Question

mathematics 1a - g cr sem a aguirre
exponential functions
which is the graph of $f(x) = 4\left(\frac{1}{2}\
ight)^x$?
four graphs of exponential functions are shown, each on a coordinate grid with x and y axes, and the graphs are curves representing exponential decay.

Explanation:

Step1: Find the y-intercept

To find the y - intercept, we set \(x = 0\) in the function \(f(x)=4(\frac{1}{2})^{x}\). Using the property \(a^{0}=1\) (where \(a=\frac{1}{2}\) and \(a
eq0\)), we have \(f(0)=4\times(\frac{1}{2})^{0}=4\times1 = 4\). So the graph should pass through the point \((0,4)\).

Step2: Analyze the end - behavior and shape

The function \(y = a(b)^{x}\) is an exponential function. Here, \(a = 4\) and \(b=\frac{1}{2}\), since \(0\lt b=\frac{1}{2}\lt1\), the function is a decreasing exponential function. As \(x\to+\infty\), \(y = 4(\frac{1}{2})^{x}\to0\) (because a positive number less than 1 raised to a large positive power approaches 0), and as \(x\to-\infty\), \(y = 4(\frac{1}{2})^{x}\to+\infty\) (because a positive number less than 1 raised to a large negative power is equivalent to the reciprocal of that number raised to a large positive power, so it approaches infinity).

Now we check the graphs:

• The first graph (top - left): When \(x = 0\), the y - value is 6 (from the graph), which is not 4. So this is not the graph of \(f(x)\).
• The second graph (top - middle): When \(x = 0\), the y - value is 4 (matches the y - intercept we calculated). Also, it is a decreasing exponential function (as \(x\) increases, \(y\) decreases towards 0), which matches the behavior of \(y = 4(\frac{1}{2})^{x}\).
• The third graph (top - right): When \(x = 0\), the y - value is 2 (from the graph), which is not 4. So this is not the graph of \(f(x)\).
• The fourth graph (bottom - left): When \(x = 0\), the y - value is 6 (from the graph), which is not 4. So this is not the graph of \(f(x)\).

Answer:

The Middle Graph (top - middle graph)