Question

The vertex form of the quadratic function f(x) = -6x² - 60x - 151 is

f(x)= a(x - h)² + k.
What is the value of a?
What is the value of h?
What is the value of k?

Answer 1

Vertex Form

This is one way of writing a quadratic function:

`f(x)= a(x-h)^2+k`

• (h,k) = vertex
• a = vertical stretch

Solving the Question

We're given:

`f(x) = -6x^2 - 60x - 151`

To write a quadratic function in vertex form, we must complete the square.

⇒ Put parentheses around the first two terms containing x and x²:

`f(x) = (-6x^2 - 60x) - 151`

⇒ Factor out -6 (keeping the x's in the parentheses):

`f(x) = -6(x^2+10x) - 151`

⇒ To complete the square, add, inside the parentheses, the square of half the coefficient of x.

• ⇒ In this case, the coefficient of x is 10.
• ⇒ Half of this value is 5.
• ⇒ The square of 5 is 25.
• ⇒ Add this inside the parentheses:

`f(x) = -6(x^2+10x+25) - 151`

⇒ Now, we cannot randomly introduce a new value into a function. To balance the +25, subtract -6(25) outside the parentheses:

`f(x) = -6(x^2+10x+25) - 151-(-6*25)\\f(x) = -6(x^2+10x+25) - 151-(-150)\\f(x) = -6(x^2+10x+25) - 151+150\\f(x) = -6(x^2+10x+25) - 1`

⇒ Finally, complete the square.

• Remember:
  `(a+b)^2=a^2+2ab+b^2`
• In this case, a is x, and b is 5:

`f(x) = -6(x+5)^2 - 1`

Answer

`f(x) = -6(x+5)^2 - 1`

• a = -6
• h = -5
• h = -1