📝 Graph translations horizontal vertical shifts (16 MCQs)

📖 From Calculus • 1. Basics before calculus • 16 questions available

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What is Graph translations horizontal vertical shifts?

Definition:
Graph translations involve shifting the entire graph of a function horizontally or vertically without changing its shape, where $y = f(x-h)$ shifts right by $h$ units, $y = f(x+h)$ shifts left, $y = f(x)+k$ shifts up by $k$ , and $y = f(x)-k$ shifts down.

Example:
For $f(x)=x^2$ , $y=(x-2)^2$ shifts right 2 units, and $y=x^2+3$ shifts up 3 units.

Reason:
Translations model real-world movements like shifting a camera view or adjusting a baseline, making graphs adaptable to new initial conditions.

Easy

Medium

Hard

📝 All Graph translations horizontal vertical shifts MCQs

Q1. If the graph of $y = f(x)$ is reflected about the y‑axis, which transformation yields the resulting equation?

y = f(-x)

✅

y = -f(x)

y = f(x-1)

y = f(x)+1

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Reflecting about the y‑axis replaces each x‑coordinate with its negative, so the new equation is $y = f(-x)$ . The other options describe different transformations such as reflection about the x‑axis or translations, which do not match the described reflection.

Q2. What is the effect of replacing $x$ by $(x-3)$ in the function $y = f(x)$ ?

A.Shift right 3 units

B.Shift left 3 units ✅

C.Shift up 3 units

D.Shift down 3 units

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Replacing $x$ with $(x-3)$ moves every point three units to the right because the input must increase by three to produce the same output as before. The other choices correspond to different directions or vertical movements, which are not produced by this substitution.

Q3. Consider the function $y = \sqrt{x}$ . Which sequence of transformations produces the graph of $y = 2\sqrt{-(x+4)}+1$ ?

A.Reflect about the y‑axis, shift left 4, vertical stretch by 2, shift up 1 ✅

B.Shift right 4, reflect about the y‑axis, vertical stretch by 2, shift up 1

C.Reflect about the x‑axis, shift left 4, vertical stretch by 2, shift up 1

D.Reflect about the y‑axis, shift right 4, vertical stretch by 2, shift down 1

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: First reflect $\sqrt{x}$ about the y‑axis to get $\sqrt{-x}$ . Then shift the graph left 4 units, giving $\sqrt{-(x+4)}$ . A vertical stretch by factor 2 multiplies the output, and finally adding 1 translates the graph upward. This matches the target expression.

Q4. Which of the following graphs represents a horizontal translation of $y = (x-2)^2$ by 3 units to the left?

y = (x-5)^2

y = (x+1)^2

✅

y = (x-2)^2

y = (x-8)^2

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Moving the graph left by 3 replaces $x$ with $(x+3)$ . Substituting into $(x-2)^2$ yields $(x+3-2)^2 = (x+1)^2$ . The other options correspond to different shifts or no shift at all.

Q5. Suppose the graph of $y = f(x)$ is first reflected about the x‑axis and then translated upward by 4 units. Which equation represents the resulting graph?

y = -f(x)+4

y = -f(x-4)

y = f(-x)+4

✅

y = -f(x-4)+4

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Reflecting about the x‑axis changes the sign of the output, giving $-f(x)$ . Adding 4 to the whole expression then shifts the graph upward by 4 units, resulting in $y = -f(x)+4$ . The option $y = f(-x)+4$ incorrectly reflects about the y‑axis.

Q6. In Table 0.2.3, which entry corresponds to the transformation $y = -\sqrt{x}$ ?

y = -\sqrt{x}

✅

y = \sqrt{-x}

y = -\sqrt{-x}

y = \sqrt{x}

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The entry $y = -\sqrt{x}$ explicitly shows the negative sign applied to the square‑root function, indicating a reflection about the x‑axis. The other entries involve a reflection about the y‑axis or no sign change.

Q7. The function $g(x) = 3\sqrt{2 - x}$ can be obtained from $h(x) = \sqrt{x}$ by a combination of reflections and translations. Which ordered pair $(a,b)$ correctly describes the horizontal shift (a) and vertical shift (b) applied after reflecting about the y‑axis?

(-2,0)

(2,0)

✅

(-2,0)

(2,0)

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Reflecting $\sqrt{x}$ about the y‑axis gives $\sqrt{-x}$ . Translating this graph right 2 units replaces $x$ with $(x-2)$ , producing $\sqrt{-(x-2)} = \sqrt{2-x}$ . No vertical shift occurs, so the ordered pair is $(2,0)$ .

Q8. Compare the graphs of $y = f(x)$ and $y = f(2x-4)$ . Which statement correctly describes the combined effect on the original graph?

A.Horizontal compression by factor 2 and shift right 2 units ✅

B.Horizontal stretch by factor 2 and shift left 4 units

C.Shift right 2 units then horizontal compression by factor 2

D.Horizontal compression by factor 2 then shift right 2 units

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Rewrite $f(2x-4)$ as $f\big(2(x-2)\big)$ . This shows a horizontal shift right 2 (replace $x$ by $x-2$ ) followed by a compression by a factor of 2 (multiply the input by 2). The order of operations matches option A.

Q9. If a point $(5,-3)$ lies on the graph of $y = f(x)$ , what point must lie on the graph of $y = -f(2 - x)$ ?

(1,-3)

(-3,-3)

(-3,3)

✅

(1,3)

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: For the new graph we need $2 - x' = 5$ giving $x' = -3$ . Then $y' = -f(5) = -(-3) = 3$ . Hence the point $(-3,3)$ must appear on the transformed graph.

Q10. Which of the following sequences will transform the graph of $y = x^{2}$ into the graph of $y = -(x+1)^{2}+2$ ?

A.Shift left 1, reflect about the x‑axis, shift up 2

B.Reflect about the x‑axis, shift left 1, shift up 2

C.Shift left 1, shift up 2, reflect about the x‑axis

D.Shift left 1, reflect about the x‑axis, shift up 2 ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: First shift the parabola left 1 unit to obtain $(x+1)^{2}$ . Reflecting about the x‑axis changes the sign, giving $- (x+1)^{2}$ . Finally, adding 2 moves the graph upward, resulting in $- (x+1)^{2}+2$ .

Q11. Two functions are defined by $y = f(x) = (x-3)^{2}$ and $y = g(x) = -(x+2)^{2}$ . Which transformation relates the graph of $g$ to the graph of $f$ ?

A.Reflect about the y‑axis and shift right 5

B.Reflect about the x‑axis and shift right 5

C.Reflect about the x‑axis and shift left 5 ✅

D.Reflect about the y‑axis and shift left 5

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Starting from $f(x)$ , reflect about the x‑axis to get $- (x-3)^{2}$ . Then shift the resulting graph left 5 units, replacing $x$ by $(x+5)$ , which yields $- (x+2)^{2}$ . This matches $g(x)$ .

Q12. Given that the graph of $y = f(x)$ passes through $(1,2)$ , which of the following points will be on the graph of $y = f(-x+4)$ ?

(3,2)

✅

(-3,2)

(1,2)

(5,2)

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Set $-x+4 = 1$ to find the x‑coordinate that produces the same output: $-x = -3$ so $x = 3$ . The y‑value remains 2, giving the point $(3,2)$ on the transformed graph.

Q13. What is the effect of multiplying the entire function by $-1$ , i.e., forming $y = -f(x)$ ?

A.Reflection about the x‑axis

B.Reflection about the y‑axis

C.Shift up 1 unit

D.Shift down 1 unit ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: Multiplying the output by $-1$ flips every y‑coordinate to its opposite, which is precisely a reflection of the original graph across the x‑axis. The other options describe different transformations that are not produced by this operation.

Q14. A function $h$ is defined by $h(x) = \sqrt{x}$ . After applying a horizontal translation of 5 units left and a vertical translation of 3 units up, the resulting function is $k(x)$ . Which expression correctly represents $k(x)$ ?

\sqrt{x+5}+3

✅

\sqrt{x-5}+3

\sqrt{x+5}-3

\sqrt{x-5}-3

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Translating left 5 replaces $x$ with $(x+5)$ . Adding 3 to the whole function then moves the graph upward by 3 units. Thus the new function is $\sqrt{x+5}+3$ .

Q15. The graph of $y = f(x)$ is reflected about the line $y = x$ , then translated 2 units down. Which of the following equations represents the new graph?

y = f(x) - 2

y = f^{-1}(x) - 2

✅

y = f(x) + 2

y = f^{-1}(x) + 2

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Reflecting about $y = x$ interchanges the roles of $x$ and $y$ , producing the inverse function $f^{-1}(x)$ . Translating the resulting graph down by 2 units subtracts 2 from the output, giving $y = f^{-1}(x) - 2$ .

Q16. If the graph of $y = f(x)$ is shifted 3 units right and then reflected about the x‑axis, which equation represents the resulting graph?

y = -f(x-3)

✅

y = -f(x+3)

y = f(-(x-3))

y = -f(-(x-3))

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Shifting right 3 replaces $x$ by $(x-3)$ , yielding $f(x-3)$ . Reflecting about the x‑axis multiplies the output by $-1$ , giving the final equation $y = -f(x-3)$ . The other options involve incorrect signs or additional reflections.

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