📝 Exponential functions properties and graphs (12 MCQs)

📖 From Calculus • 1. Basics before calculus • 12 questions available

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What is Exponential functions properties and graphs?

Definition:
Exponential functions are of the form $f(x) = a^x$ with $a>0, a \neq 1$ , having properties: $a^0 = 1$ , $a^{x+y} = a^x a^y$ , increasing if $a>1$ , decreasing if $0<a<1$ , and with horizontal asymptote $y=0$ .

Example:
$f(x)=2^x$ passes through (0,1), (1,2), (2,4), and is increasing; $g(x)=(1/2)^x$ is decreasing but also passes through (0,1).

Reason:
Exponential functions model growth/decay in populations, finance, and physics, with graphs that show rapid change, making them indispensable in applied mathematics.

Easy

Medium

Hard

📝 All Exponential functions properties and graphs MCQs

Q1. If $b^2 = 9$ for an exponential function $f(x)=b^x$ with $b>0$ , what is $f(-3)$ ?

\frac{1}{27}

✅

B.27

\frac{1}{9}

D.9

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Since $b^2=9$ gives $b=3$ . The value at $-3$ is $b^{-3}=3^{-3}=1/27$ . The other options correspond to incorrect manipulations of the exponent or base. Therefore the correct choice is $\frac{1}{27}$ .

Q2. Reflecting the graph of $y=(\tfrac12)^x$ about the y‑axis yields which function, and how does its growth for positive $x$ compare to the original?

y=(\tfrac12)^{-x}

y=2^{x}

✅

y=(\tfrac12)^{x}

y=2^{-x}

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Replacing $x$ by $-x$ gives $(\tfrac12)^{-x}=2^{x}$ . The new function $2^{x}$ increases as $x$ increases, whereas the original $(\tfrac12)^{x}$ decreases. Hence the reflected graph grows faster for positive $x$ . The other choices either leave the function unchanged or give a decreasing version.

Q3. For the exponential function $f(x)=b^{x}$ with $b=3$ , what is the horizontal asymptote of its graph?

y=1

y=3

y=0

✅

y=\infty

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Exponential graphs of the form $b^{x}$ approach zero as $x\to -\infty$ and never cross the x‑axis, making $y=0$ the horizontal asymptote regardless of the base $b$ (provided $b>0$ and $b\neq1$ ). The other options describe lines that are not asymptotes for this function.

Q4. The point $(4,16)$ lies on the graph of $y=b^{x}$ . What is the value of the base $b$ ?

A.4

B.16

\\sqrt{2}

D.2 ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: Substituting the point gives $b^{4}=16$ . Taking the fourth root yields $b=16^{1/4}=2$ . The other options either misinterpret the exponent or give incorrect roots. Thus the base is $2$ .

Q5. If an exponential function $q(x)=b^{x}$ satisfies $q(-1)=0.5$ , what is the base $b$ ?

A.0.5

B.2 ✅

\tfrac12

D.4

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The equation $b^{-1}=0.5$ implies $1/b=0.5$ , so $b=2$ . Option B correctly reflects this value. Options A and C are the reciprocals, and D is unrelated.

Q6. Which statement correctly describes the relative growth of $y=2^{x}$ and $y=3^{x}$ for $x>0$ ?

2^{x}

grows faster than

3^{x}

3^{x}

grows faster than

2^{x}

✅

C.Both grow at the same rate

D.Both decrease as

x

increases

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Because the base 3 is larger than the base 2, the function $3^{x}$ increases more rapidly for positive $x$ . This is evident from comparing values such as $2^{2}=4$ versus $3^{2}=9$ . The other options either reverse the relationship or incorrectly claim decreasing behavior.

Q7. Evaluate \ $\\displaystyle\\lim_{x\\to\\infty}\\left(\\tfrac12\\right)^{x}\$ .

A.1

B.\

\\infty\

C.\

-\\infty\

D.0 ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: The base \ $\\tfrac12\$ lies between 0 and 1, so raising it to larger and larger exponents drives the value toward zero. Formally, \ $\\left(\\tfrac12\\right)^{x}=e^{x\\ln(\\tfrac12)}\$ with \ $\\ln(\\tfrac12)<0\$ , yielding a limit of 0 as \ $x\\to\\infty\$ . The other choices misinterpret the behavior of a decaying exponential.

Q8. Which expression represents a vertical stretch of the graph of \ $y=b^{x}\$ by a factor of 3?

y=b^{3x}

y=(3b)^{x}

✅

y=3\\,b^{x}

y=b^{x+3}

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Multiplying the entire function by 3 scales every output value by 3, which is a vertical stretch. The expression \ $3\\,b^{x}\$ captures this effect. Options A and D change the exponent, altering the shape differently, while option B changes the base, also producing a different graph.

Q9. Replacing \ $x\$ by \ $x-2\$ in the equation \ $y=b^{x}\$ results in which transformation of the graph?

A.Shift right by 2 units ✅

B.Shift left by 2 units

C.Shift up by 2 units

D.Shift down by 2 units

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The substitution \ $x\\rightarrow x-2\$ moves every point of the original graph two units to the right because the function now reaches the same output value at an \ $x\$ that is two larger than before. Horizontal translations affect the input variable, not the output, so the vertical shift options are incorrect.

Q10. Let \ $f(x)=b^{x}\$ and \ $g(x)=c^{x}\$ with \ $b>c>1\$ . Which inequality correctly compares their derivatives at \ $x=0\$ ?

f&#039;(0)<g&#039;(0)

f&#039;(0)>g&#039;(0)

✅

f&#039;(0)=g&#039;(0)

D.Cannot be determined without specific values

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The derivative of \ $b^{x}\$ is \ $\\ln(b)\\,b^{x}\$ ; at \ $x=0\$ this simplifies to \ $\\ln(b)\$ . Since \ $b>c\$ and the natural logarithm is increasing, \ $\\ln(b)>\\ln(c)\$ . Therefore \ $f'(0)>\\!g'(0)\$ . The other options either reverse the inequality or claim equality without justification.

Q11. In the decay model \ $N(t)=N_{0}\\left(\\tfrac12\\right)^{t/5}\$ , what is the half‑life of the quantity?

A.2.5

B.10

C.1

D.5 ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: The expression \ $\\left(\\tfrac12\\right)^{t/5}\$ indicates that every increase of 5 time units halves the quantity. Therefore the half‑life, the time required for the amount to reduce to one‑half its initial value, is 5 units. The other numbers correspond to different intervals but not the half‑life.

Q12. Solve for \ $x\$ in the equation \ $4^{x}=8\$ .

A.2

B.0.5

C.1.5 ✅

D.3

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Rewrite both sides with base 2: \ $4^{x}=(2^{2})^{x}=2^{2x}\$ and \ $8=2^{3}\$ . Equating exponents gives \ $2x=3\$ , so \ $x=\\tfrac{3}{2}=1.5\$ . Option C matches this value; the other options arise from incorrect exponent handling.

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