The equation of the parabola \( y=2 x^{2}-8 x+13 \) in vertex form is a.) \( y=2(x-2)^{2}+5 \) b.) \( y=2(x+2)^{2}+5 \) c.) \( y=2(x-2)^{2}-5 \) d.) \( y=2(x+2)^{2}-5 \)

To convert the quadratic equation \( y = 2x^2 - 8x + 13 \) into vertex form, we can complete the square. 1. Factor out the coefficient of \( x^2 \): \( y = 2(x^2 - 4x) + 13 \). 2. Next, complete the square inside the parentheses: - Take half of \(-4\) (which is \(-2\)), square it (getting \(4\)), and adjust the equation: \( y = 2((x - 2)^2 - 4) + 13 \). 3. Simplify: \( y = 2(x - 2)^2 - 8 + 13 \) leads to \( y = 2(x - 2)^2 + 5 \). So, the vertex form is \( y = 2(x - 2)^{2} + 5 \), which corresponds to option a.) \( y = 2(x - 2)^{2} + 5 \). If you ever need to convert a quadratic into vertex form, remember that completing the square can seem tricky, but it’s all about pairing the \( x \) terms and managing that constant with precision. Just like cooking, a little adjustment can bring the flavors (or the numbers) to perfection! For further exploration, you can delve into the concept of parabolas and their properties. Textbooks or online resources about quadratic functions usually cover vertex form, and their applications in real-life contexts like projectile motion or design can add that extra zest to understanding!