Question

a company made a change in its advertising. suppose the companys monthly revenue (in thousands of dollars) x months after the change can be modeled with an exponential function. the graph of this function is shown below. use the model to answer the parts to the right. (a) what is the y-intercept of this graph? the y-intercept represents the select. (b) for the first 10 months, as time increases, the monthly revenue select. (c) give the equation of the asymptote. y = choose the statement that best describes the meaning of the asymptote. the company will only stay in business for 20 months. the companys monthly revenue does not rise above 200 thousand dollars per month. the companys monthly revenue does not fall below 50 thousand dollars per month.

Explanation:

Part (a)

Step 1: Identify y-intercept

The y-intercept occurs where \( x = 0 \) (time = 0 months, right after the advertising change). From the graph, when \( x = 0 \), the revenue (y - value) is 50 (thousand dollars). The y - intercept represents the initial monthly revenue (when \( x = 0 \), i.e., the revenue at the time the advertising change was made).

Step 2: Interpret y - intercept

The y - intercept of a revenue - time function at \( x = 0 \) is the revenue at the start (when the advertising change was implemented). So the y - intercept is 50, and it represents the company's monthly revenue when the advertising change was made (at \( x = 0 \) months).

Part (b)

Step 1: Analyze the graph's trend

Looking at the graph of the exponential function for the first 10 months (as \( x \) increases from 0 to 10), we can see that the curve is increasing but approaching a horizontal line (the asymptote). So, as time (x) increases in the first 10 months, the monthly revenue increases towards the asymptote (but is still increasing).

Part (c)

Step 1: Identify the asymptote

The horizontal asymptote of an exponential function (from the graph) is the horizontal line that the graph approaches as \( x \) gets larger. From the graph, we can see that the graph approaches \( y = 200 \).

Step 2: Interpret the asymptote

• The option "The company will only stay in business for 20 months" is incorrect because the asymptote has nothing to do with the duration of the business.
• The option "The company's monthly revenue does not rise above 200 thousand dollars per month" is correct because the graph approaches \( y = 200 \) as \( x \) increases, meaning the revenue gets closer and closer to 200 thousand dollars but never exceeds it (for large \( x \)).
• The option "The company's monthly revenue does not fall below 50 thousand dollars per month" is incorrect because the y - intercept is 50, but the graph starts at 50 and increases, so it's not a lower bound in the sense of not falling below, it's the starting point.

Answer:

s:
(a) The y - intercept is \(\boldsymbol{50}\). It represents the company's monthly revenue when the advertising change was made (at \( x = 0 \) months).
(b) As time increases, the monthly revenue \(\boldsymbol{\text{increases (towards the asymptote)}}\).
(c) The equation of the asymptote is \( y=\boldsymbol{200} \). The correct statement is "The company's monthly revenue does not rise above 200 thousand dollars per month."