Question

name: date: period: 3
cca lesson 5.3.2 homework review & preview day 1
5-101. for each table below, find the missing entries and write an equation.
a.
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$$\begin{tabular}{l|c|c|c|c|c|c|c} month (x) & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\\ \\hline population (y) & 2 & 8 & 32 & 128 & 512 & 2048 & 8192 \\\\ \\end{tabular}$$

(handwritten notes: ×4, ×4)
b.
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$$\begin{tabular}{l|c|c|c|c|c|c|c} year (x) & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\\ \\hline population (y) & 5 & 6 & 7.2 & & & & \\\\ \\end{tabular}$$

Explanation:

Part a.

Step1: Identify growth factor

Check ratio of consecutive $y$-values: $\frac{8}{2}=4$, $\frac{32}{8}=4$. Growth factor $b=4$. Initial value $a=2$ (when $x=0$).

Step2: Verify missing entries

$x=3$: $y=2\times4^3=2\times64=128$
$x=4$: $y=2\times4^4=2\times256=512$
$x=5$: $y=2\times4^5=2\times1024=2048$
$x=6$: $y=2\times4^6=2\times4096=8192$

Step3: Write exponential equation

General form: $y=a(b)^x$. Substitute $a=2, b=4$.
$y=2(4)^x$

Part b.

Step1: Identify growth factor

Check ratio of consecutive $y$-values: $\frac{6}{5}=1.2$, $\frac{7.2}{6}=1.2$. Growth factor $b=1.2$. Initial value $a=5$ (when $x=0$).

Step2: Calculate missing entries

$x=3$: $y=5\times(1.2)^3=5\times1.728=8.64$
$x=4$: $y=5\times(1.2)^4=5\times2.0736=10.368$
$x=5$: $y=5\times(1.2)^5=5\times2.48832=12.4416$
$x=6$: $y=5\times(1.2)^6=5\times2.985984=14.92992$

Step3: Write exponential equation

General form: $y=a(b)^x$. Substitute $a=5, b=1.2$.
$y=5(1.2)^x$

Answer:

Part a.

Missing entries (already verified):

• $x=3$: $128$, $x=4$: $512$, $x=5$: $2048$, $x=6$: $8192$

Equation: $y=2(4)^x$

Part b.

Missing entries:

• $x=3$: $8.64$, $x=4$: $10.368$, $x=5$: $12.4416$, $x=6$: $14.92992$

Equation: $y=5(1.2)^x$