To convert the function f(x) = 2(x – 12)^2 + 3 from vertex form to standard form, we will follow a few simple algebraic steps.
Vertex form of a quadratic function is expressed as:
f(x) = a(x – h)^2 + k
where (h, k) is the vertex of the parabola. In this case, our vertex is (12, 3) and a = 2.
Now, let’s expand the expression:
- Start by expanding the squared term:
- (x – 12)^2 expands to x^2 – 24x + 144.
Substituting this back into the function gives:
f(x) = 2(x^2 – 24x + 144) + 3
- Distribute the 2 across the terms in the parentheses:
- This becomes f(x) = 2x^2 – 48x + 288 + 3.
Finally, combine like terms:
f(x) = 2x^2 – 48x + 291
So, in standard form, the quadratic function is:
f(x) = 2x^2 – 48x + 291
This standard form clearly shows the coefficient of the squared term and the linear term, which makes it easier to analyze the graph of the function.