In Exercises 1–8, state whether the given statement is true or false.
(0, 5) is the midpoint of the line segment joining and (3, 11).
The equation is the equation of a circle with center (2, 3) and radius 5.
If a graph is symmetric with respect to the x-axis and the y-axis, then it must be symmetric with respect to the origin.
If a graph is symmetric with respect to the origin, then it must be symmetric with respect to the x-axis and the y-axis.
In the graph of the equation of the line the slope is 4 and the y-intercept is 9.
If the slope of a line is 2, then the slope of any line perpendicular to it is
The slope of a vertical line is undefined.
The equation is the equation of a circle with center and radius 5.
In Exercises 9–14, find
the distance between the point P and Q.
the coordinates of the midpoint of the line segment PQ.
the slope of the line containing the points P and Q.
Show that the points and C(3, 0) are the vertices of a right triangle.
Show that the points and D(10, 5) are the vertices of a rhombus.
Which of the points and (4, 5) is closer to the origin?
Which of the points and (5, 10) is closer to the point
Find a point on the x-axis that is equidistant from the points and (4, 7).
Find a point on the y-axis that is equidistant from the points and
In Exercises 21–24, specify whether the given graph has axis or origin symmetry (or neither).
In Exercises 25–34, sketch the graph of the given equation. List all intercepts and describe any symmetry of the graph.
In Exercises 35–37, find the standard form of the equation of the circle that satisfies the given conditions.
Center radius 5
Diameter with endpoints (5, 2) and
Center touching the y-axis
In Exercises 38–42, describe and sketch the graph of the given equation.
In Exercises 43–47, find the slope–intercept form of the equation of the line that satisfies the given conditions.
Passing through (1, 2) with slope
x-intercept 2, y-intercept 5
Passing through (1, 3) and
Passing through (1, 3), perpendicular to the x-axis
Determine whether the lines in each pair are parallel, perpendicular, or neither.
and
and
and
and
Write the point–slope form of the equation of the line shown in the figure that is
Parallel to the line in the figure with x-intercept 4.
Perpendicular to the line in the figure with y-intercept 1.
In Exercises 49–58, graph each relation. State the domain and range of each. Determine which relation determines y as a function of x.
In Exercises 59–76, let and Find each of the following.
x if
x if
In Exercises 77–82, graph each function and state its domain and range. Determine the intervals over which the function is increasing, decreasing, or constant.
In Exercises 83–86, use transformations to graph each pair of functions on the same coordinate axes.
In Exercises 87–92, state whether each function is odd, even, or neither. Discuss the symmetry of each graph.
In Exercises 93–96, express each function as a composition of two functions.
In Exercises 97–100, determine whether the given function is one-to-one. If the function is one-to-one, find its inverse and sketch the graph of the function and its inverse on the same coordinate axes.
In Exercises 101 and 102, assume that f is a one-to-one function. Find and find the domain and range of f.
For the graph of a function f shown in the figure,
Write a formula for f as a piecewise function.
Find the domain and range of f.
Find the intercepts of the graph of f.
Draw the graph of
Draw the graph of
Draw the graph of
Draw the graph of
Draw the graph of
Draw the graph of
Draw the graph of
Explain why f is one-to-one.
Draw the graph of
Scuba diving. The pressure P on the body of a scuba diver increases linearly as she descends to greater depths d in seawater. The pressure at a depth of 10 feet is 19.2 pounds per square inch, and the pressure at a depth of 25 feet is 25.95 pounds per square inch.
Write the equation relating P and d (with d as the independent variable) in slope–intercept form.
What is the meaning of the slope and the y-intercept in part (a)?
Determine the pressure on a scuba diver who is at a depth of 160 feet.
Find the depth at which the pressure on the body of the scuba diver is 104.7 pounds per square inch.
Waste disposal. The Sioux Falls, South Dakota, council considered the following data regarding the cost of disposing of waste material.
Year | 1996 | 2000 |
---|---|---|
Waste (in pounds) | 87,000 | 223,000 |
Cost | $54,000 | $173,000 |
Assume that the cost C (in dollars) is linearly related to the waste w (in pounds).
Write the linear equation relating C and w in slope–intercept form.
Explain the meaning of the slope and the intercepts of the equation in part (a).
Suppose the city has projected 609,000 pounds of waste for the year 2008. Calculate the projected cost of disposing of the waste for 2008.
Suppose the city’s projected budget for waste collection in 2012 is $1 million. How many pounds of waste can be handled?
Checking your speedometer. A common way to check the accuracy of your speedometer is to drive one or more measured (not odometer) miles, holding the speedometer at a constant 60 miles per hour and checking your watch at the start and end of the measured distance.
How does your watch show you whether the speedometer is accurate? What mathematics is involved?
Why shouldn’t you use your odometer to measure the number of miles?
Playing blackjack. Chloe, a bright mathematician, read a book on counting cards in blackjack and went to Las Vegas to try her luck. She started with $100 and discovered that the amount of money she had at time t hours after the start of the game could be expressed by the function
What amount of money had Chloe won or lost in the first two hours?
What was the average rate at which Chloe was winning or losing money during the first two hours?
Did she lose all of her money? If so, when?
If she played until she lost all of her money, what was the average rate at which she was losing her money?
Volume discounting. Major cola distributors sell a case (containing 24 cans) of cola to a retailer at a price of $4. They offer a discount of 20% for purchases over 100 cases and a discount of 25% for purchases over 500 cases. Write a piecewise function that describes this pricing scheme. Use x as the number of cases purchased and f(x) as the price paid.
Air pollution. The daily level L of carbon monoxide in a city is a function of the number of automobiles in the city. Suppose where x is the number of automobiles (in hundred thousands). Suppose further that the number of automobiles in a given city is growing according to the formula where t is time in years measured from now.
Form a composite function describing the daily pollution level as a function of time.
What pollution level is expected in five years?
Toy manufacturing. After being in business for t years, a toy manufacturer is making GI Jimmy toys per year. The sale price p in dollars per toy has risen according to the formula Write a formula for the manufacturer’s yearly revenue as
A function of time t.
A function of price p.
Height and weight of NBA players The table shows the roster for the National Basketball Association (NBA) team Boston Celtics for 2012.
Name | Height (in inches) | Weight (in pounds) |
---|---|---|
Leandro Barbosa | 75 | 194 |
Brandon Bass | 80 | 250 |
Avery Bradley | 74 | 180 |
Jason Collins | 84 | 255 |
Kevin Garnett | 83 | 253 |
Jeff Green | 81 | 235 |
Courtney Lee | 77 | 200 |
Fab Melo | 84 | 255 |
Paul Pierce | 79 | 235 |
Rajon Rondo | 73 | 186 |
Jared Sullinger | 81 | 260 |
Jason Terry | 74 | 180 |
Chris Wilcox | 82 | 235 |
Use a graphing utility to find the line of regression relating the variables x (height in inches) and y (weight in pounds).
Use a graphing utility to plot the points and graph the regression line.
Use the line in part (a) to predict the weight of an NBA player whose height is 76 inches.