Question

graph the following function.
$y = 3 \cdot 4^x$
show your work here

Explanation:

Step1: Find the y - intercept

To find the y - intercept, we set \(x = 0\) in the function \(y=3\cdot4^{x}\). Substituting \(x = 0\) into the function, we know that any non - zero number to the power of \(0\) is \(1\), so \(y = 3\cdot4^{0}=3\times1 = 3\). So the y - intercept is the point \((0,3)\).

Step2: Find another point

Let's choose \(x = 1\). Substitute \(x = 1\) into the function \(y = 3\cdot4^{x}\). Then \(y=3\times4^{1}=3\times4 = 12\). So we have the point \((1,12)\).

Step3: Analyze the shape of the function

The function \(y = 3\cdot4^{x}\) is an exponential function of the form \(y = a\cdot b^{x}\) where \(a = 3>0\) and \(b = 4>1\). So the function is an exponential growth function. As \(x\to-\infty\), \(4^{x}\to0\), so \(y = 3\cdot4^{x}\to0\), which means the x - axis ( \(y = 0\)) is a horizontal asymptote.

To graph the function, we plot the points we found \((0,3)\) and \((1,12)\), and also note the horizontal asymptote \(y = 0\). Then we draw a smooth curve through the points, approaching the horizontal asymptote as \(x\) decreases.

Answer:

To graph \(y = 3\cdot4^{x}\):

1. Plot the y - intercept \((0,3)\) and the point \((1,12)\).
2. Draw a smooth exponential growth curve through these points, with the x - axis (\(y = 0\)) as a horizontal asymptote.