Answer:
The equation that represents the function is f(x) = -2x² - 4x + 7.
Step-by-step explanation:
What is function?
A function is a relation between two sets of elements, where each element in the first set (called the domain) is associated with exactly one element in the second set (called the range).
According to question:
We know that the vertex of the quadratic function is at (-1,9), which means that the axis of symmetry is x = -1. Therefore, the x-coordinate of the two points (-2,7) and (0,7) must be equidistant from the axis of symmetry.
The distance between x=-1 and x=-2 is 1, and the distance between x=-1 and x=0 is also 1. Therefore, the quadratic function must have a symmetric form with respect to the axis x=-1. It must be a quadratic function that has the vertex form:
f(x) = a(x - (-1))² + 9
where "a" is the coefficient that determines whether the parabola opens upward or downward. To find "a", we can use one of the points that the function passes through. Let's use the point (-2,7):
f(-2) = a(-2 - (-1))² + 9 = 7
Simplifying this equation, we get:
a + 9 = 7
a = -2
Therefore, the quadratic function is:
f(x) = -2(x + 1)² + 9
Expanding this equation, we get:
f(x) = -2(x² + 2x + 1) + 9
f(x) = -2x² - 4x + 7
So, the answer is f(x) = -2x² - 4x + 7.