To write the quadratic function in the form f(x) = a(x – h)² + k, we need to complete the square for the given function.
The original function is: f(x) = x² – 8x – 20.
Step 1: Organize the quadratic and linear terms. We can rewrite the function as:
f(x) = (x² – 8x) – 20
Step 2: Complete the square on the expression x² – 8x. To do this, we take the coefficient of x, which is -8, halve it to get -4, and then square it to get 16. We add and subtract 16 inside the parentheses:
f(x) = (x² – 8x + 16 – 16) – 20
This simplifies to:
f(x) = (x – 4)² – 16 – 20
f(x) = (x – 4)² – 36
Now, we have the function in the desired form:
f(x) = 1(x – 4)² – 36, where a = 1, h = 4, and k = -36.
Step 3: Identify the vertex of the graph. The vertex of a quadratic function in the form f(x) = a(x – h)² + k is given by the point (h, k).
Thus, the vertex of the graph f(x) = (x – 4)² – 36 is:
(4, -36).
This means the vertex is located at the point (4, -36) on the graph of the function.