the graph of a cubic function has a local min at (5,-3) and a point of symmetry at (0,4). At what point does a local maximum occur?
i don't quite know how to figure this one out, any help?
To determine the point where the local maximum occurs for the cubic function, we can use the symmetry of the graph.
Given that the point (0,4) is the point of symmetry, we can infer that there is another point on the graph that is equidistant from the y-axis but mirrored across it. This implies that there is another point (let's call it P) such that it has the same y-coordinate (4) as the point of symmetry, but with a negative x-coordinate.
Since the local min is at (5,-3), and the point of symmetry is at (0,4), we can deduce that the distance between the two points (0,4) and (5,-3) is the same as the distance between the two points (0,4) and P.
Using the distance formula, we can calculate the distance between (0,4) and (5,-3):
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d = sqrt((5 - 0)^2 + (-3 - 4)^2)
d = sqrt(25 + 49)
d = sqrt(74)
Now, to find the coordinates of point P, we need to mirror the x-coordinate of (0,4) across the y-axis. The mirrored x-coordinate will be the same distance away from the y-axis as the x-coordinate of (5,-3):
x-coordinate of P = -5
So, the point P will have coordinates (-5, 4). This point represents the local maximum of the cubic function.
Therefore, the local maximum occurs at (-5, 4).