Question

Which option gives f(x) = x² - 4x - 12 rewritten in vertex form as well as the function's correct axis of symmetry?

f(x) = (x -2)² - 16, x = -16
f(x) = (x + 2)(x - 6), x = 6
f(x) = (x + 2)(x - 6), x = -2
f(x) = (x - 2)² - 16, x = 2

Answer 1

The function f(x) = x² - 4x - 12 rewritten in vertex form is f(x) = (x - 2)² - 16, and the axis of symmetry for this function is x = 2.

Step-by-step explanation:

To rewrite the function f(x) = x² - 4x - 12 in vertex form, we need to complete the square. This involves the following steps:

1. Factor out the coefficient of the x² term, if it is not 1 (which it is in this case).
2. Rearrange the equation to isolate the x-terms: x² - 4x.
3. Find the number that completes the square for x² - 4x, which is the square of half of the coefficient of x. This number is (-4/2)² = 4.
4. Add and subtract the determined number inside the brackets: (x² - 4x + 4) - 4.
5. Rewrite the function in vertex form: f(x) = (x - 2)² - 16.

The axis of symmetry for a parabola in the form (x - h)² is always x = h, which in this case is x = 2.

Therefore, the correct answer is f(x) = (x - 2)² - 16, x = 2.

Answer 2

Part 1)
`f(x)=(x-2)^(2)-16`

Part 2) The function's correct axis of symmetry is x=2

Option f(x) = (x - 2)² - 16, x = 2

Step-by-step explanation:

we have

`f(x)=x^(2) -4x-12`

This is a vertical parabola open upward

The vertex is the minimum

Part 1) Convert to vertex form

Complete the square. Remember to balance the equation by adding the same constants to each side.

`f(x)=(x^(2) -4x+2^2)-12-2^2`

`f(x)=(x^(2) -4x+4)-16`

Rewrite as perfect squares

`f(x)=(x-2)^(2)-16` ----> function in vertex form

Te vertex is the point (2,-16)

Part 2) Find the axis of symmetry

we know that

The equation of the axis of symmetry of a vertical parabola is equal to the x-coordinate of the vertex

The vertex is the point (2,-16)

therefore

The function's correct axis of symmetry is x=2