A quadratic function f is given. f(x) = −4x^2− 20x + 1 Express f in standard form.

Answer 1

We say a quadratic EQUATION is in standard form if it fits the form: ax² + bx + c = 0

But we also say that a quadratic FUNCTION is in standard form if it fits the form f(x) = a(x - h)² + k

Is that confusing or what??? (I didn't make the rules)

So, to get it in the vertex form, we complete the square:

Step 1) Drop the f(x)

-4x² - 20x + 1

Step 2) Move the constant term off to the side

-4x2 - 20x + 1

Step 3) Factor out what is in front of the x²:

-4(x² + 5x ) +1

Step 4) Add in a term to make the left side a perfect square and then subtract that from the constant

To do this, take the term in front of the "x" (which is "5") and divide by 2 then square it

-4(x² + 5x + 25/4) +1

But you can't just add in a number because you feel like it - you are changing the equation. So to compensate, we will subtract out that same amount. But, realize that what you REALLY did is subtract 25 because of the "-4" in front of the parenthesis. So we mist add 25 to cmpensate:

-4(x² + 5x + 25/4) +1 + 25

Step 5) write the left side as a square and combine the right side:

-4(x + 5/2)² + 26

Step 6) Put back the f(x)

f(x) = -4(x + 5/2)² + 26