Question

identifying key characteristics of polynomial functions
select the number of x-intercepts: 3
enter the coordinates of the x-intercepts: (0, 0), (8, 0), (18, 0)
decide whether the function has extrema:
extremum: relative minimum, relative maximum, absolute minimum, absolute maximum
options: add extremum, no extrema

Explanation:

Step1: Analyze x-intercepts

The graph intersects the x - axis at three points: (0, 0), (8, 0), and (18, 0), so the number of x - intercepts is 3, and their coordinates are as given.

Step2: Analyze extrema

Looking at the graph, we can see that there are peaks (relative maxima) and valleys (relative minima). For example, there is a relative maximum between x = 0 and x = 8, a relative minimum between x = 8 and x = 18, and another relative maximum? Wait, no, the graph has a relative maximum, a relative minimum, and since the ends go to -∞, there is no absolute maximum. But the key here is to identify the extrema. The graph has a relative maximum (the peak) and a relative minimum (the valley). Also, let's check the extrema types. The relative maximum: let's see the y - values. The peak has a positive y - value, and the valley has a negative y - value. Also, since the function goes to -∞ as x approaches the ends (left and right), there is no absolute minimum (because it can go to -∞) and no absolute maximum (but wait, the graph has a peak, but as x→±∞, the function goes to -∞? Wait, no, looking at the graph, on the left end (x→ -∞), the graph goes down (since at x = 0, it's 0, and left of x = 0, it goes down), and on the right end (x→ +∞), it goes down. So the graph has a relative maximum (the peak) and a relative minimum (the valley). Also, let's check the extrema options. The options are Relative minimum, Relative maximum, Absolute minimum, Absolute maximum. Since the function can go to -∞, there is no absolute minimum (because absolute minimum would be the lowest point, but it can go lower). There is no absolute maximum (because the peak is a relative maximum, but the function doesn't have a highest point overall? Wait, no, the peak is a relative maximum, and since the function goes to -∞ on both ends, the relative maximum is a local peak, and the relative minimum is a local valley. Also, is there an absolute minimum? No, because as x→±∞, y→ -∞, so there's no absolute minimum. Is there an absolute maximum? The peak is a relative maximum, and since the function doesn't have a highest point (but wait, the peak is the highest point in its neighborhood, but overall, since the function goes to -∞, the peak is a relative maximum. Also, the valley is a relative minimum. But let's check the graph again. The graph has a relative maximum (the first peak), a relative minimum (the valley between 8 and 18), and then another part? Wait, no, the x - intercepts are at 0, 8, 18. So between 0 and 8: the graph goes up to a peak (relative maximum), then down to (8, 0), then down to a valley (relative minimum) between 8 and 18, then up to (18, 0), then down again. So the extrema are: a relative maximum (at the peak) and a relative minimum (at the valley). Also, since the function goes to -∞ as x→±∞, there is no absolute maximum (because it can't have a highest point if it goes to -∞? Wait, no, the peak is a positive y - value, and the ends go to -∞, so the peak is a relative maximum, and the valley is a relative minimum. Also, is there an absolute minimum? No, because the function can go to -∞, so there's no absolute minimum. Is there an absolute maximum? The peak is a relative maximum, and since the function doesn't have a highest point (but the peak is the highest in its local area), so the relative maximum exists, and the relative minimum exists. Also, let's check the options. The dropdown for extremum: let's say we first identify the relative maximum. Wait, but maybe the problem is to select the extrema. But also, the x - intercepts are already given as (0,0), (8,0), (18,0) with 3 x - intercepts. For the extrema, the function has a relative maximum (the peak) and a relative minimum (the valley). But let's check the graph's extrema. The relative maximum: let's see the coordinates. The peak is between x = 0 and x = 8, and the valley is between x = 8 and x = 18. Also, the absolute minimum: since the function goes to -∞, there is no absolute minimum. The absolute maximum: the peak is a relative maximum, and since the function doesn't have a highest point (but the peak is the highest in its neighborhood), so the relative maximum is present, and the relative minimum is present. But maybe the problem is to select the extrema. However, the main task here is to identify the key characteristics. But the question seems to be about identifying the x - intercepts (which are 3, at (0,0), (8,0), (18,0)) and the extrema. But maybe the user is asking to confirm the x - intercepts and the extrema. But since the x - intercepts are already filled as 3 with (0,0), (8,0), (18,0), which matches the graph. For the extrema, the function has a relative maximum (the peak) and a relative minimum (the valley). Also, since the function goes to -∞ as x→±∞, there is no absolute maximum and no absolute minimum (except that the relative minimum is a local minimum, and relative maximum is a local maximum).

Answer:

The number of x - intercepts is 3 with coordinates (0, 0), (8, 0), (18, 0). The function has a relative maximum (the peak) and a relative minimum (the valley). (If we have to choose from the dropdown, for example, the relative maximum and relative minimum exist. But based on the graph, the extrema include a relative maximum and a relative minimum. Also, there is no absolute maximum (since the function goes to -∞) and no absolute minimum (since it goes to -∞).)