Final answer:
To find the standard form of the quadratic function with vertex (-3, 4) and a point (3, 40), first use the vertex form and plug in the point to find 'a. After calculation, 'a' equals 1, leading to the vertex form f(x) = (x + 3)^2 + 4. Expanding this gives the standard form f(x) = x^2 + 6x + 13.
Step-by-step explanation:
To find the standard form of the quadratic function with a given vertex and a point, we can use the vertex form of a quadratic function and then convert it to the standard form. The vertex form of a quadratic function is:
f(x) = a(x - h)^2 + k
where (h, k) is the vertex of the parabola. In this case, the vertex is (-3, 4), so our function starts as:
f(x) = a(x + 3)^2 + 4
We are also given a point on the graph (3, 40). We can substitute this into our function to solve for the value of a:
40 = a(3 + 3)^2 + 4
40 = a(6)^2 + 4
40 = 36a + 4
36 = 36a
a = 1
Substituting the value of a back into the vertex form, we get:
f(x) = (x + 3)^2 + 4
To convert this to standard form, we need to expand it:
f(x) = x^2 + 6x + 9 + 4
f(x) = x^2 + 6x + 13
So, the standard form of the quadratic function is f(x) = x^2 + 6x + 13.