📝 Symmetry tests for graphs (13 MCQs)

📖 From Calculus • 1. Basics before calculus • 13 questions available

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What is Symmetry tests for graphs?

Definition:
Symmetry tests for graphs determine if a graph is symmetric about the y-axis (even function, $f(-x)=f(x)$ ), the origin (odd function, $f(-x)=-f(x)$ ), or the x-axis (not a function usually), by substituting $-x$ or $-y$ .

Example:
For $f(x)=x^4$ , $f(-x)=(-x)^4=x^4=f(x)$ , so symmetric about y-axis; for $f(x)=x^3$ , $f(-x)=-x^3=-f(x)$ , so symmetric about origin.

Reason:
Symmetry simplifies graphing by reducing effort to half, and reveals underlying properties like evenness/oddness, useful in Fourier series and integrals.

Easy

Medium

Hard

📝 All Symmetry tests for graphs MCQs

Q1. Replacing y by -y in an equation of a curve tests for symmetry about which axis?

A.x‑axis ✅

B.y‑axis

C.origin

D.none

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The substitution y→-y reflects points across the x‑axis. If the equation remains unchanged, every point (x,y) on the curve has a counterpart (x,-y), which is precisely the definition of symmetry about the x‑axis. Hence the correct axis is the x‑axis.

Q2. If a curve is symmetric about the y‑axis, which of the following must be true about the point (3, 4) on the curve?

A.The point (‑3, 4) is also on the curve ✅

B.The point (3, ‑4) is also on the curve

C.The point (‑3, ‑4) is also on the curve

D.None of these points must be on the curve

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Symmetry about the y‑axis means reflecting across that axis: (x,y) ↔ (‑x,y). Therefore, any point (3,4) on the curve forces the reflected point (‑3,4) to lie on the curve as well. The other options involve reflections across the x‑axis or the origin, which are not guaranteed.

Q3. Consider the curve defined by y = √x. After replacing x by –x, the resulting equation is not equivalent. What symmetry does this indicate?

A.Symmetric about the y‑axis

B.Symmetric about the x‑axis

C.Symmetric about the origin

D.No symmetry about the y‑axis ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: Replacing x with –x tests symmetry about the y‑axis. The original equation y = √x is defined only for x ≥ 0, while the transformed equation y = √(‑x) is undefined for positive x, so the two equations are not equivalent. Hence the curve lacks symmetry about the y‑axis.

Q4. Using Theorem 0.2.3, which substitution would you apply to test if the curve x = y³ is symmetric about the origin?

A.Replace x with –x only

B.Replace y with –y only

C.Replace both x and y with their negatives ✅

D.Replace x with y

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Theorem 0.2.3(c) states that origin symmetry is checked by substituting x→‑x and y→‑y simultaneously. Applying this to x = y³ gives –x = (‑y)³, which simplifies back to the original equation, confirming the test. Therefore the correct substitution involves both variables.

Q5. A curve satisfies the condition that replacing y by –y yields an equivalent equation, but replacing x by –x does not. Which symmetry does the curve possess?

A.Symmetric only about the x‑axis ✅

B.Symmetric only about the y‑axis

C.Symmetric about the origin

D.No symmetry

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: If the equation remains unchanged after y→‑y, the curve is symmetric about the x‑axis. Failure of equivalence after x→‑x indicates the curve is not symmetric about the y‑axis, and consequently not about the origin either. Thus the only guaranteed symmetry is about the x‑axis.

Q6. Compare the curves x = y² and x = –y². Which statement correctly describes their symmetry relationship?

A.They are reflections across the y‑axis ✅

B.They are identical curves

C.Both are symmetric about the x‑axis

D.One is symmetric about the origin while the other is not

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The graph of x = y² lies to the right of the y‑axis, while x = –y² lies to the left. Reflecting the first across the y‑axis (changing x to –x) produces the second, showing they are mirror images about the y‑axis. The other options mischaracterize their positions.

Q7. Given the equation x² = y, determine which symmetries are present using substitution tests.

A.Symmetric about the x‑axis only

B.Symmetric about the y‑axis only ✅

C.Symmetric about the origin only

D.Symmetric about both axes

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Replacing x with –x leaves x² unchanged, so the equation is unchanged, indicating symmetry about the y‑axis. Replacing y with –y gives x² = –y, which is not equivalent, so there is no x‑axis symmetry. Since both variables are not simultaneously negated to produce a new equation, origin symmetry does not follow.

Q8. The curve defined by x² + y² = 4 is tested for symmetry. Which set of substitutions all produce equivalent equations?

A.Only x → –x

B.Only y → –y

C.Both x → –x and y → –y together

D.All of the above ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: The circle equation is unchanged by changing x to –x (because x² stays the same), by changing y to –y (y² stays the same), and by changing both simultaneously. Hence each individual substitution and the combined substitution yield the original equation, making all three statements true.

Q9. Explain why a nonzero function f(x) cannot have a graph symmetric about the x‑axis.

A.Because f(‑x)=‑f(x) contradicts the definition

B.Because symmetry would require f(x)=‑f(x) for all x ✅

C.Because functions must pass the vertical line test

D.Because x‑axis symmetry implies multiple y values for the same x

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: If a graph were symmetric about the x‑axis, each point (x,f(x)) would have a counterpart (x,‑f(x)). For the same x, this forces f(x)=‑f(x), which can only hold if f(x)=0 for every x. Since the function is nonzero, such symmetry is impossible.

Q10. Suppose a curve satisfies both conditions: replacing x by –x yields an equivalent equation, and replacing y by –y also yields an equivalent equation. What can be concluded about its symmetry?

A.Symmetric about the origin only

B.Symmetric about both axes ✅

C.Symmetric about the x‑axis only

D.Symmetric about the y‑axis only

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: If the equation is unchanged after x→‑x, the curve is symmetric about the y‑axis. If it is also unchanged after y→‑y, it is symmetric about the x‑axis. Having both axis symmetries automatically implies symmetry about the origin as well, so the curve possesses both axes symmetries.

Q11. The curve y = x³ – x is examined for symmetry. After performing the substitution x→–x, the equation becomes y = –x³ + x. Which symmetry, if any, does this reveal?

A.Symmetric about the y‑axis

B.Symmetric about the x‑axis

C.Symmetric about the origin

D.No symmetry ✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: Testing y‑axis symmetry requires replacing x with –x. The transformed equation y = –x³ + x is not equivalent to the original y = x³ – x, so the curve lacks y‑axis symmetry. A similar test with y→‑y fails, and the combined substitution also fails, indicating the curve has no symmetry of the types considered.

Q12. A curve is known to be symmetric about the origin. Which of the following must also be true about its symmetry?

A.It must be symmetric about the x‑axis

B.It must be symmetric about the y‑axis

C.It must be symmetric about both axes

D.None of the above necessarily holds ✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: Origin symmetry means that (x,y) ↔ (‑x,‑y). This does not imply that reflecting only one coordinate yields the original curve, so the curve need not be symmetric about either axis individually. Therefore none of the other statements is guaranteed.

Q13. If a curve is symmetric about the y‑axis, what relationship holds between the points (a, b) and (c, d) on the curve?

A.c = –a and d = b ✅

B.c = a and d = –b

C.c = –a and d = –b

D.No relationship is required

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Symmetry about the y‑axis reflects a point across that axis, changing the sign of the x‑coordinate while leaving the y‑coordinate unchanged. Thus any point (a,b) on the curve guarantees the presence of its mirror (‑a,b). The other options describe reflections across different axes or the origin.

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