Question

1. find a function that passes through (1, 6) and (3, 24) where the function is...

(a) (3pts) linear

$f(x) = $
(b) (4pts) exponential

$f(x) = $

Explanation:

(a) Linear Function

Step1: Calculate slope $m$

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
$m=\frac{24-6}{3-1}=\frac{18}{2}=9$

Step2: Find y-intercept $b$

Use $y=mx+b$ with $(1,6)$:
$6=9(1)+b$
$b=6-9=-3$

Step3: Form linear function

Substitute $m=9, b=-3$ into $f(x)=mx+b$
$f(x)=9x-3$

(b) Exponential Function

Step1: Use exponential form $f(x)=ab^x$

Set up equations with points:
For $(1,6)$: $6=ab^1$
For $(3,24)$: $24=ab^3$

Step2: Solve for $b$

Divide second equation by first:
$\frac{24}{6}=\frac{ab^3}{ab}$
$4=b^2$
$b=2$ (we use positive base for standard exponential)

Step3: Solve for $a$

Substitute $b=2$ into $6=ab$:
$6=a(2)$
$a=3$

Step4: Form exponential function

Substitute $a=3, b=2$ into $f(x)=ab^x$
$f(x)=3(2)^x$

Answer:

(a) $f(x)=9x-3$
(b) $f(x)=3(2)^x$