Question

parent ( f(x) = 2^x )
for each question, identify and describe the transformation(s) (e.g., vertical shift, horizontal shift, reflection, stretch, or compression)
parent ( f(x) = 2^x )

1. ( f(x) = 2^x + 3 )

• transformation:

1. ( f(x) = 2^x - 4 )

• transformation:

1. ( f(x) = 2^{x + 2} )

• transformation:

1. ( f(x) = 2^{x - 1} )

• transformation:

1. ( f(x) = -2^x )

• transformation:

1. ( f(x) = 2^x + 5 )

• transformation:

1. ( f(x) = 3 cdot 2^x )

• transformation:

1. ( f(x) = \frac{1}{2} cdot 2^x )

• transformation:

Explanation:

1. $f(x) = 2^x + 3$

Step1: Recall vertical shift rule

For a function $y = f(x)+k$, if $k>0$, it's a vertical shift up by $k$ units. Here, $f(x)=2^x$, new function is $2^x + 3$, so $k = 3>0$.

Step2: Determine transformation

So the transformation is a vertical shift up by 3 units.

Step1: Recall vertical shift rule

For $y = f(x)+k$, if $k<0$, it's a vertical shift down by $|k|$ units. Here, $k=-4$, so $|k| = 4$.

Step2: Determine transformation

Thus, the transformation is a vertical shift down by 4 units.

Step1: Recall horizontal shift rule

For a function $y = f(x + h)$, if $h>0$, it's a horizontal shift left by $h$ units. Here, $f(x)=2^x$, new function is $2^{x+2}$, so $h = 2>0$.

Step2: Determine transformation

So the transformation is a horizontal shift left by 2 units.

Answer:

Vertical shift up by 3 units

2. $f(x) = 2^x - 4$