Given the equation of a function, identify a possible graph corresponding to the given function. Algebraic functions including linear, square root, quadratic, rational, cubic, and absolute value functions are considered. Use all properties of the graphs of functions such as range, domain, x-intercepts, y-intercepts, horizontal shifts, vertical shifts, compression, and stretching to solve the suggested problems.
If needed, Free graph paper is available. You may also want to work through a tutorial on identifying graphs of functions before taking this self-test.
Practice Questions with Graph Links & Answers
For each question, view the 4 graphs using the provided link, select the best possible answer, and check the solution dropdown.
Question 1: Identify a possible graph for \( f(x) = (x - 1)^3 \)
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Answer: graph b
Explanation: The graph is a standard cubic curve shifted horizontally 1 unit to the right.
Question 2: Identify a possible graph for \( f(x) = |x| + 2 \)
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Answer: graph a
Explanation: The basic absolute value function graph \( |x| \) is shifted vertically upward by 2 units.
Question 3: Identify a possible graph for \( f(x) = (x - 1)^2 - 1 \)
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Answer: graph d
Explanation: This is a parabola with its vertex at \((1, -1)\) opening upward, matching graph d.
Question 4: Identify a possible graph for \( f(x) = \sqrt{x - 2} + 1 \)
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Answer: graph a
Explanation: The square root function graph starts at \((2, 1)\) and extends to the right.
Question 5: Identify a possible graph for \( f(x) = -(x - 2)^3 + 1 \)
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Answer: graph c
Explanation: The cubic curve is reflected across the horizontal axis (due to the negative sign), shifted 2 units right and 1 unit up.
Question 6: Identify a possible graph for \( f(x) = -\sqrt{x - 1} - 1 \)
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Answer: graph b
Explanation: The square root graph is reflected downward and shifted 1 unit right and 1 unit down.
Question 7: Identify a possible graph for \( f(x) = -2|x| \)
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Answer: graph b
Explanation: The absolute value graph opens downward and is vertically stretched by a factor of 2.
Question 8: Identify a possible graph for \( f(x) = \dfrac{-1}{x + 2} \)
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Answer: graph d
Explanation: This is a hyperbola with a vertical asymptote at \( x = -2 \) and a reflection across the x-axis.
Question 9: Identify a possible graph for \( f(x) = (x - 2)^3 + 1 \)
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Answer: graph d
Explanation: A cubic function shifted 2 units to the right and 1 unit up, centered at \((2, 1)\).
Question 10: Identify a possible graph for \( f(x) = \dfrac{2x - 3}{x - 1} \)
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Answer: graph d
Explanation: Rewriting as \( \frac{2(x-1) - 1}{x-1} = 2 - \frac{1}{x-1} \), it represents a rational function with vertical asymptote \( x = 1 \) and horizontal asymptote \( y = 2 \).