Question

16.
what is the equation of the horizontal asymptote for following exponential function?
a) ( y = 6 )
b)

c) ( y < 6 )
d) ( y = 8 )
18
the line shown on the graph is an asymptote for the exponential curve shown on the graph.
a) true
b) false
17.
the line shown on the graph is an asymptote for the exponential curve shown on the graph.
a) false
b) true

Explanation:

Question 16

Step1: Recall horizontal asymptote of exponential functions

For an exponential function \( y = a^x + k \) (or similar forms), the horizontal asymptote is \( y = k \), which is a horizontal line. The graph of the exponential function approaches this line as \( x \to \pm\infty \) (depending on the function). From the graph, we observe the horizontal asymptote. Looking at the options, asymptotes are equations (lines), so options with inequalities (like \( y < 6 \)) are invalid. The horizontal line here is \( y = 6 \) (assuming the graph's asymptote is at \( y = 6 \), and option a is \( y = 6 \), d is \( y = 8 \) which doesn't match). Wait, maybe the graph's asymptote is \( y = 6 \)? Wait, let's check the options. Option a: \( y = 6 \), b: (maybe typo, but c is \( y < 6 \) (inequality, not an equation of a line), d: \( y = 8 \). So the horizontal asymptote is a horizontal line, so equation is \( y = 6 \) (option a? Wait, maybe the graph in 16 has horizontal asymptote \( y = 6 \). So the correct option is a) \( y = 6 \).

Step2: Eliminate invalid options

• Option c: \( y < 6 \) is an inequality, not the equation of a horizontal line (asymptote is a line, so equation is \( y = k \)), so eliminate c.
• Option d: \( y = 8 \) – if the graph's asymptote is at \( y = 6 \), then d is wrong.
• Option b: (assuming typo, but a is \( y = 6 \), which is a horizontal line equation, so a is correct.

Step1: Recall asymptote definition

An asymptote is a line that a curve approaches as \( x \to \pm\infty \). For the exponential curve in the graph, the horizontal line shown is approached by the curve, so it's a horizontal asymptote. Thus, the statement "The line shown on the graph is an asymptote for the exponential curve" is TRUE.

Step2: Confirm with exponential function properties

Exponential functions \( y = a^x + k \) have horizontal asymptote \( y = k \). The graph here shows the exponential curve approaching the horizontal line, so the line is the asymptote, so the statement is TRUE (option a).

Step1: Recall asymptote of exponential functions

For an exponential function (like \( y = a^x + k \) or \( y = -a^x + k \), but here the graph is an exponential curve (maybe \( y = a^x - 4 \) or similar). The horizontal asymptote is a horizontal line. The line shown is a horizontal line (like \( y = -4 \) or similar). The statement is "The line shown on the graph is an asymptote for the exponential curve". Let's see: exponential curves have horizontal asymptotes. The graph's curve approaches the horizontal line, so the line is an asymptote. So the statement is TRUE (option b).

Step2: Check the graph

The exponential curve (like a parabola? No, exponential function) – the curve approaches the horizontal line, so the line is an asymptote. Thus, the statement is TRUE, so option b) TRUE.

Answer:

a) \( y = 6 \)

Question 18 (Wait, the numbering: 16, 18? Wait, the second graph is 18? Wait, the user's image has 16, 18, 17? Wait, the second problem is labeled 18? Wait, the second graph is "18" (maybe typo, original is 18? Wait, the text: "18. The line shown on the graph is an asymptote for the exponential curve shown on the graph. a) TRUE b) FALSE". From the graph, the horizontal line (asymptote) – exponential functions have horizontal asymptotes. The graph shows the exponential curve approaching the horizontal line, so the line is an asymptote. So the answer is a) TRUE.