Question

determine which function(s) are exponential. select all that apply.
a.

x	y
1	1
2	4
3	9
4	16

b.

x	y
1	1
2	3
3	9
4	27

c.

x	y
1	$\frac{5}{2}$
2	$\frac{5}{4}$
3	$\frac{5}{8}$
4	$\frac{5}{16}$

d.

x	y
1	4
2	3.5
3	3
4	2.5

Explanation:

Step1: Recall exponential function property

An exponential function has a constant ratio between consecutive $y$-values when $x$ increases by 1.

Step2: Check Option A ratios

Calculate $\frac{y_{n+1}}{y_n}$:
$\frac{1}{0}$ (undefined, invalid), $\frac{4}{1}=4$, $\frac{9}{4}=2.25$, $\frac{16}{9}\approx1.78$. No constant ratio.

Step3: Check Option B ratios

Calculate $\frac{y_{n+1}}{y_n}$:
$\frac{1}{\frac{1}{3}}=3$, $\frac{3}{1}=3$, $\frac{9}{3}=3$, $\frac{27}{9}=3$. Constant ratio of 3.

Step4: Check Option C ratios

Calculate $\frac{y_{n+1}}{y_n}$:
$\frac{\frac{5}{2}}{5}=\frac{1}{2}$, $\frac{\frac{5}{4}}{\frac{5}{2}}=\frac{1}{2}$, $\frac{\frac{5}{8}}{\frac{5}{4}}=\frac{1}{2}$, $\frac{\frac{5}{16}}{\frac{5}{8}}=\frac{1}{2}$. Constant ratio of $\frac{1}{2}$.

Step5: Check Option D ratios

Calculate $\frac{y_{n+1}}{y_n}$:
$\frac{4}{4.5}\approx0.89$, $\frac{3.5}{4}=0.875$, $\frac{3}{3.5}\approx0.86$, $\frac{2.5}{3}\approx0.83$. No constant ratio.

Answer:

B.

x	y
1	1
2	3
3	9
4	27

C.

x	y
1	$\frac{5}{2}$
2	$\frac{5}{4}$
3	$\frac{5}{8}$
4	$\frac{5}{16}$