📝 Power functions y = x^n examples (14 MCQs)

📖 From Calculus • 1. Basics before calculus • 14 questions available

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What is Power functions y = x^n examples?

Definition:
Power functions are of the form $y = x^n$ , where $n$ is a constant real number, with behavior depending on $n$ : for $n>0$ , they pass through (0,0) and (1,1); for $n<0$ , they have asymptotes; and even/odd $n$ determine symmetry.

Example:
For $n=2$ , $y=x^2$ is a parabola (even, increasing for $x>0$ ); for $n=-1$ , $y=1/x$ is a hyperbola (odd, asymptotes at axes).

Reason:
Power functions are fundamental building blocks for polynomials and many natural laws, like area ( $r^2$ ) or inverse square laws ( $1/r^2$ ).

Easy

Medium

Hard

📝 All Power functions y = x^n examples MCQs

Q1. If the exponent n increases while keeping x between -1 and 1, what happens to the value of y = x^n?

A.y increases

B.y decreases ✅

C.y stays the same

D.y becomes negative

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Because |x|<1, raising it to a larger power makes its absolute value smaller. Thus the output y gets closer to zero, i.e., it decreases. The sign does not change, so the correct choice is that y decreases.

Q2. For even n, the function f(x)=x^n is even. Which of the following statements follows directly from this property?

A.Graph is symmetric about the x‑axis

B.Graph is symmetric about the y‑axis ✅

C.Graph passes through (1,‑1)

D.Graph is periodic

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: An even function satisfies f(‑x)=f(x), which geometrically means the graph is mirrored across the y‑axis. No symmetry about the x‑axis or periodicity is implied, and the point (1,‑1) does not lie on the graph for even powers. Hence the graph is symmetric about the y‑axis.

Q3. Given a fixed x with 0 < x < 1, how does the sign of the derivative f'(x)=n x^{n-1} change as n increases?

A.Derivative becomes more positive

B.Derivative becomes more negative

C.Derivative magnitude decreases ✅

D.Derivative magnitude increases

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: For 0<x<1, the factor x^{n-1} shrinks as n grows, making the whole product n x^{n-1} smaller in magnitude. Since n is positive, the derivative remains positive but its size diminishes, so the magnitude decreases.

Q4. Consider the two functions y = x^3 and y = x^5. Apart from the origin, how many additional points do their graphs intersect?

A.None

B.One

C.Two ✅

D.Infinitely many

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Setting x^3 = x^5 gives x^3(1‑x^2)=0, so x=0 or x=±1. The points (‑1,‑1) and (1,1) are additional intersections, giving two extra points beyond the origin.

Q5. Which of the following correctly describes the relative heights of y = x^2 and y = x^4 for x in the interval (0,1)?

A.y = x^2 is above y = x^4 ✅

B.y = x^4 is above y = x^2

C.Both graphs coincide

D.They cross each other

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: When 0<x<1, raising x to a higher even power makes it smaller. Therefore x^4 < x^2 throughout the interval, so the parabola y = x^2 lies above the quartic y = x^4.

Q6. As the odd exponent n increases (n = 3,5,7,…), which statement best describes the steepness of the graph of y = x^n for x > 1?

A.Steepness decreases

B.Steepness remains constant

C.Steepness increases ✅

D.Graph becomes horizontal

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: For x>1, each additional factor of x (>1) multiplies the value, so x^n grows faster as n grows. This makes the curve rise more sharply, i.e., its steepness increases with larger odd exponents.

Q7. Which pair correctly matches the parity of n with the symmetry of its graph?

A.Even n → symmetric about origin

B.Odd n → symmetric about y‑axis

C.Even n → symmetric about y‑axis ✅

D.Odd n → no symmetry

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Even powers satisfy f(‑x)=f(x), giving symmetry about the y‑axis. Odd powers satisfy f(‑x)=‑f(x), giving symmetry about the origin, not the y‑axis. Thus the correct pairing is even n with y‑axis symmetry.

Q8. For f_n(x)=x^n, the slope at x=2 is f'_n(2)=n·2^{\,n-1}. What is the ratio of the slope for n=5 to the slope for n=3?

A.05-Mar

B.20-Dec

C.51471

D.29556 ✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: Compute each slope: for n=5, 5·2^4 = 5·16 = 80; for n=3, 3·2^2 = 3·4 = 12. The ratio is 80/12, which simplifies to 20/3 but the exact fraction given among the options is 80/12.

Q9. Which graph is guaranteed to be symmetric about the y‑axis?

A.y = x^3

B.y = x^5

C.y = x^2 ✅

D.y = x

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Only even powers produce even functions, which are symmetric about the y‑axis. Among the choices, y = x^2 has an even exponent, so its graph reflects that symmetry.

Q10. If 0 < a < 1, what happens to a^n as n becomes very large?

A.It approaches infinity

B.It approaches zero ✅

C.It approaches one

D.It oscillates

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: When the base is a fraction between zero and one, repeatedly multiplying it by itself drives the product toward zero. Hence as the exponent n grows without bound, a^n tends to zero.

Q11. Based on the trend of even‑power graphs becoming flatter on (‑1,1) as n grows, what qualitative shape would you expect for y = x^{1/2} (the square‑root function) compared to y = x?

A.More curved upward near the origin

B.Flatter near the origin and steeper for large x ✅

C.Identical to y = x

D.Symmetric about the y‑axis

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Even‑power graphs flatten near the origin because the exponent >1 reduces values in (‑1,1). The square‑root exponent ½ is less than 1, which has the opposite effect: the curve is less steep near the origin and becomes steeper as x increases, matching option B.

Q12. Consider the limit $\displaystyle \lim_{n\to\infty} x^{\,n}$ for a fixed real x. Which statement is true?

A.Limit is 0 for |x|<1 and diverges for |x|>1 ✅

B.Limit is 1 for all x

C.Limit is undefined for all x

D.Limit is 0 for |x|>1 and 1 for |x|<1

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: If |x|<1, successive powers shrink toward zero; if |x|>1, they grow without bound; and if |x|=1, the value stays at 1. Thus the limit is zero for |x|<1 and diverges (tends to infinity) for |x|>1.

Q13. In the expression y = x^{n}, the constant n is called the _____ .

A.base

B.coefficient

C.exponent ✅

D.parameter

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The constant that indicates how many times the base x is multiplied by itself is the exponent. It determines the power to which x is raised, so the correct term is exponent.

Q14. Treating n as a parameter, which of the following best describes the family of curves \{y = x^{n}\} as n varies over the integers?

A.All curves are translations of each other

B.All curves intersect at (0,0) and (1,1) ✅

C.All curves share the same slope at x=0

D.All curves are rotations of y = x

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Regardless of the integer exponent, each curve passes through the origin (0,0) and the point (1,1) because 0^{n}=0 and 1^{n}=1 for any n. No other universal intersection or rotation property holds, making option B the accurate description.

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