Question

graph the equation shown below by transforming the given graph of the parent function.
$y = \frac{1}{8} \cdot 2^x$

Explanation:

Step1: Identify the parent function

The parent function here is \( y = 2^x \), which is an exponential function with a base of 2. The given graph in the image appears to be the graph of \( y = 2^x \) (passing through points like (0,1), (1,2), (2,4), (3,8) as seen from the blue dots: when \( x = 0 \), \( y = 1 \); \( x = 1 \), \( y = 2 \); \( x = 2 \), \( y = 4 \); \( x = 3 \), \( y = 8 \)).

Step2: Analyze the transformation

The given equation is \( y=\frac{1}{8}\cdot2^x \). We can rewrite \( \frac{1}{8} \) as \( 2^{-3} \) because \( 2^3 = 8 \), so \( \frac{1}{8}=2^{-3} \). Then the equation becomes \( y = 2^{-3}\cdot2^x \). Using the exponent rule \( a^m\cdot a^n=a^{m + n} \), we get \( y = 2^{x-3} \). This represents a horizontal shift of the parent function \( y = 2^x \) to the right by 3 units, or alternatively, a vertical compression by a factor of \( \frac{1}{8} \) (since multiplying the function by \( \frac{1}{8} \) compresses it vertically).

To graph \( y=\frac{1}{8}\cdot2^x \) from \( y = 2^x \):

• For the parent function \( y = 2^x \), when \( x = 0 \), \( y = 1 \); \( x = 1 \), \( y = 2 \); \( x = 2 \), \( y = 4 \); \( x = 3 \), \( y = 8 \).
• For the transformed function \( y=\frac{1}{8}\cdot2^x \), we multiply the \( y \)-values of the parent function by \( \frac{1}{8} \):
• When \( x = 0 \), \( y=\frac{1}{8}\cdot2^0=\frac{1}{8}\cdot1=\frac{1}{8} \)
• When \( x = 1 \), \( y=\frac{1}{8}\cdot2^1=\frac{1}{8}\cdot2=\frac{1}{4} \)
• When \( x = 2 \), \( y=\frac{1}{8}\cdot2^2=\frac{1}{8}\cdot4=\frac{1}{2} \)
• When \( x = 3 \), \( y=\frac{1}{8}\cdot2^3=\frac{1}{8}\cdot8 = 1 \)
• When \( x = 4 \), \( y=\frac{1}{8}\cdot2^4=\frac{1}{8}\cdot16 = 2 \)
• When \( x = 5 \), \( y=\frac{1}{8}\cdot2^5=\frac{1}{8}\cdot32 = 4 \)
• When \( x = 6 \), \( y=\frac{1}{8}\cdot2^6=\frac{1}{8}\cdot64 = 8 \)

So the key points of the transformed function will be \( (0,\frac{1}{8}) \), \( (1,\frac{1}{4}) \), \( (2,\frac{1}{2}) \), \( (3,1) \), \( (4,2) \), \( (5,4) \), \( (6,8) \), etc. This is equivalent to shifting the parent graph \( y = 2^x \) (which has points (0,1), (1,2), (2,4), (3,8)) to the right by 3 units (since when \( x = 3 \) in the transformed function, \( y = 1 \), which is the \( y \)-value of the parent function at \( x = 0 \); when \( x = 4 \), \( y = 2 \), which is the parent's \( y \)-value at \( x = 1 \), etc.).

Answer:

To graph \( y=\frac{1}{8}\cdot2^x \), we start with the parent function \( y = 2^x \) (exponential growth, passing through (0,1), (1,2), (2,4), (3,8)). Then we apply a vertical compression by a factor of \( \frac{1}{8} \) (or a horizontal shift right by 3 units). The transformed graph will pass through points like \( (0,\frac{1}{8}) \), \( (1,\frac{1}{4}) \), \( (2,\frac{1}{2}) \), \( (3,1) \), \( (4,2) \), etc., and will have the same shape as \( y = 2^x \) but compressed vertically (or shifted horizontally) to match the equation \( y=\frac{1}{8}\cdot2^x \).