Question

use transformations of the graph of f(x)=2^x to graph the given function. be sure to graph and give the equation of the asymptote. use the graph to determine the functions domain and range. if applicable, use a graphing utility to confirm your hand - drawn graphs. g(x)=2^x + 2 graph g(x)=2^x + 2 and its asymptote. use the graphing tool to graph the function as a solid curve and the asymptote as a dashed line.

Explanation:

Step1: Analyze the parent function

The parent function is \( f(x) = 2^x \). The graph of \( f(x) = 2^x \) has a horizontal asymptote at \( y = 0 \) (since as \( x \to -\infty \), \( 2^x \to 0 \)) and passes through the point \( (0, 1) \) (when \( x = 0 \), \( f(0)=2^0 = 1 \)). Its domain is \( (-\infty, \infty) \) and range is \( (0, \infty) \).

Step2: Determine the transformation

For the function \( g(x)=2^x + 2 \), we are adding 2 to the parent function \( f(x)=2^x \). This represents a vertical shift. In general, for a function \( y = f(x)+k \), if \( k>0 \), the graph of \( f(x) \) is shifted up by \( k \) units. Here, \( k = 2 \), so the graph of \( g(x) \) is the graph of \( f(x)=2^x \) shifted up by 2 units.

Step3: Find the asymptote of \( g(x) \)

The horizontal asymptote of the parent function \( f(x) = 2^x \) is \( y = 0 \). When we shift the graph up by 2 units, the horizontal asymptote also shifts up by 2 units. So the equation of the horizontal asymptote for \( g(x) \) is \( y=2 \).

Step4: Determine the domain and range of \( g(x) \)

• Domain: The domain of an exponential function of the form \( a^x + k \) (where \( a>0,a

eq1 \)) is all real numbers, because we can plug in any real number for \( x \) into the exponential function. So the domain of \( g(x)=2^x + 2 \) is \( (-\infty, \infty) \).

• Range: The range of the parent function \( f(x)=2^x \) is \( (0, \infty) \). When we shift the graph up by 2 units, all the \( y \)-values of the function are increased by 2. So the new range is \( (0 + 2, \infty)=(2, \infty) \).

Step5: Graphing (description)

To graph \( g(x)=2^x + 2 \):

1. Start with the graph of \( f(x)=2^x \). Plot some key points of \( f(x) \), e.g., \( (-2, \frac{1}{4}) \), \( (-1, \frac{1}{2}) \), \( (0, 1) \), \( (1, 2) \), \( (2, 4) \).
2. Shift each of these points up by 2 units. So the new points will be \( (-2, \frac{1}{4}+2=\frac{9}{4}) \), \( (-1, \frac{1}{2}+2=\frac{5}{2}) \), \( (0, 1 + 2=3) \), \( (1, 2 + 2=4) \), \( (2, 4 + 2=6) \).
3. Draw a smooth curve through these shifted points.
4. Draw the horizontal asymptote \( y = 2 \) as a dashed line.

Answer:

• Asymptote Equation: \( y = 2 \)
• Domain of \( g(x) \): \( (-\infty, \infty) \)
• Range of \( g(x) \): \( (2, \infty) \)
• Graph Description: The graph of \( g(x)=2^x + 2 \) is the graph of \( y = 2^x \) shifted up 2 units. It passes through points like \( (-2, \frac{9}{4}) \), \( (-1, \frac{5}{2}) \), \( (0, 3) \), \( (1, 4) \), \( (2, 6) \) and has a horizontal asymptote \( y = 2 \) (dashed line).