use completeing the square to describe the graph of f(x)=4-16x-x^2
support your answers graphically
To describe the graph of the function f(x) = 4 - 16x - x^2 using completing the square, we need to rewrite the equation in vertex form by completing the square.
Step 1: Start with the original equation: f(x) = 4 - 16x - x^2
Step 2: Rearrange the terms to have the x^2 term, the x term, and the constant term in order: f(x) = -x^2 - 16x + 4
Step 3: Take the coefficient of x term, which is -16, divide it by 2, and square the result: (-16 / 2)^2 = 64
Step 4: Add and subtract the value calculated in Step 3 inside the parenthesis. This step maintains the value of the equation while allowing us to factor a perfect square: f(x) = -(x^2 + 16x + 64 - 64) + 4
Step 5: Group the perfect square term together: f(x) = -(x^2 + 16x + 64) + 64 + 4
Step 6: Rewrite the perfect square expression as a squared binomial: f(x) = -(x + 8)^2 + 68
Now we have the equation in vertex form: f(x) = a(x - h)^2 + k, where the vertex is at point (h, k).
In this case, the vertex is (-8, 68), and the coefficient "a" is -1, which tells us that the graph is upside down.
To support these answers graphically, you can plot the graph of the equation f(x) = 4 - 16x - x^2 and identify the vertex using the completing the square method. This way, you can confirm the coordinates of the vertex and verify the shape of the graph.