Practice Questions
Professor Graziano
24 June 2025
1. If f and g are inverse functions, then f (g(x)) =
2. Find the average rate of change of f (x) = −x2 + 5x + 1 on the interval [x, x + h]. Use your result to
find the average rate of change on [−2, 2].
3. Give the center and radius of the circle given by x2 − 2x + y 2 + 4x = 4.
4. Write the equation of the quadratic that has a vertex at (−2, 1) crosses the x-axis at (0, 0).
5. Express 2t/2 using base e. Express log2 1000 using base 10.
6. Graph y = log(1 − 2x) + 1. What is the domain of the function? Now find the inverse function.
7. Solve the inequality −|t + 2| + 4 ≤ −6.
8. Find all points of intersection of y = x2 + 5x + 4 and y = x + 1.
9. A bug moves counter-clockwise along a circle of radius 2 centered at the origin. Every 40 seconds the
bug completes a rotation. Where is the bug after 5 minutes if it starts at (2, 0)? Find (x, y).
10. The local gym offers two types of memberships. The first costs 50$ each month and 75$ per private
session. The second costs 150$ each month and 50$ per private session. How many sessions does it
take to make the second membership a better deal?
11. A rectangular fence is to be built against a wall so that only 3 sides need to be built. The perimeter
needs to be 200. What dimensions should the rectangle be so that the area enclosed is a maximum?
12. Find the domain, vertical aymptotes, removable discontinuities, and horizontal asymptote of the func2
tion h(x) = (xx2+x−6
−9 .
13. The half-life of Substance X is 100 days. Initially there are 80 grams. Find the exponential function
y = abx (find a and b) that model the decay. When is there 40 grams of substance left? When is there
60 grams of substance left?
14. Sketch a graph of y = 2 sin(πx + 2π) − 2. Consider the amplitude, period, vertical tranlastion, and
phase shift in the analysis.
15. Find the exact value for each of the following: arcsin(cosπ), arcsin(sin 5π/6), sin(arctan 4/5).
16. Convert to radians: 300◦ , −225◦ , and 750◦ .
17. Find tangent, sine, and cosine of the above angles.
18. Find cosine of the angle that the line y = 58 x makes with the positive x-axis.
19. The area of a trianle is given by A = 1/2BH, where B is the length of the base and H is the length of
the height. Two side lengths are known: 5 and 8. The angle between these sides is π/6 radians. Find
the area of the triangle. [HINT: Use a trigonometric function]
20. Give the center and radius of the circle given by x2 − 2x + y 2 + 6y = 4.
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21. Graph the functions e−x , − log x, and x on the same plane. Are these inverses of one-another? Explain.
22. A bug moves counter-clockwise along a circle of radius 2 centered at the origin. Every 36 seconds the
bug completes a rotation. Where is the bug after 78 seconds if it starts at (0, 2)? [THIS IS NOT (2, 0)]
Find (x, y).
−4π
−5π
23. Convert to degrees: 22π
3 , 3 , and 4 .
24. Find cosine of the above angles.
25. sin t = −4
5 and tan t < 0. Find tan t, cos t, sec t, and cot t.
26. Graph the functions − sin x and cos x on the same plane. Express − sin x as a transform of cos x.
27. What is the domain and range of log2 x? What is log2 1? log2 16? log2 1/8? What is the limit of log x
as x → 0+ .
28. Graph tan −πx. What is the period?
29. Sketch a graph of the polynomial function h(x) = (x + 1)2 (x − 2)3 . Include x- and y-intercepts,
multiplicity, and end behavior.
30. There is a lightning rod on the top of a building. From a location 500 feet from the base of the building,
the angle of elevation to the top of the building is 36◦ . From the same location the angle of elevation
to the top of the rod is 38◦ . Find the height of the rod.
31. Find the exponential function y = abx that passes through the points (2, 12) and (4, 3).
32. Sketch a graph of the function y = −(2)−x − 1.
33. Find the domain, vertical asymptotes, removable discontinuities, and horizontal asymptote of the
2
+x−6
function h(x) = x 9−x
2 .
34. Let y = f (x) = |x|. Give the transformation that moves the vertex to (3, −1) and then reflects over
the x-axis.
35. Write an inequality using absolute value whose solution set is all points less than 4 from −1.
36. Sketch a graph of y = 2 sin( π6 x − 3π
6 ). Describe the transformation.
37. Model the height of a passenger on a ferris wheel [Find a function]. The diameter of the wheel is 20
meters. The center of the wheel is 11 meters off the ground. The loading platform is at the bottom of
the wheel. The wheel completes a rotation every minute. At t = 0 the passenger is at the bottom of
the wheel. Carefully sketch a graph of the function [amplitude, key points on horizontal axis, etc].
38. Practice every problem from every examination.
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