The equation of a parabola with two roots and a vertex located at (7,-13) can be written in the form of y=ax^2+bx+c, where a,b, and c are constants, which of the following could be a value of a+b+c
Question
The equation of a parabola with two roots and a vertex located at (7,-13) can be written in the form of y=ax^2+bx+c, where a,b, and c are constants, which of the following could be a value of a+b+c
Solution
The equation of a parabola in vertex form is given by y = a(x-h)^2 + k, where (h,k) is the vertex of the parabola. Given that the vertex is (7,-13), the equation becomes y = a(x-7)^2 - 13.
To convert this equation to the standard form y = ax^2 + bx + c, we expand the equation:
y = a(x^2 - 14x + 49) - 13 y = ax^2 - 14ax + 49a - 13
Comparing this with the standard form, we can see that a = a, b = -14a, and c = 49a - 13.
Therefore, a + b + c = a - 14a + 49a - 13 = 36a - 13.
Without knowing the value of a (which depends on the roots of the parabola), we cannot determine a specific value for a + b + c. However, we can say that it will be of the form 36a - 13 for some value of a.
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