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For EACA of the problems below, write an equati
a.

x

y

0
0.5

1
3

2
18

3
108

y=
Check that your equation works:

$2$ . For EACA of the problems below, write an equati $\newline$ a. $\newline$ \begin{tabular}{|c|c|} $\newline$ \hline $x$ & $y$ \\ $\newline$ \hline $0$ & $0$ . $5$ \\ $\newline$ \hline $1$ & $3$ \\ $\newline$ \hline $2$ & $18$ \\ $\newline$ \hline $3$ & $108$ \\ $\newline$ \hline $\newline$ \end{tabular} $\newline$ $y=$ $\newline$ Check that your equation works:

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Find the roots of factored polynomials

Full solution

Q.

2

. For EACA of the problems below, write an equati

\newline

a.

\newline

\begin{tabular}{|c|c|}

\newline

\hline

x

&

y

\\

\newline

\hline

0

&

0

.

5

\\

\newline

\hline

1

&

3

\\

\newline

\hline

2

&

18

\\

\newline

\hline

3

&

108

\\

\newline

\hline

\newline

\end{tabular}

\newline

y=

\newline

Check that your equation works:

Observe pattern in y-values: Observe the pattern in the y-values as $x$ increases. $\newline$ As $x$ increases from $0$ to $1$ , $y$ increases from $0.5$ to $3$ . As $x$ increases from $1$ to $2$ , $y$ increases from $3$ to $x$ $2$ . As $x$ increases from $2$ to $3$ , $y$ increases from $x$ $2$ to $x$ $8$ . It seems that as $x$ increases by $1$ , $y$ is multiplied by $0$ $2$ (since $0$ $3$ , $0$ $4$ , and $0$ $5$ ).
Determine function type: Determine the type of function that could model this behavior. $\newline$ The pattern suggests an exponential function because the y-value is being multiplied by a constant factor as $x$ increases. An exponential function has the form $y = ab^x$ , where $a$ is the initial value (when $x=0$ ) and $b$ is the growth factor.
Find initial value 'a': Use the initial value (when $x=0$ ) to find 'a'. $\newline$ When $x=0$ , $y=0.5$ . Therefore, the initial value 'a' is $0.5$ .
Find growth factor 'b': Use another point to find 'b'. $\newline$ We can use the point $(1, 3)$ to find 'b'. Plugging these values into the equation $y = ab^x$ gives us $3 = 0.5 \times b^1$ . Solving for $b$ , we get $b = \frac{3}{0.5} = 6$ .
Write equation with values: Write the equation using the values of $a$ and $b$ . The equation is $y = 0.5 \times 6^x$ .
Check equation with points: Check that the equation works with the other points. $\newline$ For $x=2$ , $y$ should be $18$ . Plugging $x=2$ into the equation gives us $y = 0.5 \times 6^2 = 0.5 \times 36 = 18$ , which is correct. $\newline$ For $x=3$ , $y$ should be $108$ . Plugging $x=3$ into the equation gives us $y = 0.5 \times 6^3 = 0.5 \times 216 = 108$ , which is correct.

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