To convert the quadratic function f(x) = 8x² + 4x into vertex form, we need to complete the square.
1. First, factor out the coefficient of x² from the first two terms:
f(x) = 8(x² + rac{4}{8}x)
Which simplifies to:
f(x) = 8(x² + rac{1}{2}x)
2. Next, to complete the square, we take the coefficient of x inside the parentheses (which is 1/2), divide it by 2 (getting 1/4), and square it (resulting in 1/16).
3. Add and subtract this value inside the parentheses:
f(x) = 8(x² + rac{1}{2}x + rac{1}{16} – rac{1}{16})
4. Now, rewrite the equation with the completed square:
f(x) = 8((x + rac{1}{4})² – rac{1}{16})
5. Distributing the 8 gives us:
f(x) = 8(x + rac{1}{4})² – 8( rac{1}{16})
Which simplifies to:
f(x) = 8(x + rac{1}{4})² – rac{1}{2}
So, in vertex form, the function is:
f(x) = 8(x + rac{1}{4})² – rac{1}{2}
The vertex of this parabola is at the point (-1/4, -1/2).