Precalculus Practice Test: Function Transformations
- Transformations
- 10 questions
- 15 minutes
These 10 questions cover shifts, reflections, stretches and compressions of graphs. Changes inside the parentheses act on x and work the opposite way you might expect: f(x + 3) moves the graph left, not right.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
The graph of y = f(x) is shifted 3 units left and 2 units down. Which equation describes the new graph?
Correct answer: B
A horizontal shift left by 3 replaces x with x + 3, and a shift down by 2 subtracts 2 from the output. The equation with x − 3 moves the graph right instead.
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Question 2
How is the graph of g(x) = −f(x) related to the graph of f?
Correct answer: D
Multiplying every output by −1 flips each point (x, y) to (x, −y), a reflection across the x-axis. A reflection across the y-axis would be f(−x).
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Question 3
The point (4, −6) is on the graph of y = f(x). Which point must be on the graph of y = 2f(x − 1) + 3?
Correct answer: A
The input shifts right 1, so x = 5. The output is doubled and then raised by 3: 2(−6) + 3 = −9. The point (3, −9) shifts the wrong way, and (5, −12) forgets to add 3.
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Question 4
How does the graph of y = f(2x) compare with the graph of y = f(x)?
Correct answer: C
Each output value is now reached at half the x-value, so the graph is squeezed toward the y-axis. A factor multiplying x inside the function changes the graph horizontally, not vertically.
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Question 5
The graph of y = |x| is moved so that its vertex is at (−1, 4) and it opens downward with the same shape. Which equation describes it?
Correct answer: B
A vertex at (−1, 4) needs |x + 1| + 4, and the negative sign in front makes the V open downward. The equation with x − 1 puts the vertex at (1, 4).
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Question 6
What is the vertex of the graph of g(x) = −3(x + 2)2 − 5?
Correct answer: A
In the form a(x − h)2 + k, the vertex is (h, k); here x + 2 = x − (−2), so h = −2 and k = −5. The point (2, −5) misreads the sign inside the parentheses.
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Question 7
The point (2, 8) is on the graph of y = f(x). Which point must be on the graph of y = f(−x)?
Correct answer: C
Replacing x with −x reflects the graph across the y-axis, so (x, y) becomes (−x, y). The point (2, −8) would come from −f(x).
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Question 8
What is the range of g(x) = x + 4 − 1?
Correct answer: D
The square root part is never negative, and subtracting 1 lowers every output by 1, so the smallest output is −1. The interval [−4, ∞) is the domain, not the range.
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Question 9
The graph of y = f(x) has x-intercepts −2 and 6. What are the x-intercepts of y = f(x2)?
Correct answer: A
The new graph is zero when x/2 = −2 or x/2 = 6, so x = −4 or x = 12. Dividing x by 2 inside stretches the graph horizontally, so intercepts move away from the origin, not toward it as in −1 and 3.
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Question 10
Which sequence of steps turns the graph of y = f(x) into the graph of y = f(3x − 6)?
Correct answer: B
Factor the inside: f(3x − 6) = f(3(x − 2)), so the compression is followed by a shift right of 2. Shifting right 6 after the compression gives f(3(x − 6)) = f(3x − 18).
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