A publicist is promoting a new record. The table below represents the plan for providing promo codes for free downloads of a single from the record, [tex]f(x)[/tex], in tens of thousands of codes depending on the time since posting, [tex]x[/tex], in days.
Which is the correct interpretation of the x-intercepts of the function?
A. Their difference is the number of the most promo codes in a single day.
B. Their sum is the total number of days the number of codes increased.
C. Their sum is the total number of promo codes the publicist is releasing before the new record.
D. Their difference is the number of days the first promo codes were released before the record.
Answer (2)
The x-intercepts of the function are 0 and u, where f(0) = 0 and f(u) = 0.
Approximate the function as a quadratic: f ( x ) = − 0.076 x 2 + 3.7 x .
Calculate u by solving − 0.076 x 2 + 3.7 x = 0 , which gives u ≈ 48.68 .
The difference between the x-intercepts (u - 0) represents the number of days between the first and last release of promo codes, so the answer is their difference is the number of days the first promo codes were released before the promo codes were no longer available.
Explanation
Understanding the Problem We are given a table representing the number of promo codes, f ( x ) , in tens of thousands, released x days after posting. The table provides four data points: (0, 0), (25, 45), (u, 0), and (75, -150). We need to determine the correct interpretation of the x-intercepts of the function.
Identifying the x-intercepts The x-intercepts are the points where f ( x ) = 0 . From the given data, we know that (0, 0) and (u, 0) are x-intercepts. Therefore, the x-intercepts are 0 and u . We can approximate the function f ( x ) using the given points. Since we have three points with defined x and y values, we can fit a quadratic function to these points. The points are (0,0), (25,45), and (75,-150).
Finding the Quadratic Function Let's assume the quadratic function is of the form f ( x ) = a x 2 + b x + c . Using the point (0,0), we have f ( 0 ) = a ( 0 ) 2 + b ( 0 ) + c = 0 , so c = 0 . Now we have f ( x ) = a x 2 + b x . Using the point (25,45), we have f ( 25 ) = a ( 25 ) 2 + b ( 25 ) = 625 a + 25 b = 45 . Using the point (75,-150), we have f ( 75 ) = a ( 75 ) 2 + b ( 75 ) = 5625 a + 75 b = − 150 . We can simplify the equations to: 625 a + 25 b = 45 and 5625 a + 75 b = − 150 . Dividing the first equation by 25, we get 25 a + b = 25 45 = 5 9 = 1.8 . Dividing the second equation by 75, we get 75 a + b = − 2 . Subtracting the first simplified equation from the second, we get 50 a = − 2 − 1.8 = − 3.8 , so a = 50 − 3.8 = − 0.076 . Substituting a back into the first simplified equation, we get 25 ( − 0.076 ) + b = 1.8 , so − 1.9 + b = 1.8 , and b = 1.8 + 1.9 = 3.7 . Thus, the quadratic function is approximately f ( x ) = − 0.076 x 2 + 3.7 x .
Calculating the x-intercepts To find the x-intercepts, we set f ( x ) = 0 , so − 0.076 x 2 + 3.7 x = 0 . Factoring out x , we get x ( − 0.076 x + 3.7 ) = 0 . Thus, the x-intercepts are x = 0 and − 0.076 x + 3.7 = 0 , which means 0.076 x = 3.7 , so x = 0.076 3.7 ≈ 48.68 . Therefore, u ≈ 48.68 .
Evaluating the Interpretations Now, let's evaluate the proposed interpretations:
"Their difference is the number of the most promo codes in a single day." The difference is ∣ u − 0∣ = ∣ u ∣ ≈ 48.68 . The maximum number of promo codes occurs at the vertex of the parabola. The x-coordinate of the vertex is given by x = − 2 a b = − 2 ( − 0.076 ) 3.7 = 0.152 3.7 ≈ 24.34 . The maximum number of promo codes is f ( 24.34 ) = − 0.076 ( 24.34 ) 2 + 3.7 ( 24.34 ) ≈ 45.03 . This interpretation is incorrect.
"Their sum is the total number of days the number of codes increased." The sum is u + 0 = u ≈ 48.68 . The number of codes increased from x = 0 to x ≈ 24.34 (the vertex). This interpretation is incorrect.
"Their sum is the total number of promo codes the publicist is releasing before the new record." The sum is u + 0 = u ≈ 48.68 . This represents the time (in days) when the promo codes are no longer available. This interpretation is incorrect.
"Their difference is the number of days the first promo codes were released befor c y record" This option seems to have a typo. Assuming it means 'before the promo codes were no longer available', the difference is ∣ u − 0∣ = ∣ u ∣ ≈ 48.68 . This represents the number of days between the first and last release of promo codes. This interpretation is the most plausible.
Final Answer The correct interpretation of the x-intercepts is that their difference is the number of days between the first and last release of promo codes. Assuming the typo in the last option is corrected, this is the most plausible interpretation. Therefore, the answer is: Their difference is the number of days the first promo codes were released before the promo codes were no longer available.
Examples
Understanding the x-intercepts of a function can help businesses determine the duration of a marketing campaign. For example, if a company launches a promotional offer, the x-intercepts can represent the start and end dates of the promotion. By analyzing the function representing the campaign's effectiveness, the company can optimize the timing and duration of future campaigns to maximize their impact. This helps in resource allocation and strategic planning.
The x-intercepts of the function represent the points in time when the promo codes are available. Their difference indicates the duration, measured in days, between the first release of promo codes and when they are no longer available. Therefore, the correct answer is that their difference is the number of days the first promo codes were released before the new record.
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