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

For a function of the form \( y = a\cdot f(x) \), when \( 0 < a < 1 \), it is a vertical compression of the parent function \( y = f(x) \) by a factor of \( a \). Here, \( a=\frac{1}{2} \), so we need to vertically compress the graph of \( y = 2^x \) by a factor of \( \frac{1}{2} \).

Step3: Transform key points

Let's take some key points from the parent function \( y = 2^x \). For example, when \( x = 0 \), \( y = 2^0=1 \); when \( x = 1 \), \( y = 2^1 = 2 \); when \( x = 2 \), \( y = 2^2=4 \); when \( x = 3 \), \( y = 2^3 = 8 \).

Applying the vertical compression (multiply \( y \)-coordinates by \( \frac{1}{2} \)):

• For \( x = 0 \), new \( y=\frac{1}{2}\times1=\frac{1}{2} \)
• For \( x = 1 \), new \( y=\frac{1}{2}\times2 = 1 \)
• For \( x = 2 \), new \( y=\frac{1}{2}\times4=2 \)
• For \( x = 3 \), new \( y=\frac{1}{2}\times8 = 4 \)

Now, plot these transformed points \((0,\frac{1}{2})\), \((1,1)\), \((2,2)\), \((3,4)\) and draw the curve, which will be a vertical compression of the parent exponential curve \( y = 2^x \).

Answer:

To graph \( y=\frac{1}{2}\cdot2^x \), vertically compress the graph of the parent function \( y = 2^x \) by a factor of \( \frac{1}{2} \). This involves taking key points from \( y = 2^x \) (e.g., \((0,1)\), \((1,2)\), \((2,4)\), \((3,8)\)) and multiplying their \( y \)-coordinates by \( \frac{1}{2} \) to get \((0,\frac{1}{2})\), \((1,1)\), \((2,2)\), \((3,4)\), then drawing the exponential curve through these transformed points.