Types of Graphs

• Quadratic graphs (y = ax² + bx + c) are smooth U-shaped or n-shaped curves called parabolas, with a single turning point (vertex) and up to 2 roots.
• Cubic graphs (y = ax³ + bx² + cx + d) are S-shaped wiggly curves with up to 2 turning points and up to 3 roots.
• Reciprocal graphs (y = a/x) never touch either axis — they have two asymptotes and split into two separate curve branches.
• Exponential graphs (y = aˣ) always pass through (0, 1), have one horizontal asymptote at y = 0, and either grow or decay depending on the base.
• Across all four graph types, the same three questions unlock the sketch: where does it cross the y-axis? where does it cross the x-axis? what shape/direction does it take?

What is it, really?

A quadratic graph comes from an equation of the form y = ax² + bx + c, where a is never zero (if a were zero, the x² term would vanish and it wouldn't be quadratic anymore — it'd just be a straight line). Think of a, b, and c as three separate "dials": a controls the shape and steepness, b nudges the curve sideways, and c is simply where the curve crosses the y-axis.

Picture throwing a ball straight up in the air. It starts low, rises to a peak, then falls back down — that arc shape is exactly what a quadratic graph looks like when a is negative. Flip it upside down (a is positive) and you get a valley shape instead, like a skateboard ramp. This U or n shape is called a parabola, and every quadratic graph — no matter the numbers — has this same symmetrical, smooth curve. It's never jagged, never has sharp corners, and it always has exactly one line of symmetry running vertically through its turning point.

Every quadratic graph always crosses the y-axis exactly once (because for any curve, there's only one y-value when x = 0). But it can cross the x-axis 0, 1, or 2 times — imagine sliding the U-shape up and down: sometimes it dips below the x-axis and crosses it twice, sometimes it just grazes the axis at one point (touches but doesn't cross), and sometimes it floats entirely above or below the axis and never touches it at all. The points where it does cross are called the roots (also called x-intercepts or solutions).