Exponential equations and functions — Topic practice

5 exercises · 3 automatically checked · 2 written self-checks · Untimed

Instructions and review guidance

3 questions · Shuffled order · Untimed practice

Work through the set before checking solutions. Guide question labels match the source lesson references. 2 written responses appear below the question bank, with model answers for self-review; these are not automatically graded.

Guide question labels stay the same when the order changes. Check answers when you are ready to review.

Written self-checks are below the question bank and are excluded from the automatic score.

Written responses are self-review exercises.

Saved on this browser and device; clearing browser data removes progress.

Question 2

Guide question E5.2

If Q(t) = 72 ⋅ 3^t/4, find Q(8).

Your answer

Enter an integer, decimal or fraction without units. Use up to 5 characters, or 6 including a leading minus.

Why this answer

648

In $Q(t) = 72 \cdot 3 t/4,$ every 4-unit interval contributes a factor of 3. At $t = 8,$ two such intervals have elapsed:

$Q(8) = 72 \cdot 3 8/4 = 72 \cdot 9 = 648.$

The exponent counts the number of growth periods, not the absolute time value. Review 5.03.

Review lesson: Exponential equations and functions

Question 3

Guide question E5.3

An exponential function f(t) = Ab^t, with A, b > 0, satisfies f(1) = 10 and f(3) = 40. Find f(0).

Your answer

Enter an integer, decimal or fraction without units. Use up to 5 characters, or 6 including a leading minus.

Why this answer

$f(0) = 5$

Divide the equations $Ab 3 = 40$ and $Ab = 10:$

$\frac{40}{10}$

The given condition $b > 0$ selects $b = 2.$ Then $A = 10/2 = 5,$ so $f(0) = A = 5.$ Check: $5(2) 3 = 40.$ The two-unit interval has a factor of 4; the one-unit factor is 2. Review 5.06.

Review lesson: Exponential equations and functions

Question 5

Guide question E5.5

An amount increases by 25% and then decreases by 20%. What is the combined multiplier?

Your answer

Enter an integer, decimal or fraction without units. Use up to 5 characters, or 6 including a leading minus.

Why this answer

Combined multiplier 1

The successive multipliers are 1.25 and 0.80, so the combined multiplier is $1.25(0.80) = 1.$ The final amount equals the initial amount. The second change is calculated from a different base, so adding the signed percentages, $25% - 20%,$ does not give the net change. Review the multiplier lesson and 5.09.

Review lesson: Exponential equations and functions

Question navigator and options

Answered
Correct
Incorrect
Revealed, not answered
Current

Written self-checks

Write your complete response before opening the model answer. Compare both your answer and your reasoning.

Guide question E5.1

Write a model for an initial amount of 900 that decreases by 15% each year.

Show model answerHide model answer

$P(t) = 900(0.85) t$

A 15% decrease leaves $100% - 15% = 85%$ of the previous amount each year. The yearly multiplier is 0.85, and the initial value is 900. Thus $P(t) = 900(0.85) t$ for time t in years. The base is the fraction remaining, not the fraction lost. Review 5.02.

Guide question E5.4

In P(t) = 500(1.10)^t/2, with t in years, what is the exact annual growth factor?

Show model answerHide model answer

$\sqrt{1.10}$

Rewrite $500(1.10) t/2 = 500 ((1.10) 1/2) t .$

Thus the annual multiplier is $\sqrt{1.10}$ approximately 1.04881. The annual percentage increase is approximately 4.88%, but the question asks for the factor. Dividing 10% by 2 is not exact compound-growth reasoning. Review 5.07.

The drill

Why this answer

Why this answer

Why this answer

Review practice

Written self-checks

Guide question E5.1

Guide question E5.4