Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsHard
A meteorologist is tracking the temperature in a region. The temperature T (in degrees Fahrenheit) on a particular day can be modeled by the function T(h) = -|h - 14| + 70, where h is the hour of the day (from 0 to 24). What is the maximum temperature for the day according to this model?
- A70 degrees F
- B56 degrees F
- C14 degrees F
- D84 degrees F
Show answer & explanationAnswer & explanation
Correct answer: A. 70 degrees F
The function T(h) = -|h - 14| + 70 is an absolute value function. The term -|h - 14| is always less than or equal to 0 (since |h - 14| is always non-negative, and the negative sign flips it). The maximum value of -|h - 14| occurs when |h - 14| = 0, which means h = 14. At this point, T(14) = -|14 - 14| + 70 = -0 + 70 = 70. This is the vertex (14, 70), and since the absolute value term is negative, the graph opens downwards, making the vertex a maximum.
Why the other options are wrong
- B. Incorrect. This would be 70 - 14, an incorrect interpretation of the function.
- C. Incorrect. This is the h-coordinate of the vertex (the time of maximum temperature), not the temperature itself.
- D. Incorrect. This would be 70 + 14, which would happen if the absolute value term was positive, or if there was a calculation error.
Maximum of Absolute Value Function
For an absolute value function of the form f(x) = -a|x - h| + k (where a > 0), the maximum value is 'k', occurring at the vertex (h, k).
- Graph is a 'V' shape opening downwards.
- Maximum value is the y-coordinate of the vertex.
- Vertex is (h, k) for f(x) = a|x - h| + k.
Memory trick: Absolute value functions point up or down like a V. The 'k' is the peak or lowest point!