Interpreting a Point on the Graph

Every point on a curve is really a sentence waiting to be translated. The curve is the graph of some function, so a point (a, b) on it just means: when the input is a, the output is b. To turn that into words, go to the axis labels. Whatever units and quantity sit on the horizontal axis tell you what the first coordinate counts; whatever sits on the vertical axis tells you what the second coordinate counts. Swap the labels and the exact same dot would describe a completely different situation, even with identical numbers.

This is why reading a point correctly means naming both the quantity and the unit for each coordinate, not just reporting numbers. If t is seconds and h(t) is feet above ground, a point (2, 69) says: 2 seconds after the start, the object is 69 feet up. No arithmetic is needed once you have the function value; the hard part is translating coordinates into a full sentence about the real situation.

This connects directly to reading special points like the vertex of a parabola. The vertex's height (its y-coordinate) is the maximum or minimum output the function reaches, but its x-coordinate is the input that produces that extreme output, not a separate quantity being maximized. On a profit curve P(x), the peak gives the greatest profit, and the x-value under that peak is the number of items sold to earn it, which is almost never the largest number of items the business could sell. Confusing the axis a point sits on is the single most common way these questions go wrong.