Question

graph the equation shown below by transforming the given graph of the parent function.
$y = \frac{1}{4} \cdot 2^x$

Explanation:

Step1: Identify the parent function

The parent function here is \( y = 2^x \). The given function is \( y=\frac{1}{4}\cdot2^x \), which is a vertical scaling of the parent function.

Step2: Recall the transformation rule for vertical scaling

For a function \( y = a\cdot f(x) \), if \( 0 < a < 1 \), it is a vertical compression of the graph of \( y = f(x) \) by a factor of \( a \). Here, \( a=\frac{1}{4} \), so we compress the graph of \( y = 2^x \) vertically by a factor of \( \frac{1}{4} \).

Step3: Analyze key points of the parent function

Let's take some key points of \( y = 2^x \):

• When \( x = 0 \), \( y = 2^0=1 \)
• When \( x = 1 \), \( y = 2^1 = 2 \)
• When \( x = 2 \), \( y = 2^2=4 \)
• When \( x = 3 \), \( y = 2^3 = 8 \)

Step4: Apply the vertical scaling to these points

For each point \((x,y)\) on \( y = 2^x \), the corresponding point on \( y=\frac{1}{4}\cdot2^x \) will be \((x,\frac{1}{4}y)\):

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

Step5: Graph the transformed function

To graph \( y=\frac{1}{4}\cdot2^x \), we take the graph of \( y = 2^x \) and vertically compress each point by a factor of \( \frac{1}{4} \). So the new graph will pass through the points \((0,\frac{1}{4})\), \((1,\frac{1}{2})\), \((2,1)\), \((3,2)\) etc. and will have the same general shape as \( y = 2^x \) but "shorter" (compressed) vertically.

Answer:

To graph \( y=\frac{1}{4}\cdot2^x \), vertically compress the graph of \( y = 2^x \) by a factor of \( \frac{1}{4} \). Key points after transformation: \((0,\frac{1}{4})\), \((1,\frac{1}{2})\), \((2,1)\), \((3,2)\) (and so on), with the curve maintaining the exponential growth shape but scaled down vertically.