Key features of graphs

Why features matter

• Every graph can be described by a short list of key features — the landmarks that tell you its shape and where it sits.
• Naming these features is how you connect an equation to its picture, which is exactly what this standard rewards.

Intercepts

• The $x$-intercepts are where the graph crosses the $x$-axis. Here $y = 0$, so you set $y = 0$ and solve for $x$.
  • For a curve these are also called the roots or zeros.
• The $y$-intercept is where the graph crosses the $y$-axis. Here $x = 0$, so you set $x = 0$ and read off $y$.
• A graph has at most one $y$-intercept (a function gives one $y$ for each $x$), but can have several $x$-intercepts.

Turning points

• A turning point is where a curve changes direction — from going up to going down, or the reverse.
• A minimum turning point is the lowest point of a dip; a maximum is the highest point of a hump.
• A parabola has exactly one turning point (its vertex); a cubic can have two (a maximum and a minimum) or none.

Axis of symmetry

• Some graphs have an axis of symmetry — a mirror line where one side is the reflection of the other.
• A parabola is symmetric about the vertical line through its vertex.

Asymptotes

• An asymptote is a line the graph gets closer and closer to but never touches.
• Exponential and reciprocal graphs have asymptotes; parabolas and cubics do not.

Domain and range

• The domain is the set of $x$-values the graph uses (how far left–right it extends).
• The range is the set of $y$-values it takes (how far down–up it reaches).
• Example — for $y = x^2$ the domain is all real numbers, but the range is $y \ge 0$, because a square is never negative.