Question

Analyze the function’s graph to determine which statement is true.

Over the interval [–2.5, 0.5], the local maximum is 2.
As the x-values go to positive infinity, the function’s values go to negative infinity.
The function is decreasing over the interval (–1, 0.75).
The function is negative for the interval [–2, 0].

Answer 1

I don’t have the graph — could you upload it or describe its key features (locations of peaks/valleys, x- and y-intercepts, and the left/right end behavior)? Without seeing the graph I can’t say which statement is true.

To help you check each statement yourself, here are the tests to apply to the graph:

1. "Over the interval [–2.5, 0.5], the local maximum is 2."
- A local maximum at y = 2 means there is a point x0 in (–2.5, 0.5) with f(x0) = 2, and in some open neighborhood around x0 all other f(x) ≤ 2 (and some are < 2). Check whether such a point exists inside that interval (endpoints do not count as a local max unless the graph’s context treats endpoints specially). If the highest nearby value equals 2, this statement is true.

2. "As the x-values go to positive infinity, the function’s values go to negative infinity."
- Look at the right-hand end of the graph: does it go down without bound as x→+∞ (the curve heads down toward −∞)? If instead it levels off or rises, this statement is false.

3. "The function is decreasing over the interval (–1, 0.75)."
- The function must decrease (slope negative) at every point throughout that entire open interval. If there are any places where it is flat or increasing inside (–1, 0.75), the statement is false.

4. "The function is negative for the interval [–2, 0]."
- For every x between −2 and 0 (including endpoints), f(x) must be below the x-axis (f(x) < 0). If the graph crosses or touches the x-axis anywhere in that closed interval, the statement is false.

If you post the graph or describe: x-intercepts, y-values of peaks/valleys and tail directions, I’ll check each statement and tell you which one is true.