The functions $f(x)=2^{x}, f(x)=2 \cdot 2^{x}$ , and $f(x)=\frac{1}{2} \cdot 2^{x}$ all have the same nuultiplier. $\newline$ a. What is the multiplier for these functions? Make an $x \rightarrow f(x)$ table for each of the functions, using integer value of $x$ for $x=-3$ to $x=5$ . How can you use the tables to determine the multipliers?

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Algebra 1

Compare linear and exponential growth

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Q. The functions

f(x)=2^{x}, f(x)=2 \cdot 2^{x}

, and

f(x)=\frac{1}{2} \cdot 2^{x}

all have the same nuultiplier.

\newline

a. What is the multiplier for these functions? Make an

x \rightarrow f(x)

table for each of the functions, using integer value of

x

for

x=-3

x=5

. How can you use the tables to determine the multipliers?

Define Functions and Multiplier: Define the functions and identify the multiplier. $\newline$ $f(x) = 2^x$ , $f(x) = 2 \cdot 2^x$ , and $f(x) = (\frac{1}{2}) \cdot 2^x$ all involve the base $2$ raised to the power $x$ . The multiplier in each function is the coefficient of $2^x$ . $\newline$ For $f(x) = 2^x$ , the multiplier is $1$ . $\newline$ For $f(x) = 2 \cdot 2^x$ , the multiplier is $2$ . $\newline$ For $f(x) = (\frac{1}{2}) \cdot 2^x$ , the multiplier is $f(x) = 2 \cdot 2^x$ $1$ .
Create $x \longrightarrow f(x)$ Table: Create an $x \longrightarrow f(x)$ table for each function from $x = -3$ to $x = 5$ . For $f(x) = 2^x$ : $x = -3 \longrightarrow f(x) = 2^{-3} = \frac{1}{8}$ $x = -2 \longrightarrow f(x) = 2^{-2} = \frac{1}{4}$ $x = -1 \longrightarrow f(x) = 2^{-1} = \frac{1}{2}$ $x = 0 \longrightarrow f(x) = 2^0 = 1$ $x = 1 \longrightarrow f(x) = 2^1 = 2$ $x \longrightarrow f(x)$ $0$ $x \longrightarrow f(x)$ $1$ $x \longrightarrow f(x)$ $2$ $x \longrightarrow f(x)$ $3$
Continue Table for $2\cdot2^x$ : Continue the table for $f(x) = 2\cdot2^x$ . $\newline$ $x = -3 \longrightarrow f(x) = 2\cdot2^{-3} = \frac{1}{4}$ $\newline$ $x = -2 \longrightarrow f(x) = 2\cdot2^{-2} = \frac{1}{2}$ $\newline$ $x = -1 \longrightarrow f(x) = 2\cdot2^{-1} = 1$ $\newline$ $x = 0 \longrightarrow f(x) = 2\cdot2^0 = 2$ $\newline$ $x = 1 \longrightarrow f(x) = 2\cdot2^1 = 4$ $\newline$ $x = 2 \longrightarrow f(x) = 2\cdot2^2 = 8$ $\newline$ $x = 3 \longrightarrow f(x) = 2\cdot2^3 = 16$ $\newline$ $x = 4 \longrightarrow f(x) = 2\cdot2^4 = 32$ $\newline$ $x = $5$ \longrightarrow f(x) = $2$\cdot$2$^$5$ = $64$
Continue Table for $(1/2)\cdot2^x$: Continue the table for $f(x) = (1/2)\cdot2^x$.
$x = -3 \rightarrow f(x) = (1/2)\cdot2^{-3} = 1/16$
$x = -2 \rightarrow f(x) = (1/2)\cdot2^{-2} = 1/8$
$x = -1 \rightarrow f(x) = (1/2)\cdot2^{-1} = 1/4$
$x = 0 \rightarrow f(x) = (1/2)\cdot2^0 = 1/2$
$x = 1 \rightarrow f(x) = (1/2)\cdot2^1 = 1$
$x = 2 \rightarrow f(x) = (1/2)\cdot2^2 = 2$
$x = 3 \rightarrow f(x) = (1/2)\cdot2^3 = 4$
$x = 4 \rightarrow f(x) = (1/2)\cdot2^4 = 8$
$f(x) = (1/2)\cdot2^x$$0$
Analyze Tables for Multipliers: Analyze the tables to determine the multipliers. By comparing the values of $f(x)$ for each function at the same $x$, we see that each function's output is a multiple of $2^x$. The multiplier directly affects the output values, confirming the multipliers identified in Step $1$.