Answer: To write the equation of a parabolic function in vertex form, we need to know its vertex. The vertex of a parabolic function can be found by taking the average of the x-coordinates of two given points and using the midpoint formula.
The midpoint formula for the vertex of a parabolic function is:
Vertex = (x1 + x2, y1 + y2) / 2
where (x1, y1) and (x2, y2) are two points on the parabolic function.
Using the midpoint formula, the vertex of the parabolic function that goes through the points (5,3) and (8,13) is:
Vertex = (5 + 8, 3 + 13) / 2 = (13, 8).
Now that we have the vertex of the parabolic function, we can write its equation in vertex form. The vertex form of a parabolic function is:
y = a(x - h)^2 + k
Where (h, k) is the vertex, and a is the coefficient that determines the direction and width of the parabolic function. To find the value of a, we can use one of the given points and substitute it into the equation:
y = a(x - h)^2 + k
3 = a(5 - 13)^2 + 8
Expanding and simplifying the right side of the equation:
3 = a(-8)^2 + 8
3 = 64a + 8
56 = 64a
a = 7/8.
Finally, substitute the values for the vertex and an into the equation:
y = a(x - h)^2 + k
y = 7/8(x - 13)^2 + 8
So, the equation of the parabolic function that goes through the points (5,3) and (8,13) in vertex form is y = 7/8(x - 13)^2 + 8.
Explanation: