Question

using a graph to determine key features
reset
analyze the function’s graph to determine which statement is true. hover over (or tap) a point to see its coordinates.
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 ).
over the interval ( -2.5, 0.5 ), the local maximum is 2.

Explanation:

Step1: Analyze each statement

- For "As \( x \to +\infty \), \( f(x) \to -\infty \)": The right end of the graph rises, so \( f(x) \to +\infty \), false.
- For "Decreasing on \( (-1, 0.75) \)": The graph decreases from \( x=-1 \) to the minimum (around \( x=0 \)) then increases? Wait, no—looking at the graph, from \( x=-1 \) to the vertex at \( x \approx 0 \) (where \( y=-2 \)), then increases? Wait, no, the graph has a minimum at \( (0, -2) \)? Wait, no, the point at \( x=0 \) is \( (0, -2) \), then it rises. So from \( x=-1 \) to \( x=0.75 \), does it decrease? Wait, at \( x=-1 \), \( y=0 \); at \( x=0 \), \( y=-2 \); at \( x=0.75 \), \( y \) is rising? Wait, no, the graph after \( x=0 \) rises. Wait, maybe I misread. Wait, the graph: left part (before \( x=-2 \)) goes down, then from \( x=-3 \) to \( x=-2 \) it rises to a peak at \( x=-2 \) ( \( y=2 \) ), then falls to \( x=-1 \) ( \( y=0 \) ), then falls to \( x=0 \) ( \( y=-2 \) ), then rises to \( x=1 \) ( \( y=0 \) ), then rises. So from \( x=-1 \) to \( x=0 \), it's decreasing (from \( y=0 \) to \( y=-2 \)), then from \( x=0 \) to \( x=0.75 \), it's increasing. Wait, but the interval is \( (-1, 0.75) \). So from \( -1 \) to \( 0 \), decreasing; \( 0 \) to \( 0.75 \), increasing. But maybe the statement is about the overall trend? Wait, no, let's check the third statement: "Function is negative on \( [-2, 0] \)". At \( x=-2 \), \( y=2 \) (positive); at \( x=-1 \), \( y=0 \); at \( x=0 \), \( y=-2 \) (negative). So on \( [-2, 0] \), there are positive values (at \( x=-2 \)), so false. Fourth statement: "Over \( [-2.5, 0.5] \), local max is 2". The peak at \( x=-2 \) is \( y=2 \), and in \( [-2.5, 0.5] \), that's the local maximum, so this is true. Wait, let's recheck:

1. First statement: As \( x \to +\infty \), the graph goes up, so \( f(x) \to +\infty \), false.
2. Second statement: On \( (-1, 0.75) \), from \( x=-1 \) ( \( y=0 \) ) to \( x=0 \) ( \( y=-2 \) ) it decreases, then from \( x=0 \) to \( x=0.75 \) it increases (since it goes from \( y=-2 \) to \( y \) positive at \( x=1 \)). So it's not decreasing over the entire interval, false.
3. Third statement: On \( [-2, 0] \), at \( x=-2 \), \( y=2 \) (positive), so the function is not negative here, false.
4. Fourth statement: Over \( [-2.5, 0.5] \), the local maximum is at \( x=-2 \) with \( y=2 \), which is within the interval. So this is true. Wait, but let's check the second statement again. Wait, maybe I made a mistake. Wait, the graph: after \( x=-1 \), it goes down to \( x=0 \) ( \( y=-2 \) ), then up. So on \( (-1, 0.75) \), from \( -1 \) to \( 0 \), decreasing; \( 0 \) to \( 0.75 \), increasing. So the function is not decreasing over the entire interval, so second statement is false. Third statement: \( [-2, 0] \): at \( x=-2 \), \( y=2 \) (positive), so function is positive there, so third statement is false. Fourth statement: \( [-2.5, 0.5] \): the local maximum in this interval is at \( x=-2 \), \( y=2 \), so that's true. Wait, but let's check the options again. Wait, the fourth option: "Over the interval \( [-2.5, 0.5] \), the local maximum is 2." Yes, because at \( x=-2 \) (which is in \( [-2.5, 0.5] \)), the function reaches 2, which is a local maximum. The other statements:

- First: As \( x \to +\infty \), graph rises, so \( f(x) \to +\infty \), false.
- Second: On \( (-1, 0.75) \), the function decreases from \( x=-1 \) to \( x=0 \) ( \( y \) from 0 to -2) then increases from \( x=0 \) to \( x=0.75 \) ( \( y \) from -2 to positive), so not decreasing over the entire interval, false.
- Third: On \( [-2, 0] \), at \( x=-2 \), \( y=2 \) (positive), so function is not negative here, false.
- Fourth: True, as the local max at \( x=-2 \) (in \( [-2.5, 0.5] \)) is 2.

Wait, but maybe I misread the second statement. Wait, the interval is \( (-1, 0.75) \). Let's check the coordinates: at \( x=-1 \), \( y=0 \); at \( x=0 \), \( y=-2 \); at \( x=0.75 \), let's see, the graph at \( x=1 \) is \( y=0 \), so at \( x=0.75 \), \( y \) is between -2 and 0? No, wait, the graph after \( x=0 \) rises, so from \( x=0 \) ( \( y=-2 \) ) to \( x=1 \) ( \( y=0 \) ), it's increasing. So on \( (-1, 0) \), decreasing; \( (0, 0.75) \), increasing. So the function is not decreasing over \( (-1, 0.75) \), so second statement is false. Third statement: \( [-2, 0] \): at \( x=-2 \), \( y=2 \) (positive), so function is positive, so third statement is false. Fourth statement: Over \( [-2.5, 0.5] \), the local maximum is 2 (at \( x=-2 \)), which is correct. So the true statement is the fourth one: "Over the interval \( [-2.5, 0.5] \), the local maximum is 2."

Answer:

Over the interval \([-2.5, 0.5]\), the local maximum is 2.