Question

A quadratic function f, left bracket, x, right bracketf(x) is hidden from view. You must find all intervals where f, left bracket, x, right bracketf(x) is increasing. Choose the form of the quadratic function f, left bracket, x, right bracketf(x) that you would like to see in order to answer the question most efficiently.

Form: f, left bracket, x, right bracket, equals, 2, x, squared, minus, 8, x, minus, 24f(x)=2x
2
−8x−24

Answer 1

The form of the quadratic function f(x) =2x² - 8x - 24 used to know when f(x) is f(x) = 2(x - 2)² - 32 and f(x) is increasing on the interval (-2, ∞)

Since we have a quadratic function f(x) hidden from view. To find all intervals where f(x) is increasing. To find the form of the quadratic function f(x) =2x² - 8x - 24 that you would like to see in order to answer the question most efficiently, we proceed as follows

Since we have the quadratic function f(x) = 2x² - 8x - 24, the most efficient form to know which interval in which it is increasing is the vertex form of a quadratic equation f(x) = a(x - h) + k with vertex (h, k)

So, to put f(x) = 2x² - 8x - 24 in vertex form, we complete the square.

So, f(x) = 2x² - 8x - 24

f(x) = 2(x² - 4x - 12)

Adding the square of half the coefficient of x, we have that

f(x) = 2(x² - 4x + (-4/2)² - (-4/2)² - 12)

= 2(x² - 4x + (-2)² - (-2)² - 12)

= 2[(x - 2)² - 4 - 12)

= 2[(x - 2)² - 16]

= 2(x - 2)² - 32

So, comparing with the vertex form equation, the vertex is at (- 2, - 32).

Also, since a = 2 > 0. The vertex (-2, -32) is a minimum point.

So, the quadratic function is increasing on the interval (-2, ∞)