Question

graph the equation shown below by transforming the given graph of the parent function.
$y = \frac{1}{2} \cdot 2^x$

Explanation:

Step1: Identify the parent function

The parent function here is \( y = 2^x \), which is an exponential function. The given function is \( y=\frac{1}{2}\cdot2^x \).

Step2: Analyze the transformation

The transformation from \( y = 2^x \) to \( y=\frac{1}{2}\cdot2^x \) is a vertical compression by a factor of \( \frac{1}{2} \). For a function \( y = a\cdot f(x) \), if \( 0 < a < 1 \), it is a vertical compression. Here \( a=\frac{1}{2} \), so we take each point on the graph of \( y = 2^x \) and multiply its \( y \)-coordinate by \( \frac{1}{2} \).

• For the point \( (0, 1) \) on \( y = 2^x \), after compression: \( y=\frac{1}{2}\cdot1=\frac{1}{2} \), so the point becomes \( (0,\frac{1}{2}) \).
• For the point \( (1, 2) \) on \( y = 2^x \), after compression: \( y=\frac{1}{2}\cdot2 = 1 \), so the point becomes \( (1, 1) \).
• For the point \( (2, 4) \) on \( y = 2^x \), after compression: \( y=\frac{1}{2}\cdot4=2 \), so the point becomes \( (2, 2) \).
• For the point \( (3, 8) \) on \( y = 2^x \), after compression: \( y=\frac{1}{2}\cdot8 = 4 \), so the point becomes \( (3, 4) \).

Then we plot these new points \( (0,\frac{1}{2}) \), \( (1, 1) \), \( (2, 2) \), \( (3, 4) \) and draw the curve, which should have the same shape as the parent exponential function but vertically compressed.

Answer:

To graph \( y=\frac{1}{2}\cdot2^x \), vertically compress the graph of \( y = 2^x \) by a factor of \( \frac{1}{2} \) (multiply \( y \)-coordinates of points on \( y = 2^x \) by \( \frac{1}{2} \)) and plot the transformed points (e.g., \( (0,\frac{1}{2}) \), \( (1, 1) \), \( (2, 2) \), \( (3, 4) \)) to draw the curve.