Hello bumblebee123. Here's a handy formula to remember, when working with quadratic equations and their graphs.

Given the form y = Ax^2 + Bx + C, the x-coordinate of the vertex point is always -B/(2A).

In your function, B = -10 and A = 2. Therefore:

x-coordinate of vertex = -B/(2A) = 10/(2*2) = 5/2

This formula works with all parabolas that open upward or downward.

As you know, once you have the x-coordinate of a point, you find the corresponding y-coordinate using the given function:

y-coordinate of vertex = 2(5/2)^2 - 10(5/2) - 5 = -35/2

You also know the parabola opens upward, so the vertex point (5/2,-35/2) is the lowest point on the graph. In other words, -35/2 is the smallest function output, so the range must be all Real numbers that are -35/2 or larger.

That fact is what MarkFL discussed, in post #2.

PS: If you're allowed to use graphing software, then work alongside a graph to confirm your thoughts as you go. (That will help prevent mistakes like thinking y cannot be negative in this exercise.) Cheers