9. find the vertrex about \( y=x^{2}-3 x-10 \)

To find the vertex of the parabola given by the equation \( y = x^2 - 3x - 10 \), we can use the vertex formula. The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \), where \( a = 1 \) and \( b = -3 \). Substituting, we get \( x = -\frac{-3}{2 \cdot 1} = \frac{3}{2} \). Next, we plug this back into the equation to find the y-coordinate: \[ y = \left(\frac{3}{2}\right)^2 - 3\left(\frac{3}{2}\right) - 10 = \frac{9}{4} - \frac{9}{2} - 10 = -\frac{41}{4} \] So, the vertex is at \( \left(\frac{3}{2}, -\frac{41}{4}\right) \). The vertex of a parabola is its highest or lowest point, depending on the direction it opens. In this case, since the coefficient of \( x^2 \) is positive, the parabola opens upwards, making the vertex the lowest point. It's quite like finding a hidden gem in the landscape of quadratic equations – it's the spot where the function takes a turn! When working with quadratic functions, it’s common to accidentally mix up the signs or miscalculate the coordinates, especially while substituting values. To avoid these pitfalls, double-check your arithmetic and ensure variables are accurately identified. A checklist approach can be handy: confirm values for \( a \) and \( b \), calculate the vertex correctly, and always verify by plugging back into the original equation. It's all about taking those extra steps to ensure your vertex quest is a success!