Question

state if each function is exponential. if so, state the common difference (d).
1
x | 0 | 1 | 2 | 3 | 4
f(x) | 4 | 20 | 100 | 500 | 2500
2
x | 0 | 1 | 2 | 3 | 4
f(x) | 729 | 243 | 81 | 27 | 9
3
x | 0 | 1 | 2 | 3 | 4
f(x) | 25 | 50 | 75 | 100 | 125
you try!
x | 0 | 1 | 2 | 3 | 4
f(x) | 25 | 50 | 75 | 100 | 125

Explanation:

Problem 1

Step1: Check ratio between terms

Calculate $\frac{f(x_1)}{f(x_0)},\frac{f(x_2)}{f(x_1)},\dots$
$\frac{20}{4}=5$, $\frac{100}{20}=5$, $\frac{500}{100}=5$, $\frac{2500}{500}=5$

Step2: Determine function type

Ratios are constant ($r = 5$), so it’s exponential. Common ratio $r = 5$.

Step1: Check ratio between terms

$\frac{243}{729}=\frac{1}{3}$, $\frac{81}{243}=\frac{1}{3}$, $\frac{27}{81}=\frac{1}{3}$, $\frac{9}{27}=\frac{1}{3}$

Step2: Determine function type

Ratios are constant ($r=\frac{1}{3}$), so it’s exponential. Common ratio $r=\frac{1}{3}$.

Step1: Check difference between terms

$50 - 25 = 25$, $75 - 50 = 25$, $100 - 75 = 25$, $125 - 100 = 25$

Step2: Determine function type

Differences are constant ($d = 25$), so it’s linear (arithmetic sequence), not exponential. Common difference $d = 25$.

Answer:

Exponential, common ratio $r = 5$

Problem 2