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The table represents an absolute value function [tex]f(x)[/tex].

\[
\begin{array}{cc}
x & f(x) \\
\hline
-2 & 4 \\
-1 & 3 \\
0 & 2 \\
1 & 1 \\
2 & 0 \\
3 & 1 \\
4 & 2 \\
5 & 3 \\
6 & 4 \\
\end{array}
\]

What are the vertex and range of the function?

A. Vertex (0, 2), Range: [tex]\{y \mid 0 \leq y < \infty\}[/tex]
B. Vertex (0, 2), Range: [tex]\{y \mid 2 \leq y < \infty\}[/tex]
C. Vertex (2, 0), Range: [tex]\{y \mid 0 \leq y < \infty\}[/tex]
D. Vertex (2, 0), Range: [tex]\{y \mid 2 \leq y < \infty\}[/tex]

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A medical clinic is reducing the number of incoming patients by giving vaccines before flu season. During week 5 of flu season, the clinic saw 75 patients. In week 10 of flu season, the clinic saw 50 patients. Assume the reduction in the number of patients each week is linear. Write an equation in function form to show the number of patients seen each week at the clinic.

A. [tex]f(x) = 5x + 100[/tex]
B. [tex]f(x) = -5x + 100[/tex]
C. [tex]f(x) = 25x + 75[/tex]
D. [tex]f(x) = -25x + 75[/tex]

Answer :

Final answer:

The vertex of the absolute value function is (2, 0) with a range of non-negative real numbers. The linear function that represents the decrease of patients in a clinic each week is f(x) = -5x + 100.

Explanation:

The first part of the question asks about the vertex and range of an absolute value function. By looking at the table, we observe that the function decreases until x = 2, then starts increasing. This means that the vertex of the function is at the point where the direction changes, which is at (2, 0). The range of an absolute value function is all non-negative real numbers, so the range is {y | y ≥ 0}.

The second part of the question involves a linear function, which describes a constant rate of change. Here, the number of patients is decreasing from week 5 to week 10, so this rate of change is negative. The difference in the number of patients from week 5 to week 10 is 75 - 50 = 25, over a 5-week span.

Therefore, the rate of change is -25/5 = -5 patients per week. Assuming the trend started at week 0 with 100 patients, the function form of the linear equation would be f(x) = -5x + 100. This equation shows the number of patients the clinic will have left to see in any given week.

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