Reading Graph Characteristics Off a General Equation Like y = ax² + c (AF 705)

The equation y = ax² + c is the parabola y = x² after two simple moves: a stretches or flips it, c slides it up or down. Because x² is never negative, every term ax² pushes the output away from zero in whichever direction a points. If a is positive, the parabola opens upward, bowl-side up, with its lowest point at the vertex. If a is negative, it opens downward, dome-side up, with its highest point at the vertex. The vertex itself always sits at the point (0, c), because plugging in x = 0 kills the ax² term and leaves y = c. So c alone tells you the vertex's height: positive c puts it above the x-axis, negative c puts it below.

Once you know the opening direction and the vertex height, you know almost everything about the graph without graphing it. A parabola that opens downward with a vertex below the x-axis never touches the x-axis at all, since its highest point is already underground. A parabola that opens upward with a vertex below the x-axis must cross the x-axis exactly twice, since it starts below and has to rise on both sides. Checking the signs of a and c first is faster than solving ax² + c = 0 and then interpreting the result.

This reading skill is really the transformation rules in disguise. The number a is the same vertical stretch or flip factor you use when stretching any parent graph, and c is the same vertical shift you use when sliding a graph up or down. Once you can read a and c here, you are ready to run the equation backward: given a graph, write down a and c just by looking at its direction, steepness, and vertex.