Question

graph this function. then complete the statements below.
$f(x) = 1.5(2)^x$
select points on the graph to plot them.
the graph crosses the $y$-axis at. as the value of $x$ decreases, the
functions value.

Explanation:

Step1: Find y - intercept

To find where the graph crosses the y - axis, we set \(x = 0\) in the function \(f(x)=1.5(2)^{x}\).
Substitute \(x = 0\) into the function: \(f(0)=1.5\times(2)^{0}\).
Since any non - zero number to the power of 0 is 1, \((2)^{0}=1\). So \(f(0)=1.5\times1 = 1.5\). So the graph crosses the y - axis at \((0, 1.5)\).

Step2: Analyze the behavior as x decreases

The function \(f(x)=1.5(2)^{x}\) is an exponential function of the form \(y = ab^{x}\), where \(a = 1.5>0\) and \(b = 2>1\). For an exponential function \(y=ab^{x}\) with \(a>0\) and \(b > 1\), as \(x\) decreases (moves to the left along the x - axis), we can consider the general form. Let's take some negative values of \(x\), for example, if \(x=- 1\), then \(f(-1)=1.5\times(2)^{-1}=1.5\times\frac{1}{2}=0.75\); if \(x = - 2\), then \(f(-2)=1.5\times(2)^{-2}=1.5\times\frac{1}{4}=0.375\). As \(x\) becomes more negative (decreases), the value of \(2^{x}=\frac{1}{2^{\vert x\vert}}\) gets closer to 0, and since \(a = 1.5>0\), the function's value approaches 0 (gets smaller and smaller, approaching 0). So as the value of \(x\) decreases, the function's value approaches 0 (or decreases towards 0).

Answer:

The graph crosses the y - axis at \((0, 1.5)\). As the value of \(x\) decreases, the function's value approaches 0 (or decreases towards 0).