Question Complete the square to re-write the quadratic function in vertex form: \[ y=x^{2}-7 x+2 \] Answer Attempt lout of 2

To complete the square for the quadratic function \(y = x^2 - 7x + 2\), we focus on the \(x\) terms. Start by taking half of the coefficient of \(x\) (which is \(-7\)), giving us \(-\frac{7}{2}\), and then square it, yielding \(\left(-\frac{7}{2}\right)^2 = \frac{49}{4}\). Next, we rewrite the equation by adding and subtracting \(\frac{49}{4}\): \[ y = \left(x^2 - 7x + \frac{49}{4}\right) - \frac{49}{4} + 2 \] This simplifies to: \[ y = \left(x - \frac{7}{2}\right)^2 - \frac{49}{4} + \frac{8}{4} \] Finally, we have: \[ y = \left(x - \frac{7}{2}\right)^2 - \frac{41}{4} \] And there you have it! The quadratic is now in vertex form. Did you know that completing the square is not just a method for rewriting quadratics but is also a key step in deriving the quadratic formula? This handy technique helps unveil the underlying structure of quadratic equations, equipping you to tackle them with ease. So, the next time you find yourself faced with a stubborn quadratic, remember the power of completing the square! In real-world applications, vertex form is particularly useful in optimization problems. For instance, if you're modeling the trajectory of a baseball, the vertex represents the peak height and the time it takes to reach that peak. By expressing the function in vertex form, you can easily identify those critical points, helping you analyze various situations like maximizing profits or minimizing costs in business settings!