The equation of the parabola \( y=2 x^{2}+12 x+17 \) in vertex form is b.) \( y=2(x-3)^{2}+1 \) b.) \( y=2(x+3)^{2}-1 \) d.) \( y=2(x-3)^{2}-1 \) d. \( y=2)^{2}+1 \)

To convert the equation \( y=2x^{2}+12x+17 \) into vertex form, we can complete the square. First, factor out the 2 from the \( x^2 \) and \( x \) terms: \[ y = 2(x^2 + 6x) + 17 \] Next, complete the square inside the parentheses. The term to add and subtract is \( \left(\frac{6}{2}\right)^{2} = 9 \): \[ y = 2(x^2 + 6x + 9 - 9) + 17 \] \[ y = 2((x + 3)^{2} - 9) + 17 \] \[ y = 2(x + 3)^{2} - 18 + 17 \] \[ y = 2(x + 3)^{2} - 1 \] So the correct vertex form is \( y=2(x+3)^{2}-1 \). Now, let’s talk about parabolas! Did you know they’ve been around since ancient times? The term "parabola" comes from the Greek word for "comparison," and they describe all sorts of phenomena, from the paths of projectiles to the reflective properties of satellite dishes! If you're ever stuck on a math problem involving parabolas, remember to take a step back and look for patterns! Also, check your signs when completing the square—it's a common area to trip up! A small mistake in the sign can lead you completely off track in your equation.