Question

an online retailer developed two exponential functions to model the weekly usage of two coupon codes where x is the number of weeks since the start of the year. the equation represents the weekly usage of code 1, and the table represents the weekly usage of code 2.

code 1 usage	code 2 usage
		0	962
		4	240
		8	67
		12	25
		16	15
		20	13
	which statement correctly interprets the end behavior of the two functions?

a. the weekly usage of both coupons is decreasing and approaching a horizontal asymptote as x gets larger.

b. the weekly usage of both coupons is decreasing, but only the usage of code 2 is approaching a horizontal asymptote as x gets larger.

c. the weekly usage of code 1 is increasing, and the weekly usage of code 2 is decreasing.

d. the weekly usage of both coupons is decreasing, but only the usage of code 1 is approaching a horizontal asymptote as x gets larger.

Explanation:

1. Analyze Code 1 (\(f(x) = 800(0.75)^x + 10\)):

• The base of the exponential term, \(0.75\), is between \(0\) and \(1\), so as \(x\) (number of weeks) increases, \((0.75)^x\) decreases towards \(0\). Thus, \(800(0.75)^x\) decreases towards \(0\), and \(f(x)=800(0.75)^x + 10\) decreases towards \(10\) (horizontal asymptote \(y = 10\)) as \(x\) gets larger.

1. Analyze Code 2 (table of \(g(x)\)):

• From the table, as \(x\) (weeks) increases (from \(0\) to \(20\)), \(g(x)\) values decrease (962 → 240 → 67 → 25 → 15 → 13). As \(x\) gets very large, looking at the trend, \(g(x)\) seems to approach a horizontal asymptote (since the decrease slows down, approaching a stable value, likely around \(13\) or lower as \(x\) increases further, but the key is the end - behavior of decreasing and approaching an asymptote? Wait, no—wait, for Code 1, the function has a horizontal asymptote because it's a transformed exponential function. For Code 2, the table shows values decreasing, but does it have a horizontal asymptote? Wait, no—wait, the options: Let's re - evaluate.
• Wait, Code 1: The function \(f(x)=800(0.75)^x+10\) is an exponential decay function (since \(0\lt0.75\lt1\)) with a horizontal asymptote at \(y = 10\) (because as \(x

ightarrow\infty\), \(800(0.75)^x
ightarrow0\), so \(f(x)
ightarrow10\)). So as \(x\) increases, \(f(x)\) decreases towards \(10\).

• Code 2: From the table, as \(x\) increases (from \(0\) to \(20\)), \(g(x)\) values are decreasing (962, 240, 67, 25, 15, 13). Now, for the end - behavior: Does \(g(x)\) have a horizontal asymptote? The values are decreasing but not approaching a fixed non - zero value in the same way as Code 1? Wait, no—wait, the options: Let's check the options again.
• Option D: "The weekly usage of both coupons is decreasing, but only the usage of code 1 is approaching a horizontal asymptote as \(x\) gets larger."
• Wait, why? Because Code 1 is a mathematical exponential function with a horizontal asymptote (\(y = 10\)). Code 2 is a set of data points. As \(x\) gets larger, the data for Code 2 is decreasing, but there's no indication that it's approaching a horizontal asymptote (the values are just decreasing, but we don't have a function for Code 2, just a table. However, Code 1 has a defined exponential function with a horizontal asymptote. So both are decreasing (Code 1: exponential decay, Code 2: data decreasing), but only Code 1 has a horizontal asymptote.

Answer:

D. The weekly usage of both coupons is decreasing, but only the usage of code 1 is approaching a horizontal asymptote as \(x\) gets larger.