Question

graph the equation shown below by transforming the given graph of the parent function. ( y = 3 cdot 2^x )

Explanation:

Step1: Identify the parent function

The parent function of \( y = 3\cdot2^{x} \) is \( y = 2^{x} \). The graph provided appears to be the graph of \( y = 2^{x} \) (since for \( x = 0 \), \( y = 1 \); \( x = 1 \), \( y = 2 \); \( x = 2 \), \( y = 4 \); \( x = 3 \), \( y = 8 \), which matches the points on the given graph).

Step2: Analyze the transformation

The given function is \( y = 3\cdot2^{x} \), which is a vertical stretch of the parent function \( y = 2^{x} \) by a factor of 3. A vertical stretch by a factor of \( a \) (where \( a>1 \)) of a function \( y = f(x) \) transforms it to \( y = a\cdot f(x) \). For each point \( (x, y) \) on the parent function \( y = 2^{x} \), the corresponding point on \( y = 3\cdot2^{x} \) will be \( (x, 3y) \).

Step3: Transform key points

• For \( x = 0 \): Parent function \( y = 2^{0}=1 \), so transformed point is \( (0, 3\times1)=(0, 3) \).
• For \( x = 1 \): Parent function \( y = 2^{1}=2 \), so transformed point is \( (1, 3\times2)=(1, 6) \).
• For \( x = 2 \): Parent function \( y = 2^{2}=4 \), so transformed point is \( (2, 3\times4)=(2, 12) \) (but note the y - axis in the graph goes up to 10, so maybe we use the visible points. The given graph has points like (0,1), (1,2), (2,4), (3,8). After transformation:
• (0, 3)
• (1, 6)
• (2, 12) (but since the graph's y - max is 10, maybe we can also consider the shape. The graph of \( y = 3\cdot2^{x} \) will have the same general shape as \( y = 2^{x} \) but steeper, with each y - value multiplied by 3.

To graph \( y = 3\cdot2^{x} \), we take the graph of \( y = 2^{x} \) (the given graph) and vertically stretch each point by a factor of 3. So the new graph will pass through points like (0, 3), (1, 6), (2, 12) (adjusting for the graph's scale) and will have the same exponential growth shape but taller at each x - value.

Answer:

To graph \( y = 3\cdot2^{x} \), vertically stretch the graph of the parent function \( y = 2^{x} \) (given) by a factor of 3. This means for each point \((x,y)\) on \( y = 2^{x} \), plot the point \((x, 3y)\). For example, the point \((0,1)\) on \( y = 2^{x} \) becomes \((0,3)\), \((1,2)\) becomes \((1,6)\), \((2,4)\) becomes \((2,12)\) (or adjust for the graph's scale), resulting in an exponential curve that is steeper than the parent function's graph.