Function transformations and inverses
Shift, stretch, and reflect a parent graph, and build an inverse function by swapping x and y and re-solving.
Question 1
Precalculus practice question 1
Precalculus practice topics
Question sets mix direct skills with unfamiliar applications so method selection becomes part of the practice.
Polynomial and rational behavior
Read end behavior from the leading term and locate horizontal, vertical, and slant asymptotes.
Trigonometric functions
Work with radians and the unit circle, and read amplitude, period, phase shift, and midline from an equation.
Conics, sequences, and limits
Identify a conic from its equation, work with recursive and explicit sequences, and evaluate limits algebraically.
Precalculus practice questions with worked answers
These 8 questions are printed in full on this page, with every step of the arithmetic written out. Cover the options, solve the question on paper first, and only then open the worked answer to compare your method with the one shown.
Question 1: The graph of y = x^2 is shifted 3 units right and 5 units down. What is the new equation?
- y = (x - 3)^2 - 5
- y = (x + 3)^2 - 5
- y = (x - 3)^2 + 5
- y = (x - 5)^2 - 3
Show the worked answer
- A horizontal shift changes the x inside the bracket, and it works the opposite way round.
- Moving 3 right means replacing x with (x - 3), giving y = (x - 3)^2.
- A vertical shift is added outside, and down means subtract 5.
- The result is y = (x - 3)^2 - 5. Check the vertex: it moves from (0, 0) to (3, -5).
Answer: y = (x - 3)^2 - 5
Question 2: If f(x) = (2x + 1)/5, what is the inverse function f inverse of x?
- (5x - 1)/2
- (5x + 1)/2
- 5/(2x + 1)
- (2x - 1)/5
Show the worked answer
- Write y = (2x + 1)/5 and swap the roles of x and y: x = (2y + 1)/5.
- Multiply both sides by 5: 5x = 2y + 1.
- Subtract 1 and divide by 2: y = (5x - 1)/2.
- Check with x = 3: f inverse of 3 = 7, and f(7) = (14 + 1)/5 = 3.
Answer: (5x - 1)/2
Question 3: Describe the end behavior of f(x) = -2x^3 + 5x.
- As x grows large f falls to negative infinity, and as x goes to negative infinity f rises to positive infinity
- f rises to positive infinity at both ends
- f falls to negative infinity at both ends
- As x grows large f rises to positive infinity, and as x goes to negative infinity f falls to negative infinity
Show the worked answer
- End behavior is decided by the highest power term, -2x^3.
- The power 3 is odd, so the two ends go opposite ways.
- Test a big positive x: at x = 10, -2(1000) + 50 = -1950, which is far below zero.
- Test a big negative x: at x = -10, -2(-1000) - 50 = 1950, far above zero.
Answer: As x grows large f falls to negative infinity, and as x goes to negative infinity f rises to positive infinity
Question 4: What is the horizontal asymptote of f(x) = (6x^2 - x)/(3x^2 + 4)?
- y = 2
- y = 0
- y = 6
- There is no horizontal asymptote
Show the worked answer
- The top and the bottom both have degree 2, so compare the leading coefficients.
- The leading coefficients are 6 and 3, so the ratio is 6/3 = 2.
- Check with a large value: at x = 100, the top is 60000 - 100 = 59900 and the bottom is 30000 + 4 = 30004.
- 59900 / 30004 is about 1.996, which is closing in on 2, so y = 2.
Answer: y = 2
Question 5: Find the exact value of cos(5 pi / 6).
- negative square root of 3 over 2
- square root of 3 over 2
- negative one half
- one half
Show the worked answer
- 5 pi / 6 is 150 degrees, because pi radians = 180 degrees and 180 x 5/6 = 150.
- 150 degrees sits in the second quadrant, where cosine is negative.
- The reference angle is 180 - 150 = 30 degrees, and cos(30 degrees) = root(3)/2.
- Apply the negative sign: cos(150 degrees) = -root(3)/2, about -0.866.
Answer: negative square root of 3 over 2
Question 6: For y = 4 sin(3x) + 1, what are the amplitude, the period, and the midline?
- Amplitude 4, period 2 pi / 3, midline y = 1
- Amplitude 3, period 2 pi / 4, midline y = 1
- Amplitude 4, period 3, midline y = 0
- Amplitude 1, period 2 pi / 3, midline y = 4
Show the worked answer
- In y = A sin(Bx) + D the amplitude is the size of A, so the amplitude is 4.
- The period is 2 pi divided by B, and here B = 3.
- 2 pi / 3 is the period, so the wave repeats about every 2.09 units.
- D shifts the wave up, so the midline is y = 1, and the curve runs from 1 - 4 = -3 up to 1 + 4 = 5.
Answer: Amplitude 4, period 2 pi / 3, midline y = 1
Question 7: A circle has the equation (x - 4)^2 + (y + 3)^2 = 49. What are its center and radius?
- Center (4, -3), radius 7
- Center (-4, 3), radius 7
- Center (4, -3), radius 49
- Center (-4, 3), radius 49
Show the worked answer
- The standard circle is (x - h)^2 + (y - k)^2 = r^2 with center (h, k).
- (x - 4)^2 gives h = 4.
- (y + 3)^2 is the same as (y - (-3))^2, so k = -3.
- r^2 = 49, so r = 7. The center is (4, -3) and the radius is 7.
Answer: Center (4, -3), radius 7
Question 8: Find the limit of (x^2 - 4)/(x - 2) as x approaches 2.
- 4
- 0
- 2
- The limit does not exist
Show the worked answer
- Substituting x = 2 gives 0/0, so the expression must be simplified first.
- Factor the top: x^2 - 4 = (x - 2)(x + 2).
- Cancel the shared (x - 2) factor, leaving x + 2 for every x except 2.
- Now substitute: 2 + 2 = 4, so the limit is 4.
Answer: 4
How the three Precalculus difficulty levels differ
The difficulty buttons above change what a question asks of you, not just the size of the numbers. Each example below is taken from the question set on this page.
| Level | What it tests | Example question | Time target |
|---|---|---|---|
| Foundation | One skill at a time, with the numbers kept small enough to check in your head. | The graph of y = x^2 is shifted 3 units right and 5 units down. What is the new equation? | About 1 minute |
| Exam | The wording of a real test paper: pick the method first, then carry out two or three steps. | If f(x) = (2x + 1)/5, what is the inverse function f inverse of x? | 2 to 3 minutes |
| Challenge | Two ideas combined, or a result you have to interpret after the calculation ends. | For y = 4 sin(3x) + 1, what are the amplitude, the period, and the midline? | 4 to 5 minutes |
How to practice Precalculus effectively
Begin without notes and explain your choice before checking. For every miss, identify whether the cause was a definition, setup, calculation, interpretation, or time decision. Re-solve the question from a blank page, then return to the same skill in a mixed set tomorrow.
What your Precalculus answer review should show
A useful review shows more than the correct option. Compare the method with your first attempt, locate the earliest incorrect decision, and write one rule that would prevent the same error in a new Precalculus problem.
Move from mixed Precalculus questions to a complete course
This page targets flexible Precalculus question practice. When you need a syllabus-aligned sequence with unit selection, use Precalculus practice by unit and return here later for mixed retrieval.
Questions about Precalculus practice
When should I change the difficulty?
Move up after you can solve several questions accurately without hints and explain the method. Move down for one short set when errors show that a definition or setup is still uncertain.
How often should I practice?
Short sessions on several days usually build stronger recall than one long session. Revisit missed Precalculus skills the next day, then mix them with older topics later in the week.
Where can I review the lessons in order?
Use Precalculus practice by unit for a syllabus-aligned sequence with unit selection, practice, and a complete answer review.
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Stuck on a Precalculus question?
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