Question

1. is this function linear, quadratic, or exponential?

x	y
2	20
3	40
4	80
5	160

options: linear, quadratic, exponential

1. each of these functions grows as x gets larger and larger. which function eventually exceeds the others?

$f(x)=10x + 5$
$g(x)=2(3)^x$
$h(x)=6x^2 - x$

Explanation:

First Problem: Classify the table function

Step1: Check linear (constant Δy)

Calculate differences in y-values:
$20-10=10$, $40-20=20$, $80-40=40$, $160-80=80$
Differences are not constant, so not linear.

Step2: Check exponential (constant ratio)

Calculate ratios of consecutive y-values:
$\frac{20}{10}=2$, $\frac{40}{20}=2$, $\frac{80}{40}=2$, $\frac{160}{80}=2$
Ratios are constant (2).

Step1: Recall growth rate hierarchy

Exponential functions ($b^x, b>1$) grow faster than quadratic ($x^2$) and linear ($x$) functions as $x\to\infty$.

Step2: Match functions to types

• $f(x)=10x+5$: linear
• $h(x)=6x^2-x$: quadratic
• $g(x)=2(3)^x$: exponential

Exponential outpaces the others long-term.

Answer:

exponential

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Second Problem: Identify fastest-growing function