To determine which description best describes the graph of the function [tex]\( f(x) = 4(1.5)^x \)[/tex], we need to evaluate the function at some key points and understand how the values change as [tex]\( x \)[/tex] increases.
1. First, evaluate the function at [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 4(1.5)^0 = 4 \times 1 = 4
\][/tex]
So, the graph passes through the point [tex]\((0, 4)\)[/tex].
2. Next, evaluate the function at [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 4(1.5)^1 = 4 \times 1.5 = 6
\][/tex]
3. Evaluate the function again at [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 4(1.5)^2 = 4 \times (1.5 \times 1.5) = 4 \times 2.25 = 9
\][/tex]
From these evaluations, we observe the following:
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 4 \)[/tex]
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = 6 \)[/tex]
- At [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 9 \)[/tex]
Each time [tex]\( x \)[/tex] increases by 1, the function value [tex]\( f(x) \)[/tex] is multiplied by a factor of 1.5. For instance:
[tex]\[
\frac{f(1)}{f(0)} = \frac{6}{4} = 1.5
\][/tex]
[tex]\[
\frac{f(2)}{f(1)} = \frac{9}{6} = 1.5
\][/tex]
Thus, the correct description is:
- "The graph passes through the point [tex]\((0, 4)\)[/tex], and for each increase of 1 in the [tex]\( x \)[/tex]-values, the [tex]\( y \)[/tex]-values increase by a factor of 1.5."
Therefore, the best description is:
```
The graph passes through the point [tex]$(0, 4)$[/tex], and for each increase of 1 in the [tex]$x$[/tex]-values, the [tex]$y$[/tex]-values increase by a factor of 1.5.
```
