To convert the quadratic equation f(x) = 8x² + 4x into vertex form, we need to complete the square. Vertex form is generally expressed as:
f(x) = a(x – h)² + k
where
- (h, k) is the vertex of the parabola, and
- a is a coefficient that affects the width and direction of the parabola.
Let’s break it down step by step:
- **Factor out the coefficient of x²:**
First, we’ll factor out the coefficient of the x² term from the first two terms in the equation:
f(x) = 8(x² + rac{1}{2}x)
To complete the square, we need to find the value that makes x² + rac{1}{2}x a perfect square trinomial. We take half of the coefficient of x, which is rac{1}{2}, divide it by 2 to get rac{1}{4}, and then square it to get:
rac{1}{4}² = rac{1}{16}
Next, we add and subtract this value inside the parentheses:
f(x) = 8igg(x² + rac{1}{2}x + rac{1}{16} – rac{1}{16}igg)
Now, we can reorganize the expression inside the parentheses:
f(x) = 8igg((x + rac{1}{4})² – rac{1}{16}igg)
Distributing the 8, we adjust the constants:
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}
Thus, the vertex form of the quadratic equation f(x) = 8x² + 4x is:
f(x) = 8(x + rac{1}{4})² – rac{1}{2}
In this form, you can see that the vertex of the parabola is at igg( - rac{1}{4}, - rac{1}{2}igg).
This transformation makes it clear how the quadratic behaves and where its vertex lies, providing a clearer picture of the graph of the function.