Question

the table shows a function ( b ):

( x )	4	6

if this was an exponential function, find its formula.

round to 3 decimal places. answer: ( b(x) = )

if this was a linear function, find its formula.

round to 3 decimal places. answer: ( b(x) = )

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Explanation:

Step1: Define exponential form

The general exponential function is $b(x) = ab^x$, where $a$ is the initial value, $b$ is the base.

Step2: Set up equations from table

Substitute $x=4, b(x)=13$: $13 = ab^4$
Substitute $x=6, b(x)=6.37$: $6.37 = ab^6$

Step3: Solve for base $b$

Divide the second equation by the first:
$\frac{6.37}{13} = \frac{ab^6}{ab^4}$
$0.49 = b^2$
$b = \sqrt{0.49} = 0.7$

Step4: Solve for $a$

Substitute $b=0.7$ into $13 = ab^4$:
$13 = a(0.7)^4$
$13 = a(0.2401)$
$a = \frac{13}{0.2401} \approx 54.144$

Step5: Define linear function form

The general linear function is $b(x) = mx + c$, where $m$ is slope, $c$ is intercept.

Step6: Calculate slope $m$

$m = \frac{b(6)-b(4)}{6-4} = \frac{6.37-13}{6-4}$
$m = \frac{-6.63}{2} = -3.315$

Step7: Solve for intercept $c$

Substitute $x=4, b(x)=13, m=-3.315$ into $b(x)=mx+c$:
$13 = -3.315(4) + c$
$13 = -13.26 + c$
$c = 13 + 13.26 = 26.260$

Answer:

Exponential formula: $b(x) = 54.144(0.7)^x$
Linear formula: $b(x) = -3.315x + 26.260$