To express the quadratic function y = 2x² + 8x + 1 in vertex form, we will use the technique of completing the square. The vertex form of a quadratic function is given by:
y = a(x – h)² + k
Here, (h, k) is the vertex of the parabola.
Step 1: Factor out the coefficient of x² from the first two terms:
y = 2(x² + 4x) + 1
Step 2: Complete the square: To complete the square, we take the coefficient of x, divide it by 2, and square it. The coefficient is 4; dividing it by 2 gives 2, and squaring it gives 4. We add and subtract this value inside the parentheses:
y = 2(x² + 4x + 4 – 4) + 1
Step 3: Rearrange the equation: Rewrite the equation by factoring the perfect square trinomial:
y = 2((x + 2)² – 4) + 1
Step 4: Distributing: Now, distribute the 2:
y = 2(x + 2)² – 8 + 1
Step 5: Simplify: Finally, combine like terms:
y = 2(x + 2)² – 7
So, the vertex form of the function y = 2x² + 8x + 1 is y = 2(x + 2)² – 7. The vertex of this parabola is at the point (-2, -7).