Chapter 1-1 - Functions: Algebraically, Numerically, Graphically, and Verballypp. 5-7, Questions 1-5 (Odd):1. Archery Problem 1: An archer climbs a tree near the edge of a cliff, then shoots an arrow high into the air. The arrow goes up, then comes back down, going over the cliff and landing in the valley, 30 m below the top of the cliff. The arrow's height, y in meters above the top of the cliff depends on the time, x in seconds, since the archer released it. Figure 1-1 g shows the height as a function of time.
a. What was the approximate height of the arrow at 1 second? At 5 seconds?
how do you explain the fact that the height is negative at 5 seconds?
b. At what two times was the arrow at 10 m above the ground? At what time does the arrow land in the valley below the cliff?
c. How high was the archer above the ground at the top of the cliff when she released the arrow?
d. Why can you say that altitude is a function of time? Why is time not a function of altitude?
e. What is the domain of the function? What is the corresponding range?
2. Gas Temperature and Volume Problem: When you heat a fixed amount of gas, it expands increasing its volume. In the late 1700s French chemist Jacques Charles used numerical measurements of the temperature and volume of a gas to find a quantitative relationship between these two variables. Suppose that these temperatures and volumes had been recorded for a fixed amount of oxygen.
a. On graph paper, plot V a s a function of T. (See chart of C degree to V(L). Choose scales that go at least from T = - 300 to T = 400.
b. You should find, as Charles did, that the points lie in a straight line! Extend the line backward until it crosses the T-axis. The temperature you get is called absolute zero: the temperature at which supposedly all molecular motion stops. Based on your graph, what temperature in degrees Celsius is absolute zero? Is this the number you recall from science courses?
c. Extending a graph beyond all given data, as you did in 2b is called EXTRAPOLATION. Extra means beyond and pol- comes from pole or end. Extrapolate the graph to T = 400 and predict what the volume would be at 400 degrees C.
3. Mortgage Payment Problem: People who buy houses usually get a loan to pay for most of the house and pay on the resulting mortgage each month. Suppose you get a $50,000 loan and pay it back at $550. 34 per month. with an interest rate of 12% per year (1% per month). Your balance, B dollars, after n monthly payments is given by the algebraic equation
B = 50,000 (1.01 n) + 550.34/0.01 (1-1.01N)
a. Make a table of your balances at the end of each 12 months for the first 10 years of the mortgage. To save time, use the table feature of your grapher to do this.
b. How many months will it take you to pay off the entire mortgage? Show how you get your answer.
c. Plot on your grapher the graph of B as afunctio nof n from n = 0 until the mortgage is paid off. Sketch the graph on your paper.
d. True or false: After half the payments hav ebeen made, half the original balance remains to be paid. Show that your conclusion agrees with your graph from part c.
e. Give the domain and range of this function. Explain why the domain contains only integers.
5. Stove Heating Element Problem: When you turn on the heating element of an electric stove, the temperature increases rapidly at first, then levels off. Sketch a reasonable graph showing temperature as a function of time. Show the horizontal asymptote. Indicate on the graph the domain and range.
pp. 13-15 Questions 1-41 (Odd)Chapter 1-3 through 1-7 Chapter 1 Review Questions pp. 44-48 R1-C2R1. Punctured Tire Problem: For parts a-d, suppose that your car runs over a nail.
the tire's air pressure, y, measured in psi (pounds per square inch) , decreases with time, x, measured in minutes, as the air leaks out. A graph of pressure versus time is shown in Figure 1-8a. (see Graph)
a. Find graphically the pressure after 2 minutes . Approximately how many minutes can you drive before the pressure reaches 5 psi?
b. The algebraic equation for the funcitno in Figure 1-8 a is y = 3.5 x 0.7 x.
Make a table of numerical values of pressure for times of 0, 1, 2, 3 and 4 minutes.
c. Suppose the equation in part b gives reasonable answers until the pressure drops to 5 psi.
At that pressure, the tire comes loose from the rim and the pressure drops to 0. What is the domain of the function described by this equation? What is the corresponding range?
d. The graph in Figure 1-8 a gets closer and closer to the x-axis but never quite touches.
What special name is givne to the x-axis in this case?
e. Earththquake Problem 1: Earthquakes happen when rock plates slide past each other. The stresss pbetween paltes taht build up over a numbe rof year sis relived by the quake in a few seconds. Then the stress starts building up again. Sketch a reasonable graph showing stress as a function of time.
(See Kobe picture)
R2. For parts a-e, name the kind of function for each equation given.
a. f(x) = 3x = 7
b. f(x) = x3 + 7x2 - 12 x + 5
c. f(x) = 1.3x
d. F(x) = x1.3
e. f(x) rational function : x/-/x2 - 2x + 3
f. Name a pair fo real world variables that vcoudl be related by the function in part a.
g. if the domain of the function in part a is 2
< x
<> what is the range?
h. In a flu epidemic, the number of people infected depends on time. Sketch a reasonable graph of the number of people infected as a function of time. What kind of function has a graph that most closely resembles the one you drew?
i. For Figures 1-8 b through 1-8d, what kind of function has the graph shown?
1.8 b
1.8 c
1.8-d
j. Explain how you know that the relation in Figure 1-8e is a function, but the relation in Figure 1-8f is not a function.
1-8e 1-8f
R3. a. For functions f and g in Figure 1-8g, identify the how the pre-image function f (dashed) was transformed to get the image function g (solid). Write an equation for g(x) in terms of x given that the equation of f is f(x) = square root of (4-x2)
Confirm the result by plotting the image and the pre-image on the same screen.
b. If g(x) = 3f(x-4), explain how function f was transformed to get function g. Using the pre-image in Figure 1-8h, sketch the graph of g on a copy of this figure.
R4. Height and Weight Problem: For parts a-e, the weight of a growing child depends of his or her height, and the height depends of his or or her age. Assume that the child is 20 inches when born and grows 3 inches per year.
a. Write an equation fo rh(t) (in inches) as a function of t (in years).
b. Assume that the weight functino W is givne by the power function W(h(t) = 0.0054 h(t)2.5. find h(5) and use the reusl to calcuate the predicted weight of the child at age 5.
c. Plot the graph of y = W(h(t)). Sketch the result.
d. Assuem the height of the child increases at a constant rate. Does the weight of the child also seem to increase at a constant rate? Explain how you arrived at your answer.
e. What is a reasonable domain for t for the function W * h?
Two Linear Functions Problem 2: For parts f-i, let functinons f and g be defined by
f(x) = x - 2 for 4
< x
< 8.
g(x) = 2x - 3 for 2
< x
< 6
f. Plot the graphs of f, g and f(g(x) on the same screen. Sketch the results.
g. Find f(g(4)).
h. Show that f(g(3) is undefined, even though g(3) is defined.
i. Calculate the domain of the composite functionn g * g and show that it agrees with the graph you plotted in part f.
R5. Figure 1-81 shows the graph of f(x) = x2 + 1 in the domain = -1
< x
< 2..
a. On a copy of the figure, sketch the graph of the inverse relation. Explain why the inverse is not a function.
b. Plot the graphs of f and its inverse relations in the same screen using parametric equations.
Also plot the line y = x. How are the graphs of f and its inverse relations related to the line y = x? Hwo are the domain and range of the inverse relation related to the domain and range of function f?
c. Write an equation for the inverse of y = x2 + 1 by interchanging the variables. Sovle the new equation for y in terms of x. How does this solution reveal that there are two different y-values
for some x values?
d. On a copy of Figure 1-8j, sketch the graph of the inverse relation. What property does the function graph have that allows you to conclude that the fucntion is invertible?
What are the vertical lines at x = -3 and at x = 3 called?
See figure 1-8j?
e. Spherical Balloon Problem: Recall that the volume V(x), measured in in3, of a sphere is given by V(x) = 4/3 pi x x3 where x is the radius of the sphere in inches.
Find V(5). Find x if V(x) = 100. Write an equation for V-1 (x). Explain why both the V function and the V-1 function are examples of power functions. Describe a kind of problem for which the V-1 equation would be more useful than the V equation.
f. Sketch the graph of a one-to-one function. Explain why it is invertible.
R6. On four copies of y = f(x) in Figure 1-8K, sketch the graphs of these four functions: y = -f(x), y = f(-x), y = f(x) and y = f(x). Figure 1-8k.
b. Function f in part a is defined piecewise:
...
Plot the two branches of this function as y1 and y2 on your grapher. Does the graph agree with Figure 1-8k? Plot y = f(x). by plotting y 3 = y1 f(x) abd y4 = y2f(x). Does the graph agree with your answer to the corresponding portion of part a?
c. Explain why functions with the property f(-x) = -f(x) are called odd functions and functions with the property f(-x) = f(x) are called even functions.
d. Plot the graph of f(x) = 0.2 x 2 - x-3/x-3
Use a friendly window that includes x = 3 as a grid point. Sketch the result. Name the feature that appears at x = 3.
R7. In Section 1-7, you started a precal journal. In what ways do you think that keeping this journal will help you? How coudl you use the completed journal at the end of the course?
What is your responsibiility throughout the year to ensure that writing the journal has been a worthwhile project?
CONCEPT Problems:
C1. Four Transformation Problem: Figure 1-8l shows a pre-image function f(dashed) and a transformed image function g (solid). Dilations and translations were performed in both directions to get the g graph.
Figure out what the transformations were. Write an equation for g(x) in terms of f. Let f(x) = x2 with domain -2 < x < 2. Plot the graph of g on your grapher. Does your grapher agree with the figure?
C2. Sine Function Problem: If you enter y1 = sin (x) on your grapher and plot the graph, the result resembles Figure 1-8m. (your grapher should be in radian mode). the function is called the sine function ....
a. The Sine function is an example of a periodic function. Why do you think this name is given to the sine function?
b. The period of a periodic function is the difference in x positions from one point on the graph to the point where the graph first starts repeating itself. Approximately what does the period of the sine function seem to be?
c. is the sine function an odd function, an even function, or neither? How can you tell?
d. On a copy of Figure 1-88m, skthc a vertical dilation of the sine function graph by a factor of 5. What is the equation for this dilated functions? Check your answer by plotting the sine graph and the transformed function and check your answer by plotting both functions on your grapher.
e. Figure 1-8 n shows a two-step transformation of the sine graph in Figure 1-8m. Name the two transformations. Write an equation for the transformed function and check your answer by plotting both functions on your grapher.
f. Let f(x) = sin x. What transformation would g(x) = sin (1/2 x) be? Confirm your answer by plotting both functions on your grapher.
