Answer: In 2328, answer the questions about the given function (a) Is the point on the

The intercepts of the equation are

True or False The point is on the graph of the equation x = 2y - 2

A set of points in the xy-plane is the graph of a function if and only if every line intersects the graph in at most one point.

If the point is a point on the graph of then

Find a so that the point is on the graph of f1x2 = ax2 + 4

True or False A function can have more than one y-intercept

True or False The graph of a function always crosses the y-axis.

True or False The y-intercept of the graph of the function y = f1x2, whose domain is all real numbers, is f102.

Use the given graph of the function to answer parts (a)(n) (a) Find and (b) Find and (c) Is positive or negative? (d) Is positive or negative? (e) For what values of x is (f) For what values of x is (g) What is the domain of (h) What is the range of (i) What are the x-intercepts? (j) What is the y-intercept? (k) How often does the line intersect the graph? (l) How often does the line intersect the graph? (m) For what values of x does (n) For what values of x does f1x2 = -2?

Use the given graph of the function to answer parts (a)(n). (a) Find and (b) Find and (c) Is positive or negative? (d) Is positive or negative? (e) For what values of x is (f) For what values of x is (g) What is the domain of (h) What is the range of (i) What are the x-intercepts? (j) What is the y-intercept? (k) How often does the line intersect the graph? (l) How often does the line intersect the graph? (m) For what value of x does (n) For what value of x does f1x2 = -2?

In Problems 1122, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin

In Problems 1122, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin

In Problems 1122, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin

In Problems 1122, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin

In Problems 1122, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin

In Problems 1122, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin

In Problems 1122, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin

In Problems 1122, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin

In Problems 1122, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin

In Problems 1122, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin

In Problems 1122, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin

In Problems 1122, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin

In Problems 2328, answer the questions about the given function (a) Is the point on the graph of (b) If what is What point is on the graph of (c) If what is x? What point(s) are on the graph of (d) What is the domain of (e) List the x-intercepts, if any, of the graph of (f) List the y-intercept, if there is one, of the graph of

In Problems 2328, answer the questions about the given function (a) Is the point on the graph of (b) If what is What point is on the graph of (c) If what is x? What point(s) are on the graph of (d) What is the domain of (e) List the x-intercepts, if any, of the graph of (f) List the y-intercept, if there is one, of the graph of

In Problems 2328, answer the questions about the given function (a) Is the point on the graph of (b) If what is What point is on the graph of (c) If what is x? What point(s) are on the graph of (d) What is the domain of (e) List the x-intercepts, if any, of the graph of (f) List the y-intercept, if there is one, of the graph of

In Problems 2328, answer the questions about the given function (a) Is the point on the graph of a1, f? 3 5 b f1x2 = x2 + 2 x + 4 f. f. f? f? f1x2 = 2, x = 4, f1x2? f? 13, 142 f? f1x2 = x + 2 x - 6 f. f. f? f? f1x2 = -2, x = -2, f1x2? f? 1-1, 22 f? f1x2 = -3x2 + 5x f. f. f? f? f1x2 = -1, x = -2, f1x2? f? 1-1, 22 f? f1x2 = 2x2 - x - 1 (b) If what is What point is on the graph of (c) If what is x? What point(s) are on the graph of (d) What is the domain of (e) List the x-intercepts, if any, of the graph of (f) List the y-intercept, if there is one, of the graph of

In Problems 2328, answer the questions about the given function (a) Is the point on the graph of (b) If what is What point is on the graph of (c) If what is x? What point(s) are on the graph of (d) What is the domain of (e) List the x-intercepts, if any, of the graph of (f) List the y-intercept, if there is one, of the graph of

In Problems 2328, answer the questions about the given function (a) Is the point on the graph of (b) If what is What point is on the graph of (c) If what is x? What point(s) are on the graph of (d) What is the domain of (e) List the x-intercepts, if any, of the graph of (f) List the y-intercept, if there is one, of the graph of f.

Free-throw Shots According to physicist Peter Brancazio, the key to a successful foul shot in basketball lies in the arc of the shot. Brancazio determined the optimal angle of the arc from the free-throw line to be 45 degrees. The arc also depends on the velocity with which the ball is shot. If a player shoots a foul shot, releasing the ball at a 45-degree angle from a position 6 feet above the floor, then the path of the ball can be modeled by the function where h is the height of the ball above the floor, x is the forward distance of the ball in front of the foul line, and is the initial velocity with which the ball is shot in feet per second. Suppose a player shoots a ball with an initial velocity of 28 feet per second. (a) Determine the height of the ball after it has traveled 8 feet in front of the foul line. (b) Determine the height of the ball after it has traveled 12 feet in front of the foul line. (c) Find additional points and graph the path of the basketball. (d) The center of the hoop is 10 feet above the floor and 15 feet in front of the foul line. Will the ball go through the hoop? Why or why not? If not, with what initial velocity must the ball be shot in order for the ball to go through the hoop? Source: The Physics of Foul Shots, Discover, Vol. 21, No. 10, October 2000

Granny Shots The last player in the NBA to use an underhand foul shot (a granny shot) was Hall of Fame forward Rick Barry who retired in 1980. Barry believes that current NBA players could increase their free-throw percentage if they were to use an underhand shot. Since underhand shots are released from a lower position, the angle of the shot must be increased. If a player shoots an underhand foul shot, releasing the ball at a 70-degree angle from a position 3.5 feet above the floor, then the path of the ball can be modeled by the function , where h is the height of the ball above the floor, x is the forward distance of the ball in front of the foul line, and is the initial velocity with which the ball is shot in feet per second. (a) The center of the hoop is 10 feet above the floor and 15 feet in front of the foul line. Determine the initial velocity with which the ball must be shot in order for the ball to go through the hoop. (b) Write the function for the path of the ball using the velocity found in part (a). (c) Determine the height of the ball after it has traveled 9 feet in front of the foul line. (d) Find additional points and graph the path of the basketball. Source:The Physics of Foul Shots, Discover,Vol. 21, No. 10, October 2000

Motion of a Golf Ball A golf ball is hit with an initial velocity of 130 feet per second at an inclination of 45 to the horizontal. In physics, it is established that the height h v h1x2 = - 136x2 v2 + 2.7x + 3.5 v h1x2 = - x + 6 44x2 v2 + of the golf ball is given by the function where x is the horizontal distance that the golf ball has traveled. h1x2 = -32x2 1302 + x Applications and Extensions (a) Determine the height of the golf ball after it has traveled 100 feet. (b) What is the height after it has traveled 300 feet? (c) What is the height after it has traveled 500 feet? (d) How far was the golf ball hit? (e) Use a graphing utility to graph the function (f) Use a graphing utility to determine the distance that the ball has traveled when the height of the ball is 90 feet. (g) Create a TABLE with and To the nearest 25 feet, how far does the ball travel before it reaches a maximum height? What is the maximum height? (h) Adjust the value of until you determine the distance, to within 1 foot, that the ball travels before it reaches a maximum height.

Cross-sectional Area The cross-sectional area of a beam cut from a log with radius 1 foot is given by the function where x represents the length, in feet, of half the base of the beam. See the figure. A1x2 = 4x21 - x2 , Tbl TblStart = 0 Tbl = 25. h = h1x2. A(x ) 4x1  x 2 x 1 (a) Find the domain of A. (b) Use a graphing utility to graph the function (c) Create a TABLE with and for Which value of x maximizes the crosssectional area? What should be the length of the base of the beam to maximize the cross-sectional area

Cost of Trans-Atlantic Travel A Boeing 747 crosses the Atlantic Ocean (3000 miles) with an airspeed of 500 miles per hour. The cost C (in dollars) per passenger is given by where x is the ground speed (a) Use a graphing utility to graph the function (b) Create a TABLE with and (c) To the nearest 50 miles per hour, what ground speed minimizes the cost per passenger?

Effect of Elevation on Weight If an object weighs m pounds at sea level, then its weight W (in pounds) at a height of h miles above sea level is given approximately by (a) If Amy weighs 120 pounds at sea level, how much will she weigh on Pikes Peak, which is 14,110 feet above sea level? (b) Use a graphing utility to graph the function Use m = 120 pounds. W = W1h2. W1h2 = ma 4000 4000 + h b 2 TblStart = 0 Tbl = 50. C = C1x2. 1airspeed ; wind2. C1x2 = 100 + x 10 + 36,000 x (c) Create a Table with and to see how the weight W varies as h changes from 0 to 5 miles. (d) At what height will Amy weigh 119.95 pounds? (e) Does your answer to part (d) seem reasonable? Explain

The graph of two functions, f and g, is illustrated. Use the graph to answer parts (a)(f). TblStart = 0 Tbl = 0.5 x yf(x) y 2 4 (6, 1) (4, 1) (2, 1) (2, 2) (6, 0) (4, 3) (3, 2) (5, 2) 2 4 2 yg(x) (a) (b) (c) (d) (e) (f) a f g 1f b142 # g2122 1f - g2162 1g - f2162 1f + g2122 1f + g2142

Describe how you would proceed to find the domain and range of a function if you were given its graph. How would your strategy change if you were given the equation defining the function instead of its graph?

How many x-intercepts can the graph of a function have? How many y-intercepts can the graph of a function have?

Is a graph that consists of a single point the graph of a function? Can you write the equation of such a function?

Match each of the following functions with the graph that best describes the situation. (a) The cost of building a house as a function of its square footage (b) The height of an egg dropped from a 300-foot building as a function of time (c) The height of a human as a function of time (d) The demand for Big Macs as a function of price (e) The height of a child on a swing as a function of time

Match each of the following functions with the graph that best describes the situation. (a) The temperature of a bowl of soup as a function of time (b) The number of hours of daylight per day over a 2-year period (c) The population of Florida as a function of time (d) The distance travelled by a car going at a constant velocity as a function of time (e) The height of a golf ball hit with a 7-iron as a function of time

Consider the following scenario: Barbara decides to take a walk. She leaves home, walks 2 blocks in 5 minutes at a constant speed, and realizes that she forgot to lock the door. So Barbara runs home in 1 minute. While at her doorstep, it takes her 1 minute to find her keys and lock the door.Barbara walks 5 blocks in 15 minutes and then decides to jog home. It takes her 7 minutes to get home. Draw a graph of Barbaras distance from home (in blocks) as a function of time

Consider the following scenario: Jayne enjoys riding her bicycle through the woods. At the forest preserve, she gets on her bicycle and rides up a 2000-foot incline in 10 minutes. She then travels down the incline in 3 minutes. The next 5000 feet is level terrain and she covers the distance in 20 minutes. She rests for 15 minutes. Jayne then travels 10,000 feet in 30 minutes. Draw a graph of Jaynes distance traveled (in feet) as a function of time.

The following sketch represents the distance d (in miles) that Kevin was from home as a function of time t (in hours). Answer the questions based on the graph. In parts (a)(g), how many hours elapsed and how far was Kevin from home during this time? (c) From to (d) From to (e) From to (f) From to (g) From to (h) What is the farthest distance that Kevin was from home? (i) How many times did Kevin return home?

The following sketch represents the speed (in miles per hour) of Michaels car as a function of time t (in minutes). v t = 4.2 t = 5.3 t = 3.9 t = 4.2 t = 3 t = 3.9 t = 2.8 t = 3 t = 2.5 t = 2.8 d(t) (2, 3) (2.5, 3) (2.8, 0) (3, 0) (4.2, 2.8) (5.3, 0) (3.9, 2.8) t v (t) (2, 30) (4, 30) (4.2, 0) (6, 0) (9.1, 0) (a) Over what interval of time was Michael traveling fastest? (b) Over what interval(s) of time was Michaels speed zero? (c) What was Michaels speed between 0 and 2 minutes? (d) What was Michaels speed between 4.2 and 6 minutes? (e) What was Michaels speed between 7 and 7.4 minutes? (f) When was Michaels speed constant?

Draw the graph of a function whose domain is and whose range is What point(s) in the rectangle cannot be on the graph? Compare your graph with those of other students. What differences do you see?

Is there a function whose graph is symmetric with respect to the x-axis? Explain.