Graph Properties and Symmetry

Overview

This lesson covers key properties and terminology related to the graphs of advanced functions, including symmetry, end behavior, intervals of increase/decrease, and function classifications.

Symmetry in Graphs

• Line symmetry: A graph is symmetrical about a line x = a if each side is a mirror image (e.g., parabolas).
• Point symmetry: A graph has point symmetry about (a, b) if rotating 180° around (a, b) yields the same graph (e.g., y = x³ at the origin).

End Behavior

• End behavior describes what happens to the function as x approaches positive or negative infinity.
• Example: For a standard parabola, as x → ±∞, f(x) → ∞.

Intervals of Increase and Decrease

• An interval is increasing if y-values get larger from left to right; decreasing if they get smaller.
• Use interval notation: round brackets ( ) exclude endpoints, square brackets [ ] include endpoints.
• Example: Increasing on (–∞, 0) ∪ (4, ∞); decreasing on (0, 4).

Interval Notation and Brackets

• (a, b): x ∈ (a, b), does not include a or b (open interval).
• [a, b]: x ∈ [a, b], includes both endpoints (closed interval).
• (a, b]: x ∈ (a, b], includes b, not a.
• [a, b): x ∈ [a, b), includes a, not b.
• Infinity always uses a round bracket: (a, ∞) or (–∞, b).

Even and Odd Functions

• Odd function: f(–x) = –f(x); graph is symmetric about the origin (e.g., y = x³).
• Even function: f(–x) = f(x); graph is symmetric about the y-axis (e.g., y = x²).

Continuous and Discontinuous Functions

• Continuous function: Can be drawn without lifting the pencil; no holes or breaks.
• Discontinuous functions have gaps, holes, or jumps (e.g., 1/x at x = 0).

Properties of Parent Functions (Examples)

• Reciprocal function (f(x) = 1/x):
  • Domain: x ≠ 0; Range: y ≠ 0; Asymptotes at x = 0 and y = 0.
  • No intervals of increase; decreases on (–∞, 0) and (0, ∞).
  • Point symmetry (odd function); no zeros or y-intercept.
• Exponential function (f(x) = 2ˣ):
  • Domain: all real numbers; Range: y > 0; Asymptote at y = 0.
  • Always increasing on (–∞, ∞); no zeros.
  • End behavior: as x → ∞, y → ∞; as x → –∞, y → 0.

Key Terms & Definitions

• Symmetry — Balance about a line or point; mirror image or rotational match.
• Line Symmetry — Symmetry about a vertical line x = a.
• Point Symmetry — 180° rotational symmetry about (a, b).
• End Behavior — The direction the graph heads as x → ±∞.
• Interval Notation — Bracket notation to describe increasing/decreasing intervals.
• Even Function — f(–x) = f(x); symmetric about y-axis.
• Odd Function — f(–x) = –f(x); symmetric about the origin.
• Continuous Function — No breaks, jumps, or holes in the graph.
• Discontinuous Function — Has breaks, jumps, or holes.
• Asymptote — Line that the graph approaches but never crosses.

Action Items / Next Steps

• Complete homework: Section 1.3, #12, on properties of graphs of functions.
• Study the parent function characteristics chart and be ready to identify function properties.
• Review interval notation and be able to apply it to increasing/decreasing intervals.