PreCalculus Algebra Skills - Review (Classwork)



Getting Ready for AP Calc: Algebra Skills – 1

1. Evaluate: a. [pic] for x = –3 b. –3x2 for x = 4

Simplify the following:

2. [pic] 3. [pic] + [pic] 4. 3x4 . 2x4 5. [pic]

6. (x + 4)–1 7. [pic] 8. [pic] 9. [pic]

10. [pic] 11. [pic]

Rewrite the following without using negative exponents:

12. 3x–2 13. 4x + 3y–2 14. [pic] 15. 3(a + b)–2

Write each of the following in simplest form using radicals:

16. [pic] 17. [pic] 18. [pic] 19. [pic]

Rewrite the following in the form kxr where k is a constant and r is a rational exponent:

20. [pic] 21. [pic] 22. [pic] 23. [pic] 24. [pic] 25. [pic]

Write each of the following using rational exponents and with no variables in the denominator:

26. [pic] 27. [pic]

28. Rationalize the denominator and simplify: a. [pic] b. [pic]

30. Rationalize the numerator and simplify: [pic]

Getting Ready for AP Calc: Algebra Skills – 2

Classify the following as either true or false. When possible, fix the false ones by changing the right-hand side. Assume all variables represent real numbers.

1. –42 = 16 2. If x = –4, then –x2 = 16. 3. If x = 3, then 2x2 = 36.

4. 2(x + y) = 2x + 2y 5. 2(xy) = (2x)(2y) 6. [pic]= [pic]

7. 2x2 – x2 = 2 8. x3 – x2 = x 9. 4 + 2(x + 3) = 6(x + 3)

10. 2(x + 3)2 = (2x + 6)2 11. (3x + 12)2 = 3(x + 4)2 12. 2x – (3y – 5) = 2x – 3y – 5

13. 5x – 3(x – 2) = 5x + (–3) (–x + 2) 14. 5x2 – (x – 2)2 = 5x2 + (–x + 2)2

15. x – 3(x + 4) = x2 + 4x – 3x – 12 = x2 + x – 12

16. [pic] = [pic] + [pic] 17. [pic] = [pic]+ [pic] 18. [pic] = [pic]

19. [pic] = [pic] 20. [pic] = 3x 21. = 1 + 3

22. [pic] = 3 + 23. [pic]= [pic] 24. [pic] = [pic]

25. [pic] – [pic] = [pic] = [pic] 26. [pic]+ [pic] = [pic] + [pic] = [pic]

27. (xy)2 = x2y2 28. (x + y)2 = x2 + y2 29. (3x)2 = 3x2

30. 2x + y = 2x + 2y 31. x2 . x3 = x6 32. 32 . 35 = 97

33. (x4)3 = x7 34. [pic] = [pic] 35. x–3 = –x3

36. 3x–2 = [pic] 37. ax–1 – b = [pic] 38. [pic] = [pic]

39. [pic] = [pic] 40. 3x1/2 = [pic]

41. [pic] = x 42. [pic]= x 43. [pic] = [pic]

44. [pic] = [pic] 45. [pic] = [pic] 46. [pic]= x + y

47. [pic] = [pic] 48. [pic] = [pic] 49. [pic] = x + y

50. [pic] = [pic] 51. The solution to x2 = k (k > 0) is x = [pic].

Getting Ready for AP Calc: Algebra Skills – 3

Factor the following completely. Ideally, binomial factors should have integer coefficients.

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. (2x + 3) 3 – 12(2x + 3) 2 8. [pic] 9. [pic]

10. [pic] 11. [pic] 12. [pic]

13. [pic] 14. [pic]

15. [pic]

Getting Ready for AP Calc: Algebra Skills – 4

Factor and/or simplify the following completely.

1. 4x2 – 25y2 2. 4x2 – 64y2 3. x4 – 81

4. [pic] 5. [pic] 6. [pic]

7. (x + y)2 – 4(x + y) – 21 8. 2x2 + 7x – 30 9. [pic]

10. [pic] 11. [pic]

12. [pic] 13. [pic]

14.[pic]

Getting Ready for AP Calc: Algebra Skills – 5

In 1 – 19, solve for y if the problem involves y; otherwise solve for x.

1. [pic] 2. [pic] 3. x(x + 1)(x – 2)2 = 0

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. (x – 5)2 = 0 9. (3x – 4)(x + 6) = 0

10. [pic] 11. [pic] 12. [pic]

13. [pic] 14. [pic] 15. [pic]

16. [pic] 17. [pic] 18. [pic]

19. [pic]

20. Solve for [pic] ("[pic]- prime") in terms of x and y (Note: [pic] is not the same as y; it is a different variable):

a. [pic] b. [pic]

Getting Ready for AP Calc: Algebra Skills – 6

1 . Solve and express the solution with x on the left.

a. –2x +3 < 9 b. 3x + 7 ( 5x – 1

Solve for x and express the solution set in interval notation.

2. [pic] 3. [pic]

4. Solve [pic] where 0 < a < b < c.

5. An object travels along the x-axis with a velocity given by [pic] where [pic] is time in seconds since the particle began moving and [pic] is in meters per second. For what intervals of time is the particle moving forward ("forward" means positive [pic])?

Getting Ready for AP Calc: Functions – 1

1. Suppose y = f(x) for some function f.

a. Tell in words what the statement f(7) = –3 means.

b. Write the ordered pair represented by the statement f(7) = –3.

2. If[pic],

a. Evaluate[pic]. b. Evaluate [pic]. c. Evaluate [pic].

d. Evaluate [pic]. e. Evaluate [pic]. f. Evaluate [pic].

g. Solve [pic] = 8. h. Evaluate [pic]and simplify.

i. Evaluate your answer from part (h) above for [pic] = 0.

3. A young kid standing on level ground throws a ball. The equation of the ball's path is

[pic]

where [pic] = height of the ball and [pic] = horizontal distance from the kid.

a. Evaluate[pic].

b. Evaluate [pic] (simplification not required).

c. Evaluate [pic] (simplification not required).

d. Evaluate [pic](simplification not required). What does this represent?

e. Find the value(s) of x for which [pic] = 0.

f. Find the value(s) of x for which [pic] = 10.

g. Graph this function.

h. What is the domain of the function in the context of the problem?

i. What is the range of the function in the context of the problem?

Graph the following “piecewise” functions.*

4. [pic] 6. [pic]

*There are several variations of methods to graph piecewise functions on your calculator. One way:

Y1 = –0.5x/(x ( 0)

Y2 = [pic]/(x > 0) The inequality symbols can be found under the TEST menu (2nd MATH).

5. Evaluate [pic]for f(x) = [pic], simplify, and evaluate the resulting expression for h = 0.

Getting Ready for AP Calc: Functions – 2

The graph of the function [pic] is shown. Use it to answer questions 1 – 10.

1. Evaluate the following:

a. [pic] b. [pic] c. [pic]

2. Describe what happens to ƒ as [pic] from the left. Note the arrow on the graph. (Vocabulary: the vertical line x = –4, dashed on the graph, is an asymptote. It is not part of the graph of f.)

3. Write in simplest interval notation:

a. The domain of ƒ b. The range of ƒ

4. Find the: a. y-intercept of ƒ b. root(s) of ƒ

5. Solve for x: a. [pic] = 1 b. [pic] = –5

6. Solve for x and write the answer in simplest interval notation: a. [pic] b. [pic] > 4

7. Write in interval notation the values of x for which:

a. ƒ is increasing. b. ƒ is decreasing.

8. a. f has five horizontal tangents. Where are they?

b. f has one vertical tangent. Where is it?

9. a. Does f have any relative maxima? If so, where?

b. Does f have any relative minima? If so, where?

10. a. Does f have an absolute maximum value? If so, what is it?

b. Does f have an absolute minimum value? If so, what is it?

11. Evaluate [pic].

Getting Ready for AP Calc: Functions – 3

1. Find the natural domain of each of the following:

a. [pic] b. [pic]

2. Determine, without using your calculator, whether each function is even, odd or neither.

a. [pic] b. [pic] c. [pic]

d. [pic] e. y = ex f. [pic]

g. f(x) = sin x h. f(x) = cos x i. [pic]

3. Suppose a function y = f(x) has roots x = r1, x = r2, etc. What are the roots of the new function y = af(x) where a is a constant?

4. Let y = g(x) where [pic]. Write an equation for each of the following.

a. The image of the graph of y = g(x) after a reflection over the y-axis.

b. The image of the graph of graph of y = g(x) after a reflection over the x-axis.

c. The image of the graph of graph of y = g(x) after a translation 3 units left and 2 units up.

d. The image of the graph of graph of y = g(x) after a vertical compression by a factor of 2.

e. The image of the graph of graph of y = g(x) after a horizontal dilation by a factor of 3.

5. The graph of a function y = g(x) is sketched below. On

a separate, larger ([–7, 7] x [–4, 10]) graph, sketch and label the graphs of

a. y = –g(x)

b. y = g(–x)

c. y = g(x – 2) – 1

d. y = 3g(x)

e. y = g(0.5x)

6. Find the inverse of each function:

a. f(x) = 2x – 3 b. [pic], x ( 0 c. [pic]

d. [pic] ; x ( 2 e. f(x) = ex e. f(x) = 2ln(x + 1); x > –1

f. [pic]; x ( 0 g. f(x) = sin x

Getting Ready for AP Calc: Polynomial and Rational Functions

1. For the polynomial function y = –2x4 + ax2 + bx – 3,

a. What is the degree of the function?

b. (Without using your calculator,) what is the largest number of real roots the function could have?

c. (Without using your calculator,) what is the largest number of turning points it could have?

d. (Without using your calculator,) what are the coordinates of the y-intercept of the function?

e. (Without using your calculator,) how will the tails of the graph behave?

2. Sketch a possible graph of the function [pic] without using your calculator. You may assume that a, b and c are all positive and a < b.

3. Multiple choice. For each of the following, use the choices below:

(A) Horizontal asymptote at y = 0. (B) Horizontal asymptote at y = k, k > 0.

(C) Horizontal asymptote at y = k, k < 0 (D) Slant asymptote with positive slope.

(E) Slant asymptote with negative slope. (F) Nonlinear end behavior.

Assume a, b and c are positive constants.

a. [pic] b. [pic] c. [pic]

d. [pic] e. [pic] f. [pic]

4. Describe the behavior of the tails of each of the following rational functions.

a. [pic] b. [pic] c. [pic] d. [pic]

5. Without using your calculator, sketch a graph of each of the following. Clearly identify all intercepts and asymptotes. Check your graphs with your calculator.

a. [pic] b. [pic] c. [pic]

Getting Ready for AP Calc: Exponential and Log Functions

1. Without using your calculator,

a. Sketch the graph of y = ex.

b. Sketch the graph of y = e–x.

c. Sketch the graph of y =ln x.

2. Without using your calculator, evaluate

a. e0 b. ln (1) c. eln 3 d. ln [pic] e. [pic] f. [pic]

g. [pic] h.[pic] i. [pic] j. [pic]

3. Simplify each of the following

a. [pic] b. [pic] c. ln e–2x

4. Use the properties of logarithms to expand each of the following as much as possible:

a. [pic] b. [pic] c. ln(xne–kx)

5. Express each of the following as the log of a single quantity:

a. 4lnx – 2ln3 b. [pic] c. [pic]

6. Simplify each of the following as much as possible:

a. ln e–x b. e–lnx c. e(2lnx – ln3) d. e(ln a + ln b)

7. Solve for x: alnx = b

8. Solve for t: Aekt = B

Getting Ready for AP Calc: Trig Functions

You should be able to do the following without using your calculator.

1. Evaluate the following:

a. [pic] b. [pic] c. [pic] d. tan2 e. cos [pic]

f. 4sin2 g. tan π h. tan ( i. 2sin j. 3cos 2π

k. esin 0 l. ln cos 0 m. ln |cos ( – tan (| n. esin(3(/2) o. ln (cos2()

p. [pic] q. ln (sin 2() r. ln e3

2. If [pic], evaluate [pic].

3. Find the general solution to each equation below.

a. sin θ = b. cos 2θ = [pic] c. tan θ = 1

d. sin θ = –1 e. cos θ = 0 f. sin x2 = 0

4. If A is the radian measure a first quadrant angle and cos A = k, find, in terms of A, the measures of all the angles in the interval [0, 2() that satisfy the equation cos x = –k

5. For the function [pic],

a. Find the range of the function.

b. Find the period of the function.

Getting Ready for AP Calc: Geometry and Coordinate Geometry – 1

1. Find the average rate of change of the function [pic] over the interval [1, 3].

2. An object’s position as a function of time is given by [pic], t ( 0 (x in meters, t in seconds).

a. What is the object's average velocity for the first 4 seconds? (Average velocity = Δx/Δt)

b. What is the object's average velocity from t = 1 to t = 9 seconds?

c. Write an equation for the “secant line” that intersects the graph of x(t) at the points where

t = 1 and t = 9.

d. Write an expression for the object's average velocity from t = a to t = b seconds.

e. Write an expression for the object's average velocity on the interval from t to t + h seconds.

3. The following table shows the population of New York State (in thousands) according to US Census data for the last century (year 2000 data is an estimate).

Year19001910192019301940195019601970198019902000Pop7,2689,11410,38512,58813,47914,83016,78218,24117,55817,99018,250

a. What was the average rate of change of New York's population during the 20th century?

b. Estimate of the rate of change of New York's population in the year 2000.

c. Use the average rate of change from 1980 to 2000 to write a linear equation for New York's population as a function of time.

d. Using your equation from part f, in what year should the population reach 20 million?

4. Preview of calculus: the slope of a curve at a point on the curve is defined as the slope of the line tangent to the curve at that point.

a. Sketch the graph of [pic].

b. What happens to the slope of the graph of [pic] as you move along the curve from left to right?

c. On your sketch, draw a line that is tangent to the graph of [pic] at the point where x = 4.

d. In calculus, you will learn how to find the formula for the slope of a curve at any point. For the curve [pic], the slope at any point x is given by [pic]. What is the slope of the graph of [pic] at the point where x = 4?

e. Write the equation of the tangent line to the graph of [pic] at the point where x = 4.

Getting Ready for AP Calc: Geometry and Coordinate Geometry – 2

1. Find the distance from the point (3, 5) to

a. the x-axis. b. the y-axis. c. the line x = 7. d. the line y = 7.

2. Let (x, y) be a first quadrant point. Find the distance from the point (x, y) to

a. the x-axis. b. the y-axis. c. the line x = c. d. the line y = c.

3. a. Find the distance between the points (3, 2) and (3, 7).

b. Find the distance between the points (4, 3) and (4, 10).

c. Find the distance between the points (c, y1) and (c, y2) assuming y2 > y1.

d. Find the distance between the points (x1, d) and (x2, d) assuming x2 > x1.

4. The graph of y = 1 + x2 is shown below.

a. At the point on the graph where x = 2, how high above the x-axis is the graph?

b. At the point on the graph where y = 2, how far from the y-axis is the graph?

c. At a point (x, y) on the graph, how high is the graph above the x-axis in terms of x?

d. At a point (x, y) on the graph, how far is the graph from the y-axis in terms of y?

Figure for # 4 Figure for # 5 Figure for # 6 Figure for # 7

5. The graph of y = f(x) is shown above. At a point (x, y) on the graph,

a. how high is the graph above the x-axis in terms of x?

b. how far is the graph from the y-axis in terms of y?

6. A very narrow rectangle of width (x has its base on the x-axis. The midpoint of the top of the rectangle is on the graph of y = 9 – x2. (See figure above.)

a. If the rectangle is centered on the line x = 2, find its area in terms of (x.

b. If the rectangle is centered on the line x = a, find its area in terms of a and (x.

7. A very narrow rectangle of height (y has its left side on the y-axis. The midpoint of the right side of the rectangle is on the graph of y = 13 – x2. (See figure above.)

a. If the rectangle is centered on the line y = 4, find its area in terms of (y.

b. If the rectangle is centered on the line y = b, find its area in terms of b and (y.

8. A vertical rectangle of width (x has its base on the x-axis and the midpoint of its top at the point (x, y) on the graph of y = f(x). A horizontal rectangle of height (y has its left side one the y-axis and the midpoint of its right side at the point (x, y).

a. Find the area of the vertical rectangle in terms of x and (x.

b. Find the area of the horizontal rectangle in terms of y and (y.

Getting Ready for AP Calc: Geometry and Coordinate Geometry – 3

1. A very narrow rectangle of width (x has the midpoint of its base on the graph of [pic] and the midpoint of its top on the graph of y = 1 + x2 (see figure below).

a. If the rectangle is centered on the line x = 4, find its area in terms of (x.

b. If the rectangle is centered on the line x = a, find its area in terms of a and (x.

2. A very narrow rectangle of width (x has the midpoint of its base on the graph of [pic] and the midpoint of its top on the graph of y = f(x) (see figure below). If the rectangle is centered on a vertical line at an arbitrary value of x, find its area in terms of x and (x.

3. A very narrow rectangle of width (y has the midpoint of its left side on the graph of y = 3x + 1 and the midpoint of its right side on the graph of y = (x – 1)2 (see figure below).

a. If the rectangle is centered on the line y = 4, find its area in terms of (y.

b. If the rectangle is centered on the line y = b, find its area in terms of b and (y.

4. A very narrow rectangle of width (y has the midpoint of its left side on the graph of [pic] and the midpoint of its right side on the graph of y = f(x) (see figure below). If the rectangle is centered on a horizontal line at an arbitrary value of y, find its area in terms of y and (y.

Figure for # 1 Figure for # 2 Figure for # 3 Figure for # 4

5. An annulus is the area between two concentric circles (shaded in the figure).

a. Find the exact area of the annulus in terms of R and r.

b. In calculus, it is often useful to have an approximation to the area that is valid for small values of (r = R – r.

1) Substitute R = r + (r in your answer from part a and simplify.

2) If (r is small, ((r)2 will, in comparison, be very small and can be ignored. What is the resulting approximation to the area?

3) An alternate way to visualize the area: cut the annulus along a radius and flatten it out into a long thin “rectangle.” Find the area of the rectangle in terms of r and (r.

Getting Ready for AP Calc: Geometry and Coordinate Geometry – 4

1a. Find the volume of a disk having radius r and thickness t (see diagram below, left).

b. Find the volume of a washer having outer radius R, inner radius r, and thickness t.

2. A very narrow rectangle of width (x has its base on the x-axis. The midpoint of the top of the rectangle is on the graph of y = 9 – x2. This rectangle is rotated around the x-axis to form a thin disk. I made a somewhat lame attempt to illustrate this in the figure at right. Imagine a coin on its edge with the x-axis running right through its center; half of the coin sticks up out of your paper, the other half sticks down into your desk.

a. If the rectangle is centered on the line x = 2, find the volume of the disk (coin) in terms of (x.

b. If the rectangle is centered on the line x = a, find the volume of the disk (coin) in terms of a and (x.

3. A narrow rectangle of width (x has its base on the x-axis and the midpoint of its top at the point (x, y) on the graph of y = f(x). This rectangle is rotated around the x-axis (as in the previous problem). Find the volume of the resulting disk in terms of x and (x.

4. A very narrow rectangle of width (x has the midpoint of its base on the graph of [pic] and the midpoint of its top on the graph of y = 1 + x3/2. This rectangle is rotated about the x-axis forming a washer (a coin with a hole through the center.) The x-axis goes right through the center of the washer. Again, an attempt has been made to illustrate this in the figure at right.

a. If the rectangle is centered on the line x = 4, find the volume of the resulting washer in terms of (x.

b. If the rectangle is centered on the line x = a, find the volume of the resulting washer in terms of a and (x.

5. A very narrow rectangle of width (x has the midpoint of its base on the graph of [pic] and the midpoint of its top on the graph of y = f(x). The rectangle is rotated around the x-axis to form a washer. If the rectangle is centered on a vertical line at an arbitrary value of x, find the volume of the resulting washer in terms of x and (x.

Getting Ready for AP Calc: Geometry and Coordinate Geometry – 5

Let y = f(x) be a function that is always positive for 0 ( x ( 8. We want to approximate the area under f on that interval. We will use the right rectangular approximation method, not because it’s the best, but because it’s the easiest for today’s purpose. Suppose the interval is divided into n subintervals of equal width.

1. In terms of n, what is the width of each rectangle? We call this width (x.

2. In terms of (x, what is the x-coordinate

of the right side of

a. the first rectangle, x1?

b. the second rectangle, x2?

c. the third rectangle, x3?

d. the ith rectangle, xi?

3. In terms of (x, what is the height of the

a. the first rectangle?

b. the second rectangle?

c. the third rectangle?

d. the ith rectangle?

4. Using summation notation, write an expression for the approximate area under the curve Rn.

Getting Ready for AP Calc: Geometry and Coordinate Geometry – 6

1. A rectangle with one side on the x-axis is inscribed under the parabola y = 16 – x2. Find the area of the rectangle as a function of its half-base x and then find the maximum possible area.

2. A rectangle has one vertex at the origin, one on the positive x-axis, one on the positive y-axis and the fourth on the graph of [pic]. Find the area of the rectangle as a function of x.

3. Find, in terms of x, the area of a rectangle inscribed in the ellipse [pic].

4. The graph of y = [pic]is called the Witch of Agnesi.* Find the distance from a point on the curve to the origin as a function of x alone.

5. Find the distance from any point (x, y) on the graph of xy = 24 to the point (4, 1) as a function of x.

6. Brittany is standing 12 feet east of a garage; her nose is 5 feet off the ground. Gino lobs a snowball in the general direction of Brittany’s nose from the roof of the garage. The path of the snowball is given by [pic] where y is the snowball’s height in feet above the ground and x is the snowball’s horizontal distance from the garage. Find the distance between the snowball and Brittany’s nose as a function of x.

Getting Ready for AP Calc: Geometry and Coordinate Geometry – 7

1. A rectangle of width (x has its base on the x-axis and the midpoint of its top on the graph of y = sin x. The rectangle is centered at some arbitrary value of x between 0 and (.

a. Sketch a graph of the problem.

b. Find the area of the rectangle in terms of x and (x.

2. The rectangle from problem #1 is rotated around the x-axis to form a disk. Find the volume of that disk in terms of x and (x.

3. The region R in the first quadrant is bounded by the graphs of y = cos x, y = x2 and the y-axis.

a. Sketch the region R. (You shouldn’t even need to ask what mode your calculator should be in.)

b. Find the x-coordinate of the rightmost point of region R.

c. Sketch a vertical rectangle in R having the midpoint of its base on the graph of y = x2 and the midpoint of its top on the graph of y = cos x. Let its width be (x.

d. Find the area of the rectangle from part c.

e. Find, in terms of x and (x, the volume of the washer that results when the rectangle from part c is rotated around the x-axis.

f. (This one is hard for many people to visualize.) Imagine that the rectangle from part c is the base of a prism that sticks up out of the page a distance equal to the height of the original rectangle (in other words, two faces of the prism are squares). Express the volume of this prism in terms of x and (x.

4. The area under the function y = f(x) on the interval a ( x ( b is subdivided into n approximating rectangles of equal width (x. Within each subinterval, a point [pic] is selected with which to obtain the height of the approximating rectangle. See the diagram below.

a. What is the length of (x in terms of n?

b. Write an expression in terms of [pic] and (x for the area of the ith approximating rectangle.

c Using summation notation, write an expression for the approximate area under the graph of f on the interval [a, b]

Answers

Algebra Skills – 1

1a. 36 b. –48 2. [pic] 3. [pic] 4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic] 10. [pic] 11. [pic] 12. [pic] 13. [pic] 14. [pic]

15. [pic] 16. [pic] 17. [pic] 18. [pic] 15. [pic] 15. [pic] 21. [pic]

22. [pic] 23. [pic] 24. [pic] 25. [pic] 26. [pic] 27. [pic]

28a. [pic] b. [pic] 29. [pic]

Algebra Skills – 2

Numbers 4, 16 and 27 are true. Number 41 is true if x ( 0 or if you allow imaginary numbers. Numbers 42 and 44 are true only for positive values of the variable. Number 51 is only “half-true.” The rest are simply false. They represent very common student mistakes. Don’t make these mistakes!

Algebra Skills – 3

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. (2x – 9)(2x + 3)2 8. [pic] 9. [pic]

10. [pic] 11. [pic] 12. [pic]

13. [pic] 14. [pic] 15. [pic]

Algebra Skills – 4

1. (2x + 5y)(2x – 5y) 2. 4(x – 4y)(x + 4y) 3. (x – 3)(x + 3)( x2 + 9)

4. x(x – 2)(x + 2)(x2 + 4) 5. [pic] 6. [pic]

7. (x + y – 7)(x + y + 3) 8. (2x – 5)(x + 6) 9. (2sin x – 1)(sin x – 1)

10. [pic] 11. [pic] 12. [pic]

13. [pic] 14. [pic]

Algebra Skills – 5

1. [pic] 2. [pic] 3. x = –1 ( x= 0 ( x= 2

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. x = 5 9. x = 4/3 ( x= –6

10. [pic] 11. [pic] 12. [pic]

13. x = 0 ( x = –0.5 14. [pic] 15. x = –3 ( x = 6

16. [pic] 17. x = –a ( b 18. [pic]

20. x = –3 ( x = 2 19a. [pic] b. [pic]

Algebra Skills – 6

1a. x > –3 b. x ( –4 2. (–2, 0) ( (4, () 3. (–[pic], –b] [pic] (a, b]

4.

5. [0, 2) ( (5, ()

Functions – 1

1. a. When x = 7, y = –3. b. (7, –3)

2. a. 0.8 b. [pic] c. 4a d. [pic] e. [pic] f. [pic]

g. x = 0.5. h. [pic] i. [pic]

3. a. 8.75 b. ƒ(2x) = –0.25(2x)2 + 3(2x) + 2 c. ƒ(x + h) = –0.25(x + h)2 + 3(x + h) + 2

d. [pic] e. 6 ± or appx –0.633 [pic] 12.633

f. 4 [pic] 8 h. [0, 12.633] (appx) i. [0, 11]

5. [pic]

Functions – 2

1a. 6 b. 7 c. not defined

2. As x [pic] –4 from the left, ƒ increases without bound (“goes to infinity”).

3a. (–9, –4) [pic] [–3, 10] b. [–3, () 4a. 4 b. –2, 8

5a. x ( –1.7, x ( 7.3, x = 9 b. No solution (empty set).

6a. [–3, –2) b. (–5.6, –4) [pic] [2, 7)

7a. (–9, –4), [–3, 0], [3, 5] and [8, 10] b. [0, 2) and [2, 3] and [5, 8]

8a. x = –7, x = 0, x = 3, x = 5 and x = 8 b. x = 7

9a. yes, at x = 0, = 2 and x = 5 b. yes, at x = 3 and x = 8 10a. no b. yes; –3 11. 6

Functions – 3

1a. [–3, 5] b. [–5, –3) ( (–3, 3) ( (3, 5] 2a. even b. odd c. even d. odd

e. neither d. even e. odd f. even g. even 3. Roots are unchanged.

4a. [pic] b. [pic] c. [pic]

d. [pic] e. [pic]

5a. b. d.

c. e.

6a. [pic] b. [pic] c. [pic]

d. [pic] e. [pic] f. [pic]

g. [pic] h. [pic]

Polynomial and Rational Functions

1a. 4 b. 4 c. 3 d. (0, –3) 2.

3a. D b. B c. F d. B

e. A f. C

4a. Horizontal asymptote: y = 0 b. Horizontal asymptote: y = 2

c. Slant asymptote: with slope 1 d. Horizontal asymptote: y = 0

Exponential and Log Functions

2a. 1 b. 0 c. 3 d. [pic] e. ( f. 0 g. 0

h. ( i. ( j. –( 3a. 2x – 1 b. f(x) c. –2x

4a. 3ln x – ln 2 b. [pic] c. nln |x| – kx

5a. [pic] b. [pic] c. [pic]

6a. –x b. [pic] c. [pic] d. ab 7. [pic] 8. [pic]

Trig Functions

1a. [pic] b. [pic] c. –2 d. [pic] e. [pic] f. 2 g. 0

h. undef. i. –2 j. 3 k. 1 l. 0 m. 0 n. [pic]

o. 0 p. 1 q. undef. r. 3

2. 4 3a. [pic] b. [pic]

c. [pic] d. [pic] e. [pic] f. [pic]

4. x = ( + A ( x = 2( – A 5a. [d – |a|, d + |a|] b. 6

Geometry – 1

1. –0.219 2. a. 0.5 m/sec b. 0.5 m/sec c. y – 1 = 0.25(x – 1) d. [pic] e. [pic]

3. a. 109820 people/yr b. 26,000 people/yr c. p = 17558 + 34.6(t – 1980) d. During the year 2050.

4. b. Slope decreases toward 0. d. 1/4 e. y – 2 = [pic]

Geometry – 2

1a. 5 b. 3 c. 4 d. 2 2a. y b. x c. |c – x| d. |c – y|

3a. 5 b. 7 c. y2 – y1 d. x2 – x1 4a. 5 b. 1 c. 1 + x2 d. [pic]

5a. f(x) b. f –1(y) 6a. 5(x b. [pic]

7a. 3(y b. [pic] 8a. f(x)(x b. f –1(y)(y

Geometry – 3

1a. 15(x b. [pic] 2. [pic]

3a. 2(y b. [pic] 4. [pic]

5a. [pic] b1) [pic] b2) 2(r(r b3) 2(r(r

Geometry – 4

1a. [pic] b. [pic] 2a. 25((x b. [pic] 3. [pic]

4a. 77((x b. [pic] 5. [pic]

Geometry – 5

1. [pic] 2a. (x b. 2(x c. 3(x d. i(x.

3a. f((x) b. f(2(x) c. f(3(x) d. f(i(x) 4. [pic]

Geometry – 6

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

Geometry – 7

1a. See graph at right. b. A = sin(x)(x

2. [pic]

3a. See diagram at far right.

b. x ( 0.824 c. See diagram.

d. [pic]

e. [pic]

f. [pic] g. [pic]

4a. [pic] b. [pic] c. [pic]

-----------------------

|-a | | | | | | | |

|0 | | | | | | | |

|b | | | | | | | |

|c | | | | | | | |

| | | | | | | | |

|y | | | | | | | |

| | | | | | | | |

|x | | | | | | | |

| | | | | | | | |

|y =| | | | | | | |

|sin| | | | | | | |

|x | | | | | | | |

| | | | | | | | |

|x | | | | | | | |

| | | | | | | | |

|y =| | | | | | | |

|cos| | | | | | | |

|x | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

y

x

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

y

x

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

Solution: [-a, 0) ( [c, ()

x

x – – + + +

(x + a) – + + + +

(x – b)2 + + + + +

(x – c) – – – – +

f(x) – + – – +

y

x

-a -b c

y

. . . .

. . . .

n-1

n

i

3

2

y = f(x)

8

x1

1

x

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

y

x

[pic]

xi-1

xi

y = f(x)

x1

x2

y

x

a

xi

(x

. . . .

. . . .

i

y = f(x)

b

D

Alternate reasoning for #4: For extremely large |x|, the function behaves like a cubic with positive leading coefficient: left tail goes to --"; right tail goes to -". In between, the function changes sign at the odd roots x =

[pic]

y

x

[pic]

[pic]

y

x

t

R

r

(x, y)

y

y = f(x)

x

x

y

y = 9 – x2

x

t

r

R

(r

r

[pic]

y

[pic]

x

y = (x – 1)2

y

y = 3x + 1

x

[pic]

y

[pic]

x

[pic]

y

[pic]

x

(x, y)

y

y = f(x)

x

y

y = 13 – x2

x

?

?

(x, y)

y

y = f(x)

x

?

?

(x, y)

y

y = 1 + x2

x

y

Dy = 9 – x2

x

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