) + 8, using points with w = 1 and w = 3.
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4.
y - 5 = -3( x + 2) - 6, using points with x = 1 and x = 3.
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5–6 Check that the point indicated lies on the line and find the
equation of the line in point-slope form using the given point. Multiply
out to check that the point-slope form matches the original equation.
5. The line f ( x ) = 2 x + 3 and the point (2, 7).
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6.
The line g( y ) = -2 y + 7 and the point (3, 1).
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7–12 Find equations in slope-intercept form for the following lines.
Sketch a graph indicating the original point from point-slope form.
7. The line f ( x ) = 2( x - 1) + 3.
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8. The line g ( z ) = -3( z + 1) - 3.
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9. A line passing through the point (1, 6) with slope -2.
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10. A line passing through the point (-1, 6) with slope 4.
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11. A line passing through the points (1, 6) and (4, 3).
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12. A line passing through the points (6, 1) and (3, 4).
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13–16 Check whether the following are linear functions.
13.
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14. F ( r ) = r 2 + 5.
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15. P ( q ) = 8(3 q + 2) - 6.
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16.
Q ( w ) = 8(3 q + 2) - 6( q + 4).
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17–18 Check that the following curves do not have constant slope by
computing the slopes between the points indicated. Compare with the
graphs in Exercises 5 and 6 in Section 1.2 .
17.
at z = 1, z = 2, and z = 4, as in Exercise 5 . Find the slope between z
= 1 and z = 2, and the slope between z = 2 and z = 4.
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18.
F ( r ) = r 2 + 5 at r = 0, r = 1, and r = 4, as in Exercise 6 . Find
the slope between r = 0 and r = 1, and the slope between r = 1 and r =
4.
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19–24 Solve the following equations. Check your answer by plugging in
the value you found.
19. 2 x + 3 = 7.
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20.
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21. 2 x + 3 = 3 x + 7.
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22. -3 y + 5 = 8 + 2 y .
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23. 2(5 ( x - 1) + 3 = 5(2( x - 2) + 5).
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24. 2(4( x - 1) + 3 = 5(2( x - 2) + 5).
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25-28 Solve the following equations for the given variable, treating the other letters as constant parameters.
25. Solve 2 x + b = 7 for x .
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26. Solve mx + 3 = 7 for x .
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27. Solve 2 x + b = mx + 7 for x . Are there any values of b or m for which this has no solution?
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28. Solve mx + b = 3 x + 7 for x . Are there any values of b or m for which this has no solution?
29-32 Most unit conversions are proportional relations. Find the slope and graph the relations between the following units.
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29.
Graph the length of a fish in inches on the horizontal axis and
centimeters on the vertical axis. Use the fact that 1 in. = 2.54 cm.
Mark the point corresponding to a length of 1 in.
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30.
Graph the length of a fish in centimeters on the horizontal axis and
inches on the vertical axis. Use the fact that 1 in. = 2.54 cm. Mark the
point corresponding to a length of 1 in.
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31.
Graph the mass of a fish in grams on the horizontal axis and its weight
in pounds on the vertical axis. Use the identity 1 lb = 453.6 g. Mark
the point corresponding to a weight of 1 lb.
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32.
Graph the weight of a fish in pounds on the horizontal axis and its
mass in grams on the vertical axis. Use the identity 1 lb = 453.6 g.
Mark the point corresponding to a weight of 1 lb.
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33–34 Not very many functions commute with each other. The following
problems ask you to find all linear functions that commute with the
given linear function.
33.
Find all functions of the form g ( x ) = mx + b that commute with the
function f ( x ) = x + 1. Can you explain your answer in words?
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34.
Find all functions of the form g ( x ) = mx + b that commute with the
function f ( x ) = 2 x . Can you explain your answer in words?
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35-38 Many fundamental relations express a proportional relation between
two measurements with different dimensions. Find the slopes and the
equations of the relations between the following quantities.
35. Volume = area × thickness. Find the volume V as a function of the area A if the thickness is 1.0 cm.
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36. Volume = area × thickness. Find the volume V as a function of thickness T if the area is 7.0 cm 2 .
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37.
Total mass = mass per bacterium × number of bacteria. Find total mass M
as a function of the number of bacteria b if the mass per bacterium is
5.0 × 10 -9 g.
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38.
Total mass = mass per bacterium × number of bacteria. Find total mass M
as a function of mass per bacterium m if the total number is 10 6 .
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39–42 A ski slope has a slope of -0.2. You start at an altitude of
10,000 ft.
39. Write the equation giving altitude a as a function of horizontal distance moved d .
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40. Write the equation of the line in meters.
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41. What will be your altitude when you have gone 2000 ft horizontally?
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42. The ski run ends at an altitude of 8000 ft. How far will you have gone horizontally?
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43–46 The following data give the elevation of the surface of the Great Salt Lake in Utah.
43. Graph these data.
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44. During which periods is the surface elevation changing linearly?
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45.
What was the slope between 1965 and 1975? What would the surface
elevation have been in 1990 if things had continued as they began? How
different is this from the actual depth?
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46.
What was the slope during between 1985 and 1995? What would the surface
elevation have been in 1965 if things had always followed this trend?
How different is this from the actual depth?
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47–50 Graph the following relations between measurements of a growing
plant, checking that the points lie on a line. Find the equations in
both point-slope and slope-intercept form.
47. Mass as a function of age. Find the mass on day 1.75.
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48. Volume as a function of age. Find the volume on day 2.75.
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49. Glucose production as a function of mass. Estimate glucose production when the mass reaches 20.0 g.
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50.
Volume as a function of mass. Estimate the volume when the mass reaches
30.0 g. How will the density at that time compare with the density when
a = 0.5?
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51–44 Consider the data in the following table (adapted from Parasitoids
by H. C. F. Godfray), describing the number of wasps that can develop
inside caterpillars of different weights.
51. Graph these data. Which point does not lie on the line?
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52. Find the equation of the line connecting the first two points.
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53. How many wasps does the function predict would develop in a caterpillar weighing 0.72 g?
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54.
How many wasps does the function predict would develop in a caterpillar
weighing 0.0 g? Does this make sense? How many would you really expect?
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55–58 The world record times for various races are decreasing at roughly
linear rates (adapted from Guinness Book of Records , 1990).
55.
The men’s Olympic record for the 1500 meters was 3:36.8 in 1972 and
3:35.9 in 1988. Find and graph the line connecting these. (Don’t forget
to convert everything into seconds.)
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56.
The women’s Olympic record for the 1500 meters was 4:01.4 in 1972 and
3:53.9 in 1988. Find and graph the line connecting these.
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57. If things continue at this rate, when will women finish the race in exactly no time? What might happen before that date?
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58. If things continue at this rate, when will women be running this race faster than men?
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59.
Try Exercise 58 on the computer. Compute the year when the times will
reach 0. Give your best guess of the times in the year 1900.
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60.
Graph the ratio of temperature measured in Fahrenheit to temperature
measured in Celsius for -273 = °C < 200. What happens near °C = 0?
What happens for large and small values of °C? How would the results
differ if the zero for Fahrenheit were changed to match that of Celsius?
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