Algebra 2 Quiz: Transformations Of Functions And Graphs
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Transformations Of Functions And GraphsQuestion 1 of 20

Let f(x)=x2f(x)=x^2 be the parent function. What transformation changes f(x)f(x) to g(x)=f(x)+3g(x)=f(x)+3?

Shift down 3 units
Shift up 3 units
Shift right 3 units
Vertical stretch by a factor of 3
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Algebra 2 Quiz

Algebra 2 Quiz: Transformations Of Functions And Graphs

Practice Transformations Of Functions And Graphs in Algebra 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Transformations Of Functions And Graphs, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Let f(x)=x2f(x)=x^2 be the parent function. What transformation changes f(x)f(x) to g(x)=f(x)+3g(x)=f(x)+3?

  1. Shift down 3 units
  2. Shift up 3 units (correct answer)
  3. Shift right 3 units
  4. Vertical stretch by a factor of 3
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x) + k, k·f(x), f(kx), and f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x) + k shifts the graph vertically (up if k > 0, down if k < 0), (2) k·f(x) stretches vertically if |k| > 1 or compresses if 0 < |k| < 1 (and reflects across x-axis if k < 0), (3) f(x + k) shifts horizontally—LEFT if k > 0, RIGHT if k < 0 (opposite of what you might expect!), (4) f(kx) compresses horizontally if |k| > 1 or stretches if 0 < |k| < 1 (and reflects across y-axis if k < 0). Outside the function (f(x) + k and k·f(x)) affects y-values/vertical; inside the function (f(x + k) and f(kx)) affects x-values/horizontal. Here, g(x) = f(x) + 3 adds 3 to every y-value of the parent quadratic, resulting in a vertical shift upward by 3 units—keep up the great work noticing that! Choice B correctly identifies the transformation as a shift up 3 units. A common distractor like choice A might confuse the sign, suggesting a downward shift, but since we're adding a positive 3 outside, it's definitely upward—don't worry, practicing more will make this intuitive! Transformation memory aid: think 'outside affects y, inside affects x.' Anything added/multiplied OUTSIDE f (like f(x) + 3 or 2f(x)) changes y-values (vertical effects). Anything done INSIDE the parentheses (like f(x + 3) or f(2x)) changes x-values (horizontal effects). The tricky part: horizontal shifts are opposite to the sign—f(x + 3) shifts LEFT 3 because you're subtracting 3 from x-coordinates, and f(x - 2) shifts RIGHT 2. Think: what x-value gives the original function's behavior?

Question 2

Let the parent function be f(x)=x3f(x)=x^3. What is the effect on the graph of replacing f(x)f(x) with f(2x)f(2x)?

  1. Vertical compression by a factor of 2
  2. Vertical stretch by a factor of 2
  3. Horizontal stretch by a factor of 2
  4. Horizontal compression by a factor of 2 (correct answer)
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x) + k, k·f(x), f(kx), and f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x) + k shifts the graph vertically (up if k > 0, down if k < 0), (2) k·f(x) stretches vertically if |k| > 1 or compresses if 0 < |k| < 1 (and reflects across x-axis if k < 0), (3) f(x + k) shifts horizontally—LEFT if k > 0, RIGHT if k < 0 (opposite of what you might expect!), (4) f(kx) compresses horizontally if |k| > 1 or stretches if 0 < |k| < 1 (and reflects across y-axis if k < 0). When we replace f(x) with f(2x), we're multiplying the input by 2, which means to get the same y-value, we need half the original x-value—this compresses the graph horizontally by a factor of 2. Choice A correctly identifies this as a horizontal compression by a factor of 2. Choice B incorrectly calls it a stretch (which would be f(x/2)), while C and D confuse horizontal transformations with vertical ones. Transformation memory aid: think 'outside affects y, inside affects x.' Anything done INSIDE the parentheses (like f(x + 3) or f(2x)) changes x-values (horizontal effects).

Question 3

How does replacing f(x)f(x) with f(3x)f(3x) change the graph of f(x)f(x)?

  1. It shifts the graph right 3 units
  2. It compresses the graph horizontally by a factor of 3 (all xx-values are divided by 3) (correct answer)
  3. It stretches the graph horizontally by a factor of 3 (all xx-values are multiplied by 3)
  4. It stretches the graph vertically by a factor of 3 (all yy-values are multiplied by 3)
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x) + k, k·f(x), f(kx), and f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x) + k shifts the graph vertically (up if k > 0, down if k < 0), (2) k·f(x) stretches vertically if |k| > 1 or compresses if 0 < |k| < 1 (and reflects across x-axis if k < 0), (3) f(x + k) shifts horizontally—LEFT if k > 0, RIGHT if k < 0 (opposite of what you might expect!), (4) f(kx) compresses horizontally if |k| > 1 or stretches if 0 < |k| < 1 (and reflects across y-axis if k < 0). Outside the function (f(x) + k and k·f(x)) affects y-values/vertical; inside the function (f(x + k) and f(kx)) affects x-values/horizontal. Replacing with f(3x) affects inside the function, so it's horizontal: since |3| > 1, it compresses horizontally by a factor of 3, meaning x-values are scaled by 1/3 or divided by 3 to squeeze the graph toward the y-axis. Choice B correctly identifies the transformation as compressing horizontally by a factor of 3 with x-values divided by 3. A distractor like D might confuse it with vertical effects, but remember, the '3' is inside with x, so it's horizontal, not vertical! Transformation memory aid: think 'outside affects y, inside affects x.' Anything added/multiplied OUTSIDE f (like f(x) + 3 or 2f(x)) changes y-values (vertical effects). Anything done INSIDE the parentheses (like f(x + 3) or f(2x)) changes x-values (horizontal effects). The tricky part: horizontal shifts are opposite to the sign—f(x + 3) shifts LEFT 3 because you're subtracting 3 from x-coordinates, and f(x - 2) shifts RIGHT 2. Think: what x-value gives the original function's behavior?

Question 4

Let the parent function be f(x)=x2f(x)=x^2, whose graph is the standard parabola with vertex at (0,0)(0,0). What transformation changes f(x)f(x) to g(x)=2f(x−3)+1g(x)=2f(x-3)+1?

  1. Shift left 3 units, stretch vertically by a factor of 2, then shift down 1 unit
  2. Shift right 3 units, compress vertically by a factor of 12\tfrac{1}{2}, then shift up 1 unit
  3. Shift right 3 units, stretch vertically by a factor of 2, then shift up 1 unit (correct answer)
  4. Shift left 3 units, compress horizontally by a factor of 2, then shift up 1 unit
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x) + k, k·f(x), f(kx), and f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x) + k shifts the graph vertically (up if k > 0, down if k < 0), (2) k·f(x) stretches vertically if |k| > 1 or compresses if 0 < |k| < 1 (and reflects across x-axis if k < 0), (3) f(x + k) shifts horizontally—LEFT if k > 0, RIGHT if k < 0 (opposite of what you might expect!), (4) f(kx) compresses horizontally if |k| > 1 or stretches if 0 < |k| < 1 (and reflects across y-axis if k < 0). Outside the function (f(x) + k and k·f(x)) affects y-values/vertical; inside the function (f(x + k) and f(kx)) affects x-values/horizontal. For g(x) = 2f(x-3) + 1, start with f(x-3) which shifts the parabola right by 3 units (since it's f(x - 3), making x larger to achieve the same output), then multiply by 2 to stretch vertically by a factor of 2, and finally add 1 to shift up 1 unit. Choice C correctly identifies the transformation as shifting right 3 units, stretching vertically by a factor of 2, then shifting up 1 unit. A common mistake, like in choice A, is thinking f(x-3) shifts left instead of right—remember, the shift direction is opposite the sign inside the function! Transformation memory aid: think 'outside affects y, inside affects x.' Anything added/multiplied OUTSIDE f (like f(x) + 3 or 2f(x)) changes y-values (vertical effects). Anything done INSIDE the parentheses (like f(x + 3) or f(2x)) changes x-values (horizontal effects). The tricky part: horizontal shifts are opposite to the sign—f(x + 3) shifts LEFT 3 because you're subtracting 3 from x-coordinates, and f(x - 2) shifts RIGHT 2. Think: what x-value gives the original function's behavior?

Question 5

Let f(x)=x2f(x)=x^2. The graph of gg is shown on the coordinate plane along with ff. The function gg is of the form g(x)=f(x+k).g(x)=f(x+k). What is the value of kk?

(Use the graph: ff has vertex at (0,0)(0,0) and gg has vertex at (2,0)(2,0).)

  1. k=2k=2
  2. k=−2k=-2 (correct answer)
  3. k=12k=\tfrac{1}{2}
  4. k=−12k=-\tfrac{1}{2}
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x)+kf(x) + k, k⋅f(x)k \cdot f(x), f(kx)f(kx), and f(x+k)f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x)+kf(x) + k shifts the graph vertically (up if k>0k > 0, down if k<0k < 0), (2) k⋅f(x)k \cdot f(x) stretches vertically if ∣k∣>1|k| > 1 or compresses if 0<∣k∣<10 < |k| < 1 (and reflects across x-axis if k<0k < 0), (3) f(x+k)f(x + k) shifts horizontally—LEFT if k>0k > 0, RIGHT if k<0k < 0 (opposite of what you might expect!), (4) f(kx)f(kx) compresses horizontally if ∣k∣>1|k| > 1 or stretches if 0<∣k∣<10 < |k| < 1 (and reflects across y-axis if k<0k < 0). Outside the function (f(x)+kf(x) + k and k⋅f(x)k \cdot f(x)) affects y-values/vertical; inside the function (f(x+k)f(x + k) and f(kx)f(kx)) affects x-values/horizontal. Given g(x)=f(x+k)g(x) = f(x + k) with f(x)=x2f(x) = x^2 and the graph showing g's vertex at (2,0)(2,0) compared to f's at (0,0)(0,0), this indicates a horizontal shift right by 2 units, so f(x+k)=(x+k)2f(x + k) = (x + k)^2 has vertex at (−k,0)(-k, 0); setting −k=2-k = 2 gives k=−2k = -2. Choice B correctly finds the k value as −2-2. An error like in choice A might ignore the opposite sign rule for horizontal shifts—f(x+2)f(x + 2) would shift left to (−2,0)(-2,0), not right, so k must be negative for a right shift. Transformation memory aid: think 'outside affects y, inside affects x.' Anything added/multiplied OUTSIDE f (like f(x)+3f(x) + 3 or 2f(x)2f(x)) changes y-values (vertical effects). Anything done INSIDE the parentheses (like f(x+3)f(x + 3) or f(2x)f(2x)) changes x-values (horizontal effects). The tricky part: horizontal shifts are opposite to the sign—f(x+3)f(x + 3) shifts LEFT 3 because you're subtracting 3 from x-coordinates, and f(x−2)f(x - 2) shifts RIGHT 2. Think: what x-value gives the original function's behavior?

Question 6

Determine whether the function f(x)=x4−2x2f(x)=x^4-2x^2 is even, odd, or neither.

  1. Neither, because polynomials cannot be even or odd
  2. Even, because f(−x)=f(x)f(-x)=f(x) (correct answer)
  3. Neither, because it contains both even and odd powers of xx
  4. Odd, because f(−x)=−f(x)f(-x)=-f(x)
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x) + k, k·f(x), f(kx), and f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Even functions have y-axis symmetry: f(-x) = f(x) for all x, meaning the left half of the graph is a mirror image of the right half. Examples include f(x) = x², x⁴, |x|, and x² + 3. Odd functions have origin symmetry: f(-x) = -f(x), meaning rotating the graph 180° about the origin gives the same graph. Examples include f(x) = x, x³, 1/x, and x³ - x. Most functions are neither even nor odd! To check, compute f(-x) = (-x)^4 - 2(-x)^2 = x^4 - 2x^2, which equals f(x), confirming y-axis symmetry. Choice A correctly determines it's even because f(-x) = f(x). If you chose C thinking mixed powers mean neither, that's understandable, but actually all powers here are even, making it even—odd powers would suggest possible oddness! For even/odd testing: (1) Take the given function f(x), (2) Find f(-x) by substituting -x for every x (use parentheses!), (3) Simplify completely, (4) Compare with f(x) and -f(x): if f(-x) = f(x), it's even; if f(-x) = -f(x), it's odd; if neither match, it's neither. Example: f(x) = x² - 3, so f(-x) = (-x)² - 3 = x² - 3 = f(x) → even! Graphically: even functions have y-axis as mirror line, odd functions look the same after 180° rotation.

Question 7

Let the parent function be f(x)=x3f(x)=x^3. What is the effect on the graph of replacing f(x)f(x) with g(x)=f(x+2)g(x)=f(x+2)?​

  1. Shift right 2 units
  2. Shift left 2 units (correct answer)
  3. Shift up 2 units
  4. Shift down 2 units
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x) + k, k·f(x), f(kx), and f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x) + k shifts the graph vertically (up if k > 0, down if k < 0), (2) k·f(x) stretches vertically if |k| > 1 or compresses if 0 < |k| < 1 (and reflects across x-axis if k < 0), (3) f(x + k) shifts horizontally—LEFT if k > 0, RIGHT if k < 0 (opposite of what you might expect!), (4) f(kx) compresses horizontally if |k| > 1 or stretches if 0 < |k| < 1 (and reflects across y-axis if k < 0). In g(x) = f(x + 2), we're adding 2 to the input before applying f, which shifts the graph LEFT by 2 units—to get the same output, we need an x-value that's 2 less than before. Choice B correctly identifies this as shifting left 2 units. Choice A incorrectly suggests right (remember: the sign is opposite for horizontal shifts!), while C and D confuse horizontal shifts with vertical ones. The tricky part: horizontal shifts are opposite to the sign—f(x + 3) shifts LEFT 3 because you're subtracting 3 from x-coordinates, and f(x - 2) shifts RIGHT 2. Think: what x-value gives the original function's behavior?

Question 8

Given the graph of f(x)=x2f(x)=x^2 and the graph of g(x)=f(x)+kg(x)=f(x)+k shown, the vertex of gg is at (0,−3)(0,-3). What is kk?

  1. k=3k=3
  2. k=−13k=-\tfrac{1}{3}
  3. k=13k=\tfrac{1}{3}
  4. k=−3k=-3 (correct answer)
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x)+kf(x) + k, k⋅f(x)k \cdot f(x), f(kx)f(kx), and f(x+k)f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x)+kf(x) + k shifts the graph vertically (up if k>0k > 0, down if k<0k < 0), (2) k⋅f(x)k \cdot f(x) stretches vertically if ∣k∣>1|k| > 1 or compresses if 0<∣k∣<10 < |k| < 1 (and reflects across x-axis if k<0k < 0), (3) f(x+k)f(x + k) shifts horizontally—LEFT if k>0k > 0, RIGHT if k<0k < 0 (opposite of what you might expect!), (4) f(kx)f(kx) compresses horizontally if ∣k∣>1|k| > 1 or stretches if 0<∣k∣<10 < |k| < 1 (and reflects across y-axis if k<0k < 0). Outside the function (f(x)+kf(x) + k and k⋅f(x)k \cdot f(x)) affects y-values/vertical; inside the function (f(x+k)f(x + k) and f(kx)f(kx)) affects x-values/horizontal. Here, g(x)=f(x)+kg(x) = f(x) + k shifts vertically by kk, moving the vertex from (0,0)(0,0) to (0,k)(0,k), so for (0,−3)(0,-3), kk must be -3 to shift down 3 units. Choice B correctly finds the kk value as -3. If you selected A, you might have forgotten that positive kk shifts up, but negative shifts down—keep that sign in mind! Transformation memory aid: think 'outside affects y, inside affects x.' Anything added/multiplied OUTSIDE f (like f(x)+3f(x) + 3 or 2f(x)2f(x)) changes y-values (vertical effects). Anything done INSIDE the parentheses (like f(x+3)f(x + 3) or f(2x)f(2x)) changes x-values (horizontal effects). The tricky part: horizontal shifts are opposite to the sign—f(x+3)f(x + 3) shifts LEFT 3 because you're subtracting 3 from x-coordinates, and f(x−2)f(x - 2) shifts RIGHT 2. Think: what x-value gives the original function's behavior?

Question 9

Let the parent function be f(x)=∣x∣f(x)=|x|. Which function represents shifting the graph of ff left 1 unit and down 2 units?

  1. g(x)=f(x−1)−2g(x)=f(x-1)-2
  2. g(x)=f(x+1)−2g(x)=f(x+1)-2 (correct answer)
  3. g(x)=f(x−1)+2g(x)=f(x-1)+2
  4. g(x)=f(x+1)+2g(x)=f(x+1)+2
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x) + k, k·f(x), f(kx), and f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x) + k shifts the graph vertically (up if k > 0, down if k < 0), (2) k·f(x) stretches vertically if |k| > 1 or compresses if 0 < |k| < 1 (and reflects across x-axis if k < 0), (3) f(x + k) shifts horizontally—LEFT if k > 0, RIGHT if k < 0 (opposite of what you might expect!), (4) f(kx) compresses horizontally if |k| > 1 or stretches if 0 < |k| < 1 (and reflects across y-axis if k < 0). Outside the function (f(x) + k and k·f(x)) affects y-values/vertical; inside the function (f(x + k) and f(kx)) affects x-values/horizontal. To shift left 1 unit (horizontal, inside: f(x + 1)) and down 2 units (vertical, outside: subtract 2), the function is g(x) = f(x + 1) - 2. Choice B correctly identifies the transformation as g(x) = f(x + 1) - 2. If you chose A, you might have reversed the shift sign, but recall horizontal shifts oppose the sign inside—f(x + 1) is left, f(x - 1) is right! Transformation memory aid: think 'outside affects y, inside affects x.' Anything added/multiplied OUTSIDE f (like f(x) + 3 or 2f(x)) changes y-values (vertical effects). Anything done INSIDE the parentheses (like f(x + 3) or f(2x)) changes x-values (horizontal effects). The tricky part: horizontal shifts are opposite to the sign—f(x + 3) shifts LEFT 3 because you're subtracting 3 from x-coordinates, and f(x - 2) shifts RIGHT 2. Think: what x-value gives the original function's behavior?

Question 10

Let the parent function be f(x)=1xf(x)=\frac{1}{x}. Describe the transformations that change f(x)f(x) to g(x)=f(x)−2.g(x)=f(x)-2.

  1. Shift the graph left 2 units
  2. Shift the graph down 2 units (correct answer)
  3. Compress the graph vertically by a factor of 2
  4. Shift the graph up 2 units
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x) + k, k·f(x), f(kx), and f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x) + k shifts the graph vertically (up if k > 0, down if k < 0), (2) k·f(x) stretches vertically if |k| > 1 or compresses if 0 < |k| < 1 (and reflects across x-axis if k < 0), (3) f(x + k) shifts horizontally—LEFT if k > 0, RIGHT if k < 0 (opposite of what you might expect!), (4) f(kx) compresses horizontally if |k| > 1 or stretches if 0 < |k| < 1 (and reflects across y-axis if k < 0). Since g(x) = f(x) - 2, we have the form f(x) + k where k = -2, which means we're adding -2 to all y-values, shifting every point on the graph down by 2 units. Choice A correctly identifies this as shifting the graph down 2 units. Choice B incorrectly suggests shifting up (which would be f(x) + 2), while C confuses vertical with horizontal shifts, and D mistakes it for compression. Outside the function (f(x) + k and k·f(x)) affects y-values/vertical; inside the function (f(x + k) and f(kx)) affects x-values/horizontal.

Question 11

On the coordinate plane, the graph of f(x)=∣x∣f(x)=|x| is shown along with the graph of g(x)g(x) (a V-shape) whose vertex is at (2,0)(2,0) and which has the same shape as f(x)f(x). Which function could be g(x)g(x)?

  1. g(x)=f(x+2)g(x)=f(x+2)
  2. g(x)=f(x)−2g(x)=f(x)-2
  3. g(x)=f(x−2)g(x)=f(x-2) (correct answer)
  4. g(x)=2f(x)g(x)=2f(x)
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x) + k, k·f(x), f(kx), and f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x) + k shifts the graph vertically (up if k > 0, down if k < 0), (2) k·f(x) stretches vertically if |k| > 1 or compresses if 0 < |k| < 1 (and reflects across x-axis if k < 0), (3) f(x + k) shifts horizontally—LEFT if k > 0, RIGHT if k < 0 (opposite of what you might expect!), (4) f(kx) compresses horizontally if |k| > 1 or stretches if 0 < |k| < 1 (and reflects across y-axis if k < 0). Outside the function (f(x) + k and k·f(x)) affects y-values/vertical; inside the function (f(x + k) and f(kx)) affects x-values/horizontal. The vertex moving to (2,0) without shape change indicates a horizontal shift right by 2, which f(x-2) achieves by adjusting inputs—brilliant connection! Choice C correctly identifies g(x) = f(x-2). Choice A would shift left instead, landing the vertex at (-2,0)—remember the opposite sign for horizontal shifts! Transformation memory aid: think 'outside affects y, inside affects x.' Anything added/multiplied OUTSIDE f (like f(x) + 3 or 2f(x)) changes y-values (vertical effects). Anything done INSIDE the parentheses (like f(x + 3) or f(2x)) changes x-values (horizontal effects). The tricky part: horizontal shifts are opposite to the sign—f(x + 3) shifts LEFT 3 because you're subtracting 3 from x-coordinates, and f(x - 2) shifts RIGHT 2. Think: what x-value gives the original function's behavior?

Question 12

Let the parent function be f(x)=x2f(x)=x^2. What transformation changes f(x)f(x) to g(x)=f(x+2)g(x)=f(x+2)?

  1. Shift the graph right 2 units
  2. Shift the graph left 2 units (correct answer)
  3. Shift the graph up 2 units
  4. Compress the graph horizontally by a factor of 2
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x) + k, k·f(x), f(kx), and f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x) + k shifts the graph vertically (up if k > 0, down if k < 0), (2) k·f(x) stretches vertically if |k| > 1 or compresses if 0 < |k| < 1 (and reflects across x-axis if k < 0), (3) f(x + k) shifts horizontally—LEFT if k > 0, RIGHT if k < 0 (opposite of what you might expect!), (4) f(kx) compresses horizontally if |k| > 1 or stretches if 0 < |k| < 1 (and reflects across y-axis if k < 0). Since g(x) = f(x + 2), we have x + 2 inside the function, which shifts the graph LEFT by 2 units (counterintuitively, adding inside shifts left). Choice B correctly identifies this as shifting the graph left 2 units. Choice A incorrectly suggests shifting right (which would be f(x - 2)), while C confuses horizontal with vertical shifts, and D mistakes it for compression. The tricky part: horizontal shifts are opposite to the sign—f(x + 3) shifts LEFT 3 because you're subtracting 3 from x-coordinates, and f(x - 2) shifts RIGHT 2. Think: what x-value gives the original function's behavior?

Question 13

Show the symmetry type of f(x)=x3−xf(x)=x^3-x by identifying it as even, odd, or neither.

  1. Even, because f(−x)=f(x)f(-x)=f(x)
  2. Odd, because f(−x)=−f(x)f(-x)=-f(x) (correct answer)
  3. Neither, because it has both even and odd powers
  4. Even, because it is symmetric about the origin
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x) + k, k·f(x), f(kx), and f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Even functions have y-axis symmetry: f(-x) = f(x) for all x, meaning the left half of the graph is a mirror image of the right half. Examples include f(x) = x², x⁴, |x|, and x² + 3. Odd functions have origin symmetry: f(-x) = -f(x), meaning rotating the graph 180° about the origin gives the same graph. Examples include f(x) = x, x³, 1/x, and x³ - x. Most functions are neither even nor odd! To test f(x) = x³ - x, we find f(-x) = (-x)³ - (-x) = -x³ + x = -(x³ - x) = -f(x), confirming the function is odd since f(-x) equals -f(x). Choice B correctly identifies this as odd because f(-x) = -f(x). A common error is thinking that having both even and odd powers makes a function neither even nor odd, but the actual test is whether f(-x) equals f(x) or -f(x). For even/odd testing: (1) Take the given function f(x), (2) Find f(-x) by substituting -x for every x (use parentheses!), (3) Simplify completely, (4) Compare with f(x) and -f(x): if f(-x) = f(x), it's even; if f(-x) = -f(x), it's odd; if neither match, it's neither.

Question 14

Let f(x)=xf(x)=\sqrt{x} (defined for x≥0x\ge 0). Consider g(x)=f(x3).g(x)=f\left(\frac{x}{3}\right). How does the graph of gg compare to the graph of ff?

  1. Shift right 3 units
  2. Vertical stretch by factor 33
  3. Horizontal compression by factor 33
  4. Horizontal stretch by factor 33 (correct answer)
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x)+kf(x) + k, k⋅f(x)k \cdot f(x), f(kx)f(kx), and f(x+k)f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x)+kf(x) + k shifts the graph vertically (up if k>0k > 0, down if k<0k < 0), (2) k⋅f(x)k \cdot f(x) stretches vertically if ∣k∣>1|k| > 1 or compresses if 0<∣k∣<10 < |k| < 1 (and reflects across x-axis if k<0k < 0), (3) f(x+k)f(x + k) shifts horizontally—LEFT if k>0k > 0, RIGHT if k<0k < 0 (opposite of what you might expect!), (4) f(kx)f(kx) compresses horizontally if ∣k∣>1|k| > 1 or stretches if 0<∣k∣<10 < |k| < 1 (and reflects across y-axis if k<0k < 0). Outside the function (f(x)+kf(x) + k and k⋅f(x)k \cdot f(x)) affects y-values/vertical; inside the function (f(x+k)f(x + k) and f(kx)f(kx)) affects x-values/horizontal. For g(x)=f(x/3)g(x) = f(x/3) with f(x)=xf(x) = \sqrt{x}, this is equivalent to f((1/3)x)f((1/3)x), stretching the graph horizontally by a factor of 3 since ∣1/3∣<1|1/3| < 1, meaning x-values need to be three times larger to produce the same y (e.g., g(3)=f(1)=1g(3) = f(1) = 1, while f(3)≈1.73f(3) \approx 1.73). Choice B correctly identifies this as a horizontal stretch by factor 3. A distractor like choice A confuses stretch with compression—when the coefficient inside is less than 1 in absolute value, it's a stretch, not compression; the opposite holds for ∣k∣>1|k| > 1. Transformation memory aid: think 'outside affects y, inside affects x.' Anything added/multiplied OUTSIDE f (like f(x)+3f(x) + 3 or 2f(x)2f(x)) changes y-values (vertical effects). Anything done INSIDE the parentheses (like f(x+3)f(x + 3) or f(2x)f(2x)) changes x-values (horizontal effects). The tricky part: horizontal shifts are opposite to the sign—f(x+3)f(x + 3) shifts LEFT 3 because you're subtracting 3 from x-coordinates, and f(x−2)f(x - 2) shifts RIGHT 2. Think: what x-value gives the original function's behavior?

Question 15

Let the parent function be f(x)=x2f(x)=x^2. Which function represents a shift of the graph of ff right 3 units and down 2 units?

  1. g(x)=f(x+3)+2g(x)=f(x+3)+2
  2. g(x)=f(x−3)−2g(x)=f(x-3)-2 (correct answer)
  3. g(x)=f(x+3)−2g(x)=f(x+3)-2
  4. g(x)=f(x−3)+2g(x)=f(x-3)+2
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x) + k, k·f(x), f(kx), and f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x) + k shifts the graph vertically (up if k > 0, down if k < 0), (2) k·f(x) stretches vertically if |k| > 1 or compresses if 0 < |k| < 1 (and reflects across x-axis if k < 0), (3) f(x + k) shifts horizontally—LEFT if k > 0, RIGHT if k < 0 (opposite of what you might expect!), (4) f(kx) compresses horizontally if |k| > 1 or stretches if 0 < |k| < 1 (and reflects across y-axis if k < 0). Outside the function (f(x) + k and k·f(x)) affects y-values/vertical; inside the function (f(x + k) and f(kx)) affects x-values/horizontal. To shift right 3 units, we need f(x - 3) (minus inside means right), and to shift down 2 units, we subtract 2 outside: g(x) = f(x - 3) - 2. Choice B correctly shows g(x) = f(x - 3) - 2 for a shift right 3 and down 2. Students often mix up the signs for horizontal shifts, thinking f(x + 3) shifts right, but remember: inside transformations work opposite to intuition. Transformation memory aid: think 'outside affects y, inside affects x.' Anything added/multiplied OUTSIDE f (like f(x) + 3 or 2f(x)) changes y-values (vertical effects). Anything done INSIDE the parentheses (like f(x + 3) or f(2x)) changes x-values (horizontal effects).

Question 16

Let f(x)=xf(x)=\sqrt{x}. Which function represents a horizontal shift of the graph of ff right by 3 units?

  1. g(x)=f(x+3)g(x)=f(x+3)
  2. g(x)=f(x)−3g(x)=f(x)-3
  3. g(x)=f(x−3)g(x)=f(x-3) (correct answer)
  4. g(x)=3f(x)g(x)=3f(x)
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x) + k, k·f(x), f(kx), and f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x) + k shifts the graph vertically (up if k > 0, down if k < 0), (2) k·f(x) stretches vertically if |k| > 1 or compresses if 0 < |k| < 1 (and reflects across x-axis if k < 0), (3) f(x + k) shifts horizontally—LEFT if k > 0, RIGHT if k < 0 (opposite of what you might expect!), (4) f(kx) compresses horizontally if |k| > 1 or stretches if 0 < |k| < 1 (and reflects across y-axis if k < 0). Outside the function (f(x) + k and k·f(x)) affects y-values/vertical; inside the function (f(x + k) and f(kx)) affects x-values/horizontal. To shift f(x) = √x right by 3 units, replace x with (x - 3), resulting in g(x) = f(x - 3) = √(x - 3), which moves the graph so the domain starts at x=3 instead of x=0. Choice C correctly identifies g(x) = f(x - 3) as the right shift by 3. A common error, as in choice A, is forgetting the opposite sign for horizontal shifts—f(x + 3) shifts left by 3, not right, which would make the domain start at x=-3. Transformation memory aid: think 'outside affects y, inside affects x.' Anything added/multiplied OUTSIDE f (like f(x) + 3 or 2f(x)) changes y-values (vertical effects). Anything done INSIDE the parentheses (like f(x + 3) or f(2x)) changes x-values (horizontal effects). The tricky part: horizontal shifts are opposite to the sign—f(x + 3) shifts LEFT 3 because you're subtracting 3 from x-coordinates, and f(x - 2) shifts RIGHT 2. Think: what x-value gives the original function's behavior?

Question 17

Let the parent function be f(x)=1xf(x)=\dfrac{1}{x}. What is the effect of replacing f(x)f(x) with g(x)=f(−x)g(x)=f(-x)?​

  1. Reflect across the xx-axis
  2. Reflect across the yy-axis (correct answer)
  3. Shift left 1 unit
  4. Vertical stretch by a factor of 2
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x) + k, k·f(x), f(kx), and f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x) + k shifts the graph vertically (up if k > 0, down if k < 0), (2) k·f(x) stretches vertically if |k| > 1 or compresses if 0 < |k| < 1 (and reflects across x-axis if k < 0), (3) f(x + k) shifts horizontally—LEFT if k > 0, RIGHT if k < 0 (opposite of what you might expect!), (4) f(kx) compresses horizontally if |k| > 1 or stretches if 0 < |k| < 1 (and reflects across y-axis if k < 0). In g(x) = f(-x), we're replacing x with -x inside the function, which reflects the graph across the y-axis—every point (x, y) becomes (-x, y). Choice B correctly identifies this as a reflection across the y-axis. Choice A incorrectly suggests x-axis reflection (that would be -f(x)), while C and D confuse reflections with shifts or stretches. Transformation memory aid: think 'outside affects y, inside affects x.' The negative sign inside f(-x) affects x-values, creating a y-axis reflection where left and right sides swap.

Question 18

Let the parent function be f(x)=xf(x)=\sqrt{x}. What is the effect on the graph of replacing f(x)f(x) with f(x2)f\left(\frac{x}{2}\right)?

  1. Vertical stretch by a factor of 2
  2. Shift right 2 units
  3. Horizontal compression by a factor of 2
  4. Horizontal stretch by a factor of 2 (correct answer)
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x) + k, k·f(x), f(kx), and f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x) + k shifts the graph vertically (up if k > 0, down if k < 0), (2) k·f(x) stretches vertically if |k| > 1 or compresses if 0 < |k| < 1 (and reflects across x-axis if k < 0), (3) f(x + k) shifts horizontally—LEFT if k > 0, RIGHT if k < 0 (opposite of what you might expect!), (4) f(kx) compresses horizontally if |k| > 1 or stretches if 0 < |k| < 1 (and reflects across y-axis if k < 0). When we replace f(x) with f(x/2), we're dividing the input by 2, which means to get the same y-value, we need twice the original x-value—this stretches the graph horizontally by a factor of 2. Choice A correctly identifies this as a horizontal stretch by a factor of 2. Choice B incorrectly calls it a compression (which would be f(2x)), while C confuses horizontal with vertical transformations, and D mistakes it for a shift. Inside the function (f(x + k) and f(kx)) affects x-values/horizontal—when k < 1 inside, we stretch horizontally.

Question 19

Let the parent function be f(x)=∣x∣f(x)=|x|. What transformation changes f(x)f(x) to g(x)=−f(x)g(x)=-f(x)?​

  1. Reflect across the xx-axis (correct answer)
  2. Reflect across the yy-axis
  3. Shift down 1 unit
  4. Horizontal compression by a factor of 2
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x) + k, k·f(x), f(kx), and f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x) + k shifts the graph vertically (up if k > 0, down if k < 0), (2) k·f(x) stretches vertically if |k| > 1 or compresses if 0 < |k| < 1 (and reflects across x-axis if k < 0), (3) f(x + k) shifts horizontally—LEFT if k > 0, RIGHT if k < 0 (opposite of what you might expect!), (4) f(kx) compresses horizontally if |k| > 1 or stretches if 0 < |k| < 1 (and reflects across y-axis if k < 0). In g(x) = -f(x), we're multiplying all output values by -1, which reflects the graph across the x-axis—every point (x, y) becomes (x, -y). Choice A correctly identifies this as a reflection across the x-axis. Choice B incorrectly suggests y-axis reflection (that would be f(-x)), while C and D confuse reflections with shifts or compressions. Transformation memory aid: negative sign outside the function (-f(x)) flips y-values, creating x-axis reflection; negative sign inside the function (f(-x)) flips x-values, creating y-axis reflection.

Question 20

Let the parent function be f(x)=x2f(x)=x^2. What transformation changes f(x)f(x) to g(x)=f(x)−3g(x)=f(x)-3?

  1. Shift the graph up 3 units
  2. Shift the graph down 3 units (correct answer)
  3. Shift the graph right 3 units
  4. Stretch the graph vertically by a factor of 3
Explanation: This question tests your understanding of how algebraic transformations of functions—like f(x) + k, k·f(x), f(kx), and f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x) + k shifts the graph vertically (up if k > 0, down if k < 0), (2) k·f(x) stretches vertically if |k| > 1 or compresses if 0 < |k| < 1 (and reflects across x-axis if k < 0), (3) f(x + k) shifts horizontally—LEFT if k > 0, RIGHT if k < 0 (opposite of what you might expect!), (4) f(kx) compresses horizontally if |k| > 1 or stretches if 0 < |k| < 1 (and reflects across y-axis if k < 0). Since g(x) = f(x) - 3, we have the form f(x) + k where k = -3, which means we're adding -3 to all y-values, shifting every point on the graph down by 3 units. Choice B correctly identifies this as shifting the graph down 3 units. Choice A incorrectly suggests shifting up (which would be f(x) + 3), while choices C and D confuse this with vertical stretching or horizontal shifting. Transformation memory aid: think 'outside affects y, inside affects x.' Anything added/multiplied OUTSIDE f (like f(x) + 3 or 2f(x)) changes y-values (vertical effects).