Hello Islandguy. That y is not the same as the y in your exercise, so I would pick a different symbol when writing the slope-intercept form for an arbitrary tangent line to the curve in this exercise.
Y = mx + b
But, we're not asked to report an equation for the horizontal tangent line, so actually we don't need the slope-intercept form at all. Let's just go with m=slope.
As we move along the given parabola in this exercise, x changes. As x changes, so does m. (There are lots of different tangent lines along the parabola, and each has its own slope.)
Did your calculus class mention the following?
The first derivative is a slope. That is, at each point on the curve of a function y, the value of m is y
'.
In other words, the first derivative of a function gives us the tangent line's slope (at each x in the domain). In this regard, the first derivative is a function of x itself: a function of slopes.
When we're interested in where a tangent-line slope is zero, then we're interested in where the first derivative is zero.
?