Final answer:
To convert the quadratic equation f(x)=-x²-8x-18 to vertex form, we complete the square and rewrite the equation in the form y=a(x-h)²+k. The vertex form reveals that h=-4 and k=-2, which are the coordinates of the vertex.
Step-by-step explanation:
To convert the given quadratic equation f(x) = -x² - 8x - 18 from standard form to vertex form, y = a(x - h)² + k, we need to complete the square to find the values of h and k. The vertex form of a quadratic function provides the coordinates of the vertex, where h represents the x-coordinate and k the y-coordinate of the vertex.
Step 1: Start with the given quadratic equation f(x).
f(x) = -x² - 8x - 18
Step 2: Factor out the coefficient of x² from the first two terms.
f(x) = -(x² + 8x) - 18
Step 3: Find the number that completes the square for the expression in parentheses.
The number needed to complete the square is ²², which is the square of half the coefficient of x, so (8/2)² = 16.
Step 4: Add and subtract the number inside the parentheses, considering the factor outside.
f(x) = -(x² + 8x + 16 - 16) - 18
Step 5: Rewrite as a perfect square trinomial and simplify.
f(x) = -((x + 4)² - 16) - 18
f(x) = -(x + 4)² + 16 - 18
Step 6: Combine like terms to find the values of h and k.
f(x) = -(x + 4)² - 2
The values of h and k are h = -4 and k = -2.