Key Features of a Graph: Intercepts, Turning Points and Behavior

Intercepts tell you where a graph crosses the axes. X-intercepts are where y = 0, found by setting the function equal to zero and solving. The y-intercept is where x = 0, found by plugging in zero. Turning points are where the graph switches from rising to falling or back again — local peaks and valleys. Together these points act like a skeleton: once you know where a curve crosses the axes and where it turns, you know its overall shape, even before you plot a single other point.

For a parabola, two x-intercepts and a y-intercept are enough to write the whole equation. The x-intercepts give you the factors, and the y-intercept tells you how tall or short to stretch the parabola to match. The vertex, which is the one turning point a parabola has, sits exactly halfway between the two x-intercepts horizontally. For higher-degree polynomials, the same idea stretches further: a cubic can rise, turn, fall, turn again, and rise, and tracking where it increases versus decreases (by checking the sign of change between turning points) tells you the whole story of its behavior.

This connects directly to the four representations of a function: a table, a graph, an equation, and a verbal description all encode the same intercepts and turning points, just differently. Once a function's degree is known, the maximum number of turning points is fixed — one less than the degree — which links this topic to end behavior and to counting how wild a polynomial's graph can get.