The function graph having a vertex at (3, 5) and passing through the point (5, 13) has the equation y = 2x² - 12x + 23.
The vertex form of a parabolic equation is y = a(x - h)² + k, where a, h, and k, are constants, and the vertex is at the point (h, k).
In the question, we are asked to find the function whose graph has the vertex at the (3, 5) and passes through the point (5, 13).
Assuming it to be a parabolic function, we know that the vertex form of a parabolic equation is y = a(x - h)² + k, where a, h, and k, are constants, and the vertex is at the point (h, k).
Thus, substituting h = 3 and k = 5, in the above equation as the vertex of the required function is at the point (3, 5), we get the equation:
y = a(x - 3)² + 5.
Now, since the graph passes through the point (5, 13), we substitute x = 5, and y = 13 in the above equation to get:
13 = a(5 - 3)² + 5,
or, 13 = 4a + 5,
or, 4a = 13 - 5 = 8,
or, a = 2.
Thus, substituting a = 2, h = 3, and k = 5, in the equation y = a(x - h)² + k, we get:
y = 2(x - 3)² + 5,
or, y = 2(x² - 6x + 9) + 5,
or, y = 2x² - 12x + 18 + 5,
or, y = 2x² - 12x + 23.
Thus, the function graph having a vertex at (3, 5) and passing through the point (5, 13) has the equation y = 2x² - 12x + 23.
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