The problem provides two exponential functions, $f(x) = 6 \cdot 2^x$ and $g(x) = 2 \cdot 4^x$, and their graphs. We need to determine the growth factor for each function, the average rate of change for each function over specified intervals, and which function grows faster.

1. Problem Description

The problem provides two exponential functions, $f(x) = 6 \cdot 2^x$ and $g(x) = 2 \cdot 4^x$, and their graphs. We need to determine the growth factor for each function, the average rate of change for each function over specified intervals, and which function grows faster.

2. Solution Steps

a) Growth Factor

The growth factor for an exponential function of the form $y = a \cdot b^x$ is $b$.

For $f(x) = 6 \cdot 2^x$, the growth factor is

2. For $g(x) = 2 \cdot 4^x$, the growth factor is

4.

b) Average Rate of Change for $f(x)$

From $x=0$ to $x=1$:

$f(0) = 6 \cdot 2^0 = 6 \cdot 1 = 6$

$f(1) = 6 \cdot 2^1 = 6 \cdot 2 = 12$

Average rate of change = $\frac{f(1) - f(0)}{1 - 0} = \frac{12 - 6}{1} = 6$

From $x=1$ to $x=2$:

$f(1) = 12$

$f(2) = 6 \cdot 2^2 = 6 \cdot 4 = 24$

Average rate of change = $\frac{f(2) - f(1)}{2 - 1} = \frac{24 - 12}{1} = 12$

c) Average Rate of Change for $g(x)$

From $x=0$ to $x=1$:

$g(0) = 2 \cdot 4^0 = 2 \cdot 1 = 2$

$g(1) = 2 \cdot 4^1 = 2 \cdot 4 = 8$

Average rate of change = $\frac{g(1) - g(0)}{1 - 0} = \frac{8 - 2}{1} = 6$

From $x=1$ to $x=2$:

$g(1) = 8$

$g(2) = 2 \cdot 4^2 = 2 \cdot 16 = 32$

Average rate of change = $\frac{g(2) - g(1)}{2 - 1} = \frac{32 - 8}{1} = 24$

d) Which Function Grows Faster?

$g(x)$ grows faster than $f(x)$ because it has a larger growth factor (4 versus 2). Also, the average rate of change of $g(x)$ is higher than that of $f(x)$ in the interval from $x=1$ to $x=2$.

3. Final Answer

Growth factor for $f(x)$: 2

Growth factor for $g(x)$: 4

Average rate of change for $f(x)$ from $x=0$ to $x=1$: 6

Average rate of change for $f(x)$ from $x=1$ to $x=2$: 12

Average rate of change for $g(x)$ from $x=0$ to $x=1$: 6

Average rate of change for $g(x)$ from $x=1$ to $x=2$: 24

$g(x)$ grows faster.