Question

use transformations to graph the following function. determine the domain, range, horizontal asymptote, and y-intercept of the function. f(x) = 6^x + 4 a. the graph of y = 6^x should be shifted vertically up by 4 units. b. the graph of y = 6^x should be shifted vertically down by 4 units. c. the graph of y = 6^x should be shifted horizontally left by 4 units. d. the graph of y = 6^x should be shifted horizontally right by 4 units. use the graphing tool to graph the function. (for any answer boxes shown with the grapher, type an exact answer.) the domain of f(x) is (-∞,∞). (type your answer in interval notation.) the range of f(x) is (0,∞). (type your answer in interval notation.)

Explanation:

Step1: Analyze Vertical Shift

For a function \( y = f(x)+k \), if \( k>0 \), the graph of \( y = f(x) \) is shifted vertically up by \( k \) units. Here, \( f(x)=6^{x}+4 \) is \( y = 6^{x}+4 \), so \( k = 4>0 \), meaning vertical shift up by 4 units.

Step2: Analyze Domain

The function \( f(x)=6^{x}+4 \) is an exponential function. Exponential functions of the form \( a^{x}+b \) (where \( a>0,a
eq1 \)) have domain all real numbers, so domain is \( (-\infty,\infty) \).

Step3: Analyze Range

The range of \( y = 6^{x} \) is \( (0,\infty) \). When we shift it up by 4 units, the range becomes \( (4,\infty) \) (since \( 6^{x}>0\implies6^{x}+4 > 4 \)). Wait, the original range in the problem was wrong. Let's correct:
For \( y = 6^{x} \), range is \( (0,\infty) \). For \( y=6^{x}+4 \), as \( 6^{x}>0 \), then \( 6^{x}+4>4 \), so range is \( (4,\infty) \).

Step4: Horizontal Asymptote

The horizontal asymptote of \( y = 6^{x} \) is \( y = 0 \). After vertical shift up by 4 units, the horizontal asymptote becomes \( y=4 \).

Step5: y - intercept

To find y - intercept, set \( x = 0 \). Then \( f(0)=6^{0}+4=1 + 4=5 \), so y - intercept is \( (0,5) \).

Answer:

• Transformation: A. The graph of \( y = 6^{x} \) should be shifted vertically up by 4 units.
• Domain: \( (-\infty,\infty) \)
• Range: \( (4,\infty) \) (corrected from the original wrong range in the problem)
• Horizontal Asymptote: \( y = 4 \)
• y - intercept: \( (0,5) \)