- The x-intercepts are found at \(x=0\) and \(x=50\) since \(f(x)=0\) at these points.
- The difference between these two intercepts is \(50 - 0 = 50\).
- This value of 50 represents the duration of the promo code distribution period (the number of days the codes were released).
- Therefore, the correct interpretation is that their difference is the number of days the promo codes were released. Final answer is the fourth option: \(\boxed{\text{Their difference is the number of days the first promo codes were released before the new record.}}\)
Explanation
- Identifying the X-Intercepts
The table provides data for a function \(f(x)\), where \(x\) represents the time in days since posting and \(f(x)\) represents the number of promo codes in tens of thousands. We observe the values:
\begin{itemize}
\item \(x = 0, f(x) = 0\)
\item \(x = 25, f(x) = 45\)
\item \(x = 50, f(x) = 0\)
\item \(x = 75, f(x) = -150\)
\end{itemize}
An x-intercept of a function occurs when \(f(x) = 0\). Based on the provided table, the function has x-intercepts at \(x = 0\) and \(x = 50\). The interval \([0, 50]\) represents the duration during which the promo codes were being released before the value dropped to zero.
- Evaluating the Interpretations
We have identified the two x-intercepts as \(x_1 = 0\) and \(x_2 = 50\). We need to determine the correct interpretation of these values by analyzing their sum and difference:
\begin{itemize}
\item The sum of the x-intercepts is \(0 + 50 = 50\).
\item The difference of the x-intercepts is \(50 - 0 = 50\).
\end{itemize}
Now, let's evaluate the provided statements:
\begin{itemize}
\item "Their difference is the number of the most promo codes in a single day": The maximum value (most codes) is 45, which does not match the difference of 50.
\item "Their sum is the total number of days the number of codes increased": The number of codes increases from \(x=0\) to \(x=25\), which is a duration of 25 days, not the sum of 50.
\item "Their sum is the total number of promo codes...": The sum is 50, which is a number of days, not a quantity of promo codes (which are given in tens of thousands).
\item "Their difference is the number of days the first promo codes were released before the new record": The difference is 50, which corresponds precisely to the duration from the start of the promotion (\(x=0\)) until the release ended (\(x=50\)).
\end{itemize}
Thus, the correct interpretation is that their difference is the number of days the promo codes were released.
- Conclusion
Based on the analysis, the difference between the two x-intercepts (\(50 - 0 = 50\)) represents the total number of days during which the promotion was active. This matches the interpretation that the difference corresponds to the duration of the promo code distribution period. Therefore, the last statement provided is the correct one. The final answer is the fourth option.
Examples
In marketing and analytics, understanding the time frame of a campaign is crucial. If you are launching a product, you might track sales over time. The x-intercepts of your sales growth function would tell you the duration of your campaign—from the day you launch (\(x=0\)) to the day the specific promotional interest dies out (\(x=50\)). By calculating the difference, you identify exactly how many days your campaign was effective, helping you plan future strategies more efficiently.