Question

is this function linear, quadratic, or exponential?

x | y
4 | 14
5 | $\frac{175}{8}$
6 | $\frac{63}{2}$
7 | $\frac{343}{8}$
8 | 56

linear quadratic exponential

Explanation:

Step1: Check linearity (constant Δy)

First, calculate differences between consecutive y-values:
$\frac{175}{8}-14=\frac{175}{8}-\frac{112}{8}=\frac{63}{8}$
$\frac{63}{2}-\frac{175}{8}=\frac{252}{8}-\frac{175}{8}=\frac{77}{8}$
$\frac{343}{8}-\frac{63}{2}=\frac{343}{8}-\frac{252}{8}=\frac{91}{8}$
$56-\frac{343}{8}=\frac{448}{8}-\frac{343}{8}=\frac{105}{8}$
Differences are not constant, so not linear.

Step2: Check quadratic (constant Δ(Δy))

Calculate differences of the differences:
$\frac{77}{8}-\frac{63}{8}=\frac{14}{8}=\frac{7}{4}$
$\frac{91}{8}-\frac{77}{8}=\frac{14}{8}=\frac{7}{4}$
$\frac{105}{8}-\frac{91}{8}=\frac{14}{8}=\frac{7}{4}$
Second differences are constant.

Step3: Verify no exponential growth

Calculate ratios of consecutive y-values:
$\frac{175/8}{14}=\frac{175}{112}=\frac{25}{16}$
$\frac{63/2}{175/8}=\frac{63}{2}\times\frac{8}{175}=\frac{504}{350}=\frac{36}{25}$
Ratios are not constant, so not exponential.

Answer:

quadratic