Question

which best describes the graph of the function $f(x) = 4(1.5)^x$?

• the graph passes through the point $(0, 4)$, and for each increase of 1 in the $x$-values, the $y$-values increase by 1.5.
• the graph passes through the point $(0, 4)$, and for each increase of 1 in the $x$-values, the $y$-values increase by a factor of 1.5.
• the graph passes through the point $(0, 1.5)$, and for each increase of 1 in the $x$-values, the $y$-values increase by 4.
• the graph passes through the point $(0, 1.5)$, and for each increase of 1 in the $x$-values, the $y$-values increase by a factor of 4.

Explanation:

Step1: Find y-intercept (x=0)

Substitute $x=0$ into $f(x)=4(1.5)^x$:
$f(0)=4(1.5)^0=4\times1=4$
So the graph passes through $(0, 4)$.

Step2: Analyze growth factor

For exponential functions $f(x)=a(b)^x$, when $x$ increases by 1, $f(x+1)=a(b)^{x+1}=b\times a(b)^x = b\times f(x)$. Here $b=1.5$, so y-values multiply by 1.5 (increase by a factor of 1.5) when x increases by 1.

Answer:

B. The graph passes through the point (0, 4), and for each increase of 1 in the x-values, the y-values increase by a factor of 1.5