AP Pre-Calculus
Understanding Change of Rate Graphs in AP Pre-Calculus
1. Let f be a decreasing function where the rate of change of f is increasing.
1. Let f be a decreasing function where the rate of change of f is increasing. Which of the following could be the graph of f?
(pre calculus rate of change)
Solution:
If f is decreasing, then the graph must go down from left to right. That means the slope is negative.
If the rate of change of f is increasing, then the slope is becoming less negative.
Example:
The slope may change from -5 to -3 to -1.
So the function is still decreasing, but it is decreasing more slowly. This means the graph is: decreasing and concave up
Now check the choices:
- A is increasing, so it is not correct.
- B is increasing, so it is not correct.
- C is decreasing and concave up, so it is correct.
- D is decreasing, but it is concave down because the slope becomes more negative.
Final answer: C
AP Precalculus tip:
When a problem says “decreasing function,” look for a graph that moves downward from left to right. When it says “rate of change is increasing,” look for slopes that are becoming larger. For a decreasing function, that means the slopes are negative but becoming closer to zero. This is why the correct graph is decreasing and concave up.
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