Question

graph the equation shown below by transforming the given graph of the parent function.
$y = 2^x - 1$

Explanation:

Step1: Identify Parent Function

The parent function here is \( y = 2^x \). Its graph has a y - intercept at \( (0,1) \) (since when \( x = 0 \), \( y=2^0 = 1 \)) and passes through points like \( (1,2) \), \( (2,4) \), \( (3,8) \) as seen in the given graph.

Step2: Analyze the Transformation

The given function is \( y=2^x - 1 \). This is a vertical shift of the parent function \( y = 2^x \). The general form of a vertical shift for a function \( y = f(x) \) is \( y=f(x)+k \), where if \( k<0 \), the graph shifts down by \( |k| \) units. Here, \( k=- 1 \), so we shift the graph of \( y = 2^x \) down by 1 unit.

Step3: Find Key Points After Transformation

• For the y - intercept of the parent function \( y = 2^x \), when \( x = 0 \), \( y = 1 \). After shifting down 1 unit, the y - intercept of \( y=2^x - 1 \) is \( (0,1 - 1)=(0,0) \).
• For the point \( (1,2) \) on \( y = 2^x \), after shifting down 1 unit, the point becomes \( (1,2 - 1)=(1,1) \).
• For the point \( (2,4) \) on \( y = 2^x \), after shifting down 1 unit, the point becomes \( (2,4 - 1)=(2,3) \).
• For the point \( (3,8) \) on \( y = 2^x \), after shifting down 1 unit, the point becomes \( (3,8 - 1)=(3,7) \). Also, the horizontal asymptote of \( y = 2^x \) is \( y = 0 \). After shifting down 1 unit, the horizontal asymptote of \( y=2^x - 1 \) is \( y=-1 \).

To graph \( y = 2^x-1 \), we take the graph of \( y = 2^x \) (the given parent function graph) and shift each point down by 1 unit. So the new graph will have points \( (0,0) \), \( (1,1) \), \( (2,3) \), \( (3,7) \) and the horizontal asymptote at \( y=-1 \).

Answer:

To graph \( y = 2^x-1 \), shift the graph of the parent function \( y = 2^x \) (given) down by 1 unit. Key points after transformation: \( (0,0) \), \( (1,1) \), \( (2,3) \), \( (3,7) \) with horizontal asymptote \( y = - 1 \). (Graphing involves plotting these points and drawing the curve approaching the asymptote \( y=-1 \) as \( x\to-\infty \) and increasing exponentially as \( x\to\infty \) after the shift.)