ConcepTest Section 1.6 Question 1 Graph y = x 2, y = x 3, y = x 4, y = x 5. List at least 3 observations. (Closed Book)

ConcepTest Section 1.6 Answer 1 ANSWER (a) y = x 2 and y = x 4 have the same general shape – that of a “U”. (b) y = x 3 and y = x 5 have the same general shape – that of a “seat”. (c) For x > 1, as the power increases the function grows faster. (d) When 0 x 3 > x 4 > x 5. (e) For x > 0, all functions are increasing and concave up. (f) All functions intersect at (1, 1) and (0, 0). (g) For x < 0, the functions y = x 2 and y = x 4 are decreasing and concave up. (h) For x < 0, the functions y = x 3 and y = x 5 are increasing and concave down. COMMENT: This question could also be used as an exploratory activity.

ConcepTest Section 1.6 Question 2 Graph y = x -1, y = x -2, y = x 1-3, y = x -4. List at least 3 observations. (Closed Book)

ConcepTest Section 1.6 Answer 2 ANSWER (a) y = x -2 and y = x -4 have the same general shape and they are always positive. (b) y = x -1 and y = x -3 have the same general shape. (c) As the power decreases, the function approaches 0 faster as x increases. (d) For x > 0, they are all concave up. (e) They intersect at (1, 1). (f) Each has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. (g) For x < 0, the functions y = x -2 and y = x -4 are increasing and concave up. (h) For x < 0, the functions y = x -1 and y = x -3 are decreasing and concave down. COMMENT: This question could also be used as an exploratory activity.

ConcepTest Section 1.6 Question 3 Graph y = x 1/2, y = x 1/3, y = x 1/4, y = x 1/5. What do you observe about the growth of these functions? (Closed Book)

ConcepTest Section 1.6 Answer 3 ANSWER The smaller the power of the exponent, the slower the function grows for x > 1. COMMENT: Your students may observe other properties.

ConcepTest Section 1.6 Question 4 For Problems 4 – 8, as x  ∞ which function dominates, (a) or (b)? (That is, which function is larger in the long run?) (a) 0.1x 2 (b) x

ConcepTest Section 1.6 Answer 4 ANSWER (a). Power functions with the power greater than one and with a positive coefficient grow faster than linear functions. COMMENT: You could ask about the behavior as x  – ∞ as well.

ConcepTest Section 1.6 Question 5 For Problems 4 – 8, as x  ∞ which function dominates, (a) or (b)? (That is, which function is larger in the long run?)

ConcepTest Section 1.6 Answer 5 ANSWER COMMENT: One reason for such a question is to note that global behavior may not be determined by local behavior.

ConcepTest Section 1.6 Question 6 For Problems 4 – 8, as x  ∞ which function dominates, (a) or (b)? (That is, which function is larger in the long run?) (a) 3 – 0.9 x (b) log x

ConcepTest Section 1.6 Answer 6 ANSWER (b). Note that 3 – 0.9 x has a horizontal asymptote whereas the range of ln x is all real numbers. COMMENT: Students should realize that the graph the calculator displays can be misleading.

ConcepTest Section 1.6 Question 7 For Problems 4 – 8, as x  ∞ which function dominates, (a) or (b)? (That is, which function is larger in the long run?) (a) x 3 (b) 2 x

ConcepTest Section 1.6 Answer 7 ANSWER (b). Exponential growth functions grow faster than power functions. COMMENT: You could ask about the behavior as x  – ∞ as well.

ConcepTest Section 1.6 Question 8 For Problems 4 – 8, as x  ∞ which function dominates, (a) or (b)? (That is, which function is larger in the long run?) (a) 10(2 x ) (b) 72,000x 12

ConcepTest Section 1.6 Answer 8 ANSWER (a). Exponential growth functions grow faster than power functions, no matter how large the coefficient. COMMENT: One reason for such a question is to note that global behavior may not be determined by local behavior.

ConcepTest Section 1.6 Question 9 List the following functions in order from smallest to largest as x  ∞ (that is, as x increases without bound).

ConcepTest Section 1.6 Answer 9 ANSWER (a), (c), (d), (e), (b). Notice that f(x) and h(x) are decreasing functions, with f(x) being negative. Power functions grow slower than exponential growth functions, so k(x) is next. Now order the remaining exponential functions, where functions with larger bases grow faster. COMMENT: This question was used as an elimination question in a classroom session modeled after “Who Wants to be a Millionaire?”, replacing “Millionaire” by “Mathematician”.

ConcepTest Section 1.6 Question 10 The equation y = x3 + 2x 2 – 5x - 6 is represented by which graph?

ConcepTest Section 1.6 Answer 10 ANSWER (b). The graph will have a y-intercept of -6 and not be that of an even function. COMMENT: You may want to point out the various tools students can use to solve this problem, i.e. intercepts, even/odd, identifying the zeros, etc. You could also have students identify a property in each of the other choices that is inconsistent with the graph y = x 3 + 2x 2 – 5x – 6.

ConcepTest Section 1.6 Question 11 The graph in Figure 1.26 is a representation of which function?

ConcepTest Section 1.6 Answer 11 ANSWER (b). The graph will have a x-intercept of w and –3. COMMENT: You could ask students to describe the graphs of the other equations.

ConcepTest Section 1.6 Question 12 The equation y = x 2 + 5x + 6 is represented by which graph?

ConcepTest Section 1.6 Answer 12 ANSWER (d). The graph will be a parabola with x-intercepts of –2 and –3. COMMENT: Have students identify a property in each of the other choices that is inconsistent with the graph of y = x 2 + 5x + 6.

ConcepTest Section 1.6 Question 13 Which of the graphs could represent a graph of y = ax 4 + bx 3 + cx 2 + dx + e? Here a, b, c, d, and e are real numbers, and a ≠ 0.

ConcepTest Section 1.6 Answer 13 ANSWER (a), (b), (c), and (d). COMMENT: Follow-up Question. What property could you observe in a graph to know it could not be that of a fourth degree polynomial? Answer. The graph of the function has opposite end behavior (y  ∞ as x  ∞), or the graph turns more than three times, are a few properties students might observe.

ConcepTest Section 1.6 Question 14 Which of the graphs (a) – (d) could be that of a function with a double zero?

ConcepTest Section 1.6 Answer 14 ANSWER The graphs in (b) and (c) touch the x-axis, but do not cross there. So, they have double zeros. COMMENT: Students should remember that a double zero means the function will “bounce” off the x-axis.

ConcepTest Section 1.6 Question 15

ConcepTest Section 1.6 Answer 15 ANSWER (b). The domain of f(x) is not equal to the domain of g(x). COMMENT: Students should realize that some algebraic manipulations can only be applied to functions if the domain is stated. Follow-up Question. How can you change the statement to make it true? Answer. Keep f(x) defined as in the problem, and remove x = -1 from the domain of g(x). Alternatively, replace “f(x) = g(x)” with “f(x) = g(x) if x ≠ -1”.

ConcepTest Section 1.6 Question 16

ConcepTest Section 1.6 Answer 16 ANSWER (c). The equation indicates x-intercepts at  1 and a vertical asymptote at x = 2. (c) is the only graph where this happens. COMMENT: Students should analyze the long-term behavior of the function as well as the short-term behavior (i.e. asymptotes). You could also ask the students to analyze the similarities and differences of the functions (i.e. zero, intercepts, etc.).

ConcepTest Section 1.6 Question 17

ConcepTest Section 1.6 Answer 17 ANSWER (b). The graph goes through the origin and is positive for x > 2. COMMENT: Have students identify a property in each of the other choices that is inconsistent with the graph of

ConcepTest Section 1.6 Question 18

ConcepTest Section 1.6 Answer 18 ANSWER (d). The graph goes through the origin and has vertical asymptotes at x = 1 and –2. It is also positive for 0 < x < 1, and for large x, the function is negative. COMMENT: You could have the students verbalize why they excluded choices (a), (b), and (c).

ConcepTest Section 1.6 Question 19

ConcepTest Section 1.6 Answer 19 ANSWER (c). The graph goes through the origin and has vertical asymptotes at x = 1 and –2. It is negative for 0 < x < 1, and the function is positive for large x. Alternatively the function has a horizontal asymptote y = 2. COMMENT: You could have the students verbalize why they excluded choices (a), (b), and (d).

ConcepTest Section 1.6 Question 20 Which of the following functions represents the higher of the two functions in Figure 1.28 divided by the lower of the two functions?

ConcepTest Section 1.6 Answer 20 ANSWER (d). The ratio of the higher of the two functions to the lower equals 2 at x = 0, x = 1, and x = 2. COMMENT: You might want to point out that the division is undefined when x = 3. Follow-up Question. What does the graph of the reciprocal of the ratio look like? Answer.

ConcepTest Section 1.6 Question 21 Which of the following functions represents the higher of the two functions in Figure 1.29 divided by the lower of the two functions?

ConcepTest Section 1.6 Answer 21 ANSWER (b). Notice this ratio starts at 2 and approaches 1 as x approaches 4. COMMENT: The division of the two functions is not defined when x = 4. Follow-up Question. Could the graph of the reciprocal of this ratio be one of these choices? Answer. No, the graph of the reciprocal passes through the point (0, ½).

ConcepTest Section 1.6 Question 22 Which of the following functions represents the higher of the two functions in Figure 1.30 divided by the lower of the two functions?

ConcepTest Section 1.6 Answer 22 ANSWER (b). Because as x increases the lower graph is approaching zero faster than the upper graph, their ratio will be increasing. COMMENT: The division of the two functions is not defined when x = 4. Follow-up Question. Could the graph of the reciprocal of this ratio be one of these choices? Answer. Yes, (d), because the lower function could be a quadratic, and a quadratic function divided by a linear function is linear where it is defined.

ConcepTest Section 1.6 Question 23 Which of the following functions represents the ratio of the function starting at the point (0, 8) to the function starting at the point (0, 4) as shown in Figure 1.31.

ConcepTest Section 1.6 Answer 23 ANSWER (d). The ratio is always positive, decreasing, and less than 1 after the curves cross. Alternatively, the ratio has the value 1 when x ≈ 3. COMMENT: Follow-up Question. Could the graph of the reciprocal of this ratio be one of the choices? Answer. No, none of these graphs passes through the point (0, ½).

ConcepTest Section 1.6 Question 24 Every exponential function has a horizontal asymptote. (a) True (b) False

ConcepTest Section 1.6 Answer 24 ANSWER (a). The horizontal axis is a horizontal asymptote for all exponential functions. COMMENT: You may need to remind students that an exponential function has the form y = ab x where a ≠ 0 and b > 0 but b ≠ 1.

ConcepTest Section 1.6 Question 25 Every exponential function has a vertical asymptote. (a) True (b) False

ConcepTest Section 1.6 Answer 25 ANSWER (b). Exponential functions do not have vertical asymptotes. The domain for an exponential function is all real numbers. COMMENT: You may point out that analyzing the domain of a function is another way to decide if vertical asymptotes exist.