Write the equation for the function graphed below.

Given the vertex, (2, 7), and that it is a downward-facing parabola:

We can use the vertex form of the quadratic function:

f(x) = a(x - h)^2 + k
where:

Vertex = (h, k)

“a” determines whether the graph opens up or down.

* If “a” is negative, the graph opens down.

The value of “a” also makes the parent function wider or narrower.

The value of “h” determines how far left or right the parent function is translated.

The value of “k” determines how far up or down the parent function is translated.

Given these information, we can plug in the value of the vertex into the vertex form:

f(x) = a(x - h)^2 + k

f(x) = a(x - 2)^2 + 7

Next, to solve for “a”, we can use one of the points on the graph, (1, 5), and plug these values into the equation:

f(x) = a(x - 2)^2 + 7

5 = a(1 - 2)^2 + 7

5 = a(-1)^2 + 7

5 = a(1) + 7

Subtract 7 from both sides:

5 - 7 = a(1) + 7 - 7
-2 = a

Therefore, a = -2. Now, we can establish the following quadratic function in vertex form:

f(x) = -2(x - 2)^2 + 7

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