📝 Graph reflections over x axis y axis (11 MCQs)

📖 From Calculus • 1. Basics before calculus • 11 questions available

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What is Graph reflections over x axis y axis?

Definition:
Graph reflections flip the graph over an axis: $y = -f(x)$ reflects over the x-axis (changing signs of outputs), while $y = f(-x)$ reflects over the y-axis (changing signs of inputs), mirroring the original graph.

Example:
For $f(x) = x^3$ , $y = -x^3$ is a reflection over x-axis, and $y = (-x)^3 = -x^3$ is same here, but for $f(x)=x^2$ , $y = (-x)^2 = x^2$ (no change due to evenness).

Reason:
Reflections are used in optics, signal processing, and symmetry analysis, helping to visualize inverse relationships or transformed data.

Easy

Medium

Hard

📝 All Graph reflections over x axis y axis MCQs

Q1. What is the equation of the graph obtained by reflecting $y = |x|$ across the x‑axis?

y = -|x|

✅

y = |x|

y = -x

y = x

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Reflecting across the x‑axis changes the sign of every y‑coordinate while leaving x unchanged. The absolute‑value function $y = |x|$ becomes its vertical mirror, giving $y = -|x|$ . The other options either leave the graph unchanged or represent different transformations, so A is correct.

Q2. Starting with $y = f(x)$ , the graph is first shifted 3 units left and then reflected about the y‑axis. Which expression represents the resulting function?

y = f(x+3)

y = f(-x-3)

✅

y = f(-x+3)

y = f(-x)

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: A left shift replaces x by $x+3$ ; a subsequent reflection about the y‑axis replaces x by $-x$ . Applying both operations yields $f(-(x+3)) = f(-x-3)$ . Options A, C, and D omit one of the transformations, leaving B as the only correct representation.

Q3. Which sequence of transformations yields the graph of $y = 4 - |x-2|$ from the basic graph $y = |x|$ ?

A.Translate right 2, reflect across y‑axis, translate up 4

B.Translate left 2, reflect across x‑axis, translate down 4

C.Translate right 2, reflect across x‑axis, translate up 4 ✅

D.Translate left 2, reflect across y‑axis, translate up 4

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: First move the parent graph right by 2, giving $|x-2|$ . Reflecting this across the x‑axis changes the sign, producing $-|x-2|$ . Finally, shifting the result upward by 4 units adds 4 to the y‑value, resulting in $4-|x-2|$ . Only option C follows this exact order.

Q4. How does the graph of $y = -\sqrt{x}$ compare to the graph of $y = \sqrt{x}$ ?

A.It is a horizontal reflection across the y‑axis.

B.It is a translation 1 unit upward.

C.It coincides with

y = \sqrt{x}

. ✅

D.It is a vertical reflection of

y = \sqrt{x}

across the x‑axis.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The function $\sqrt{x}$ lies above the x‑axis for $x\ge0$ . Multiplying the entire output by $-1$ flips every point vertically, producing a mirror image below the axis. This is precisely a reflection across the x‑axis. Options A and B describe different transformations, and option C is false because the graphs are distinct.

Q5. What are the coordinates of the vertex of the graph of $y = 4 - |x-2|$ ?

(0,4)

(2,4)

✅

(2,0)

(4,2)

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The expression $|x-2|$ attains its minimum value 0 when $x=2$ . Substituting this into the outer expression gives $y = 4 - 0 = 4$ . Thus the point $(2,4)$ is the vertex, the highest point of this V‑shaped graph. The other choices either misplace the x‑coordinate or the y‑coordinate.

Q6. Reflect the parabola $y = (x-1)^2$ about the line $y = x$ . Which equation represents the reflected curve?

y = (x+1)^2

x = (y+1)^2

x = (y-1)^2

✅

y = (x-1)^2

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Reflecting about $y = x$ swaps the roles of x and y. Starting with $y = (x-1)^2$ , interchange the variables to obtain $x = (y-1)^2$ . This equation describes the same set of points after the reflection. Options A and D leave the equation unchanged, and B uses the wrong sign inside the square.

Q7. If the graph of $y = |2x+4| + 2$ is first reflected about the x‑axis and then about the y‑axis, what is the resulting equation?

y = -|2x-4| - 2

✅

y = -|2x+4| - 2

y = |2x-4| + 2

y = -|2x-4| + 2

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Reflecting across the x‑axis changes the sign of y, giving $y = -|2x+4| - 2$ . A subsequent reflection across the y‑axis replaces x by $-x$ , turning $|2(-x)+4|$ into $|-2x+4| = |2x-4|$ . The final expression is therefore $y = -|2x-4| - 2$ , which matches option A.

Q8. Which of the following describes the transformation sequence that produces $y = -|2x+4| + 5$ from the parent graph $y = |x|$ ?

A.Horizontal stretch by factor 2, shift left 2, reflect across y‑axis, shift up 5

B.Horizontal compression by factor 1/2, shift left 2, reflect across x‑axis, shift up 5 ✅

C.Horizontal stretch by factor 2, shift right 2, reflect across y‑axis, shift up 5

D.Horizontal stretch by factor 2, shift left 2, reflect across x‑axis, shift up 5

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Inside the absolute value, $2x+4 = 2(x+2)$ indicates a horizontal compression by a factor of $1/2$ followed by a left shift of 2 units. The outer negative sign reflects the graph across the x‑axis, and the final $+5$ translates it upward. Option B lists these steps in the correct order.

Q9. Reflect the graph of $y = \sqrt{x}$ about the line $y = -x$ . Which expression gives the resulting curve?

y = -x^2

✅

y = -\sqrt{-x}

x = -\sqrt{-y}

y = \sqrt{-x}

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: A reflection across $y = -x$ swaps coordinates and changes both signs, turning $(x,\sqrt{x})$ into $(-\,\sqrt{x},-x)$ . Eliminating the parameter leads to the relation $y = -x^2$ , a downward‑opening parabola. The other options either retain a square‑root form or mix variables incorrectly.

Q10. Which statement correctly compares the graphs of $y = |x|$ reflected about the x‑axis and about the y‑axis?

A.Both reflections produce the same graph.

B.Reflection about the x‑axis yields

y = -|x|

; about the y‑axis yields

y = |x|

. ✅

C.Reflection about the x‑axis yields

y = -|x|

; about the y‑axis yields

y = -|x|

D.Reflection about the x‑axis yields

y = |x|

; about the y‑axis yields

y = -|x|

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Reflecting across the x‑axis changes the sign of the output, giving $y = -|x|$ , a V opening downward. Reflecting across the y‑axis replaces x by $-x$ , but because the absolute value eliminates the sign, the graph remains $y = |x|$ . Hence the two reflected graphs are different, matching option B.

Q11. A point $P(3,-2)$ is reflected across the line $y = x$ to a point $P'$ . What is the distance between $P$ and $P'$ ?

5\sqrt{2}

✅

5

2\sqrt{5}

\sqrt{13}

💡 Difficulty: medium | ✅ Correct: A

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