Question

complete the statement about the graph of the function $f(x) = 0.5(3)^x$. the graph crosses the y - axis at the point. as the value of $x$ increases, the function. the equation of the graph’s asymptote is

Explanation:

Step1: Find y - intercept

To find the y - intercept, we set \(x = 0\) in the function \(f(x)=0.5(3)^{x}\).
Substitute \(x = 0\) into the function: \(f(0)=0.5\times(3)^{0}\).
Since any non - zero number to the power of 0 is 1, \((3)^{0}=1\). So \(f(0)=0.5\times1 = 0.5\). The y - intercept is the point \((0,0.5)\) (or \((0,\frac{1}{2})\)).

Step2: Analyze the behavior as \(x\) increases

The function \(f(x)=0.5(3)^{x}\) is an exponential function of the form \(y = a(b)^{x}\), where \(a = 0.5>0\) and \(b = 3>1\).
For exponential functions with \(a>0\) and \(b > 1\), as \(x\) increases, the function value \(y\) (or \(f(x)\)) increases. So as the value of \(x\) increases, the function increases.

Step3: Find the equation of the asymptote

For exponential functions of the form \(y=a(b)^{x}+k\), the horizontal asymptote is \(y = k\). In the function \(f(x)=0.5(3)^{x}\), we can rewrite it as \(f(x)=0.5(3)^{x}+0\). So the equation of the horizontal asymptote is \(y = 0\) (the x - axis).

Answer:

The graph crosses the y - axis at the point \(\boldsymbol{(0, 0.5)}\). As the value of \(x\) increases, the function \(\boldsymbol{\text{increases}}\). The equation of the graph's asymptote is \(\boldsymbol{y = 0}\).