Idea Transcript
Math 106 Study Guide for Chapters 3 & 4 1. Find the function whose graph is a parabola with vertex and that passes through the point in vertex form: . Sketch the graph and label the vertex and any intercepts on the graph. 2. Find the equation of the quadratic function whose graph is shown. y 6
(-2, 4) 4
(0, 3) 2
x -5
5 -2
-4
-6
3. A 5.5 feet tall woman is shooting a free throw. The path of the basketball is parabolic in shape and the ball reaches its maximum height of 11.5 feet when the ball is 10 feet from the player. (a) Find the equation for the path of the ball. Let be the horizontal distance from the shooter and be the height of the ball. (Assume that the ball is 5.5 ft. high when it is released.) (b) The ball hits the front of the rim, which is 10 feet high. How far is the shooter from the rim? 4. After experimentation, two college students find that when a bottle of Washington Champagne is shaken several times, held upright and uncorked, its cork travels according to the function defined by where is the cork’s height in feet above the ground and is the time in seconds after the cork has been released. (a) After how many seconds will the cork reach its maximum height? (b) What is the maximum height of the cork? 5. A gardener has 80 feet of fencing to fence in a rectangular garden. He wants to put the fence all the way around the garden, and he wants to section it into 4 areas with fencing parallel to one side of the rectangle, as shown in the picture below. (a) Find a function that models the total area of the garden that he can fence in terms of its length .
x
(b) Find the dimensions of the largest possible total area that he can fence.
6. Consider the polynomial function (a) What is the degree of the polynomial (b) Determine the end behavior of P( x) . (c) Find the zeros of (d) Sketch a graph of
. ?
and determine the multiplicity of each. . Label any intercepts on the graph.
7. Consider the graph of the polynomial function given below. Which of the following statement(s) is (are) true? Circle all possible true statements. Explain your reasoning. a. The polynomial could have degree 4.
30
20
b. The polynomial could have degree 5. 10
c. The polynomial could have degree 6. -2
2
-10
d. The polynomial could have degree 7. -20
e. The polynomial could have degree 8.
-30
Reasoning: 8. Determine a polynomial with real coefficients which satisfies the given conditions. If no such polynomial exists, explain why. You do not need to multiply out the polynomial. (a) (b) (c)
has degree 4, leading coefficient 2, and zeros 3 (multiplicity 3) and (multiplicity 1). has degree 5, leading coefficient -2, and zeros 1 (multiplicity 1) and (multiplicity 2). has degree 3, zeros 2 (multiplicity 1) and (multiplicity 1), and has constant term of 26.
9. Given the polynomial function (a) Right-End Behavior: As x , Circle one:
(the function points up) or
, find the following.
(the function points down). Why?
(b) Left-End Behavior: As x , Circle one:
(the function points up) or
(the function points down). Why?
(c) List all possible zeros of . (d) Find all zeros (real and complex) of by factoring. List all zeros and identify any multiplicity. (e) Sketch the graph of . Label any intercepts on the graph.
10. (Text p. 268 #23-#28) For each polynomial graph at right: (a) state whether the degree of the function is even or odd; (b) use the graph to name the zeros of , then state whether their multiplicity is even or odd; (c) state the minimum possible degree of and write it in factored form; and (d) estimate the domain and range. Assume all zeros are real
11. (Text p. 268 #37-#42) Every function at right has zeros , , and . Match each to its corresponding graph using degree, end behavior, and multiplicity of each zero. (i) (ii) (iii) (iv) (v) (vi)
12. Answer the following for g ( x) x3 5x 2 2 x 12 . (a) List all possible rational zeros of g ( x) . (b) Find all the zeros of g ( x) . (c) Write g ( x) as a product of linear factors. 13. Determine all the asymptotes, intercepts and domain for each of the following functions: (a) f ( x)
x2 5 x 2 10
(b) g ( x)
5 4 x2 7 x 2 15 x 2
(c) h( x)
2 x2 x 6 x 1
14. Which, if any, of the following rational functions have the same asymptotes and intercepts as the function shown? Explain your reasoning.
15. For each part, determine a rational function satisfying the given criteria. Show work and explain why you choose each part. (a)
has zeros at asymptote .
and
, vertical asymptotes
and
, and horizontal
(b)
has zeros at asymptote .
and
, vertical asymptotes
and
, and horizontal
(c)
, different from your answer to (b), satisfying the same criteria as
.
16. Let
. Find the following, showing work as to how you found each (even if you can
do it in your head). If necessary, explain how you found an answer. Sketch a graph of label the asymptotes and intercepts on the graph.
and
(a) Domain in interval notation (b) -intercept (c) -intercept(s) (d) All asymptotes 17. Let
Find the following, showing work as to how you found each (even if you can do it
in your head). If necessary, explain how you found an answer. Sketch a graph of asymptotes and intercepts on the graph.
and label the
(a) Domain in interval notation (b) -intercept (c) -intercept(s) (d) All asymptotes 18. The figure below shows the graph of a rational function with vertical asymptotes , , and horizontal asymptote . The graph has -intercepts at and , and it passes through the point . The equation for has one of the five form shown below. Choose the appropriate form for , and then write the equation. Assume that is in simplest form.
19. Solve the inequalities. Express the answer in interval notation. (a) (b) (c)
2 20. Let f and g be quadratic polynomial functions, of the form y ax bx c , where a 1 , whose
graphs are given below. y
y
6
4
6
f(x)
g(x)
4
2
2
x -5
5
Set V x
x -5
5
-2
-2
-4
-4
-6
-6
f x . g x
(a) What is (are) the vertical asymptote(s) of V? (b) What is (are) the zero(s) of V? (c) What is the y-intercept of V? (d) Which of the following is an asymptote of V? (Circle one.) I.
II.
III.
21. Match the correct solution to the inequality, below. (a) (b) (c) (d) (e) none of these
22. Determine the domain of
√
IV. where the graph of
which is given
23. A rectangular pen is to be constructed that will enclose
square feet.
(a) Determine the length of the fence needed as a function of the width of the pen. (b) Determine the dimensions of the pen if feet of fencing are to be used. (c) Determine an interval for the possible width of the pen if less than feet of fencing is to be used. 24. Is the function 25. Suppose true? (A)
one-to-one? If so, find is a one-to-one function. Given that (B)
. which of the following CANNOT be
(C)
(D)
(E)
26. Find the inverse of each function: (a)
𝑟 𝑇 𝑟
1 0.9
(b)
where
(c) (d) 27. If the point
2 2.3
3 3.4
4 1
5 0
.
. √ is on the graph of
(A)
(B) (
)
, which if the following points MUST be on the graph of (C)
(D)
(E) None of these
28. The groundskeeper of a local high school estimates that due to heavy usage by the baseball and softball teams, the pitcher’s mound loses one-fifth of its height every month. (a) Determine the height of the mound after 3 months if it began at a height of 25 cm. (b) Determine how long until the pitcher’s mound is less than 16 cm high. 29. Graph the following functions. Label all intercepts and asymptotes. In each case, what is the domain and range? (a) (b) (c) (d) 30. Sketch the graph the following functions. Label all intercepts and asymptotes. In each case, what is the domain and range? (a) (b) 32. Simplify the following. (a) 33. Determine the domain of
(b) log 125 1 log e .
1 log 4 16 e3
?
34. Determine the x - and y -intercepts of the function g ( x) log 2 ( x 8) 1 exactly. 35. Solve the following equations: (a) (c)
(c)
36. Solve the following equations: (a) 2
2 x
1 32
x 3
(d)
(b)
(e)
(c)
175 50 1 5e3 x
(f)
(g) (h) e2 x e x 11 1 37. A population of ladybugs grows according to the limited growth model
where t is measured in weeks, t 1 .
A 400 400e0.04t
(a) How many ladybugs will there be in 20 weeks? (b) What happens to the population as t grows very large? (c) When will the population be approximately 300 ladybugs? 38. A valuable painting was purchased in 1980 for $125,000. The painting is expected to double in value in 15 years. Its value (in thousands of dollars) is modeled by the function , where is the number of years since 1990. Leave your answers in exact form. (a) Determine the value of the annual rate . (b) According to this model, when will the value of the painting be $625,000 39. A population of bacteria doubles every generation. Find an equation modeling the growth rate and determine how many generations it would take to reach over 500 bacteria. kt 40. The number of bacteria N (t ) in a culture is given by the model N (t ) 10e , where t is the time in
hours. If the culture contains 60 bacteria when t 4 hours, determine the following. (a) Determine the rate, k , of growth or decay. Leave answer in exact form. (b) After how many hours will the culture triple its initial population? Write your answer in exact form and then round to two decimal places. 41. The half-life of radioactive isotope Carbon-14 is 5730 years. If you have 2 grams remaining after 1000 years, then what was the mass of the initial sample?