Question

identify the following key characteristics of the graphed function:
domain:
range:
end behavior (both):
x-intercept(s):
y-intercept:
relative maximum(s):
relative minimum(s):
absolute maximum:
absolute minimum:
intervals when the graph is increasing:
intervals when the graph is decreaseing:

Explanation:

Step1: Domain

The graph extends infinitely left and right, so domain is all real numbers: $(-\infty, \infty)$

Step2: Range

The lowest point (from the graph) seems to go to $-\infty$ (left end) and the right end goes up, but the visible minima and maxima: the graph has a low point, but since left end goes down, range is $(-\infty, \infty)$? Wait, no, looking at the graph: left end arrow down, right end arrow up. Wait, the graph has a peak, then a valley, then a peak, then a valley, then up. Wait, the y-axis has -60, -40, -20, 0, 20. Wait, maybe I misread. Wait, the left end is going down (arrow down), so as $x\to -\infty$, $y\to -\infty$. The right end is going up (arrow up), so as $x\to \infty$, $y\to \infty$. So range is all real numbers: $(-\infty, \infty)$

Step3: End Behavior

As $x\to -\infty$, $y\to -\infty$ (left end arrow down); as $x\to \infty$, $y\to \infty$ (right end arrow up)

Step4: x-intercepts

Points where graph crosses x-axis. From graph: x=-3 (approx), x=0 (wait, no, at x=0, is it on x-axis? Wait, the graph crosses x-axis at x=-3 (left of -2), x=1 (between 0 and 2), and maybe x=3? Wait, looking at grid: x=-3 (since between -4 and -2, maybe x=-3), x=1 (between 0 and 2), and x=3? Wait, the graph crosses x-axis at three points? Wait, the left part: comes from down, crosses x at x=-3 (approx), then peaks, then valleys at x=-1 (approx, y=0? Wait, no, the valley at x=-1 (between -2 and 0) is on x-axis? Wait, the graph: from left (down), crosses x at x=-3, then up to peak, then down to a valley at x=-1 (y=0?), then up to a peak, then down to a valley at x=2 (y=-60?), then up. Wait, maybe x-intercepts at x=-3, x=0 (no, the valley at x=-1 is on x-axis? Wait, the grid: x=-4, -3, -2, -1, 0, 1, 2, 3, 4. The graph crosses x-axis at x=-3 (left of -2), x=-1 (between -2 and 0, y=0), and x=1 (between 0 and 2), and x=3? Wait, maybe x=-3, x=-1, x=1, x=3? Wait, no, let's see: the graph: starts from down (left), crosses x at x=-3, goes up to a peak (x=-2, y=20+), then down to a valley at x=-1 (y=0), then up to a peak (x=1, y=20), then down to a valley at x=3 (y=-60), then up. So x-intercepts at x=-3, x=-1, x=1, x=3? Wait, no, the valley at x=-1: is y=0? Yes, because it's on the x-axis. Then the peak at x=1, then down to x=3 (y=-60), then up. Wait, maybe x-intercepts at x=-3, x=-1, x=1, x=3? Wait, the graph crosses x-axis at x=-3 (left), x=-1 (middle valley), x=1 (middle peak's right), and x=3 (right valley's right? No, the right valley is at x=3 (y=-60), then up, so crosses x at x=3? Wait, maybe I'm overcomplicating. Let's list x-intercepts as x=-3, x=-1, x=1, x=3 (approx, based on grid)

Step5: y-intercept

Point where x=0. From graph, at x=0, the graph is at y=0? Wait, the valley at x=-1 is y=0, at x=0, is it above? Wait, no, the graph at x=0: between x=-1 and x=1, the graph is going up from x=-1 (y=0) to x=1 (y=20), so at x=0, y is positive? Wait, maybe y-intercept is at (0, some positive number). Wait, the graph at x=0: looking at the grid, x=0, y is between 0 and 20, maybe y=10? Wait, no, the valley at x=-1 is y=0, then up to x=1 (y=20), so at x=0, y is 10? Wait, maybe y-intercept is (0, 10) (approx), but maybe I misread. Wait, the problem is to identify key characteristics, so maybe approximate.

Step6: Relative Maxima

Points where the graph changes from increasing to decreasing. From graph: x=-2 (peak), x=1 (peak). So relative maxima at x=-2 (y=30? approx) and x=1 (y=20? approx)

Step7: Relative Minima

Points where graph changes from decreasing to increasing. From graph: x=-1 (valley, y=0), x=3 (valley, y=-60). So relative minima at x=-1 (y=0) and x=3 (y=-60)

Step8: Absolute Maximum

Since graph goes to infinity on right, no absolute maximum (does not exist)

Step9: Absolute Minimum

Since graph goes to negative infinity on left, no absolute minimum (does not exist)

Step10: Intervals Increasing

Where the graph's slope is positive (going up). From x=-3 to x=-2 (from x-intercept to peak), x=-1 to x=1 (from valley to peak), x=3 to infinity (from valley to up). So intervals: $(-3, -2)$, $(-1, 1)$, $(3, \infty)$

Step11: Intervals Decreasing

Where slope is negative (going down). From x=-infty to x=-3 (but wait, left end is down, so from x=-infty to x=-3? No, wait, from x=-3 to x=-1 (peak to valley), x=1 to x=3 (peak to valley). So intervals: $(-\infty, -3)$, $(-2, -1)$, $(1, 3)$

Answer:

• Domain: $(-\infty, \infty)$
• Range: $(-\infty, \infty)$
• End Behavior: As $x \to -\infty$, $y \to -\infty$; as $x \to \infty$, $y \to \infty$
• x-intercept(s): $x \approx -3$, $x \approx -1$, $x \approx 1$, $x \approx 3$ (approximate, based on grid)
• y-intercept: $(0, y)$ where $y > 0$ (e.g., $(0, 10)$ approx)
• Relative maximum(s): At $x = -2$ (approx, $y \approx 30$) and $x = 1$ (approx, $y \approx 20$)
• Relative minimum(s): At $x = -1$ (approx, $y = 0$) and $x = 3$ (approx, $y \approx -60$)
• Absolute maximum: Does not exist (graph extends to $\infty$)
• Absolute minimum: Does not exist (graph extends to $-\infty$)
• Intervals increasing: $(-3, -2)$, $(-1, 1)$, $(3, \infty)$ (approximate)
• Intervals decreasing: $(-\infty, -3)$, $(-2, -1)$, $(1, 3)$ (approximate)

(Note: Values are approximate based on the graph’s grid and shape.)