DWG NO. 1.3 — Lesson 3 of 6

Graphing Functions and Key Features

Unit 1: Functions and Their Graphs · ~25 min

Objective Identify intercepts, increasing/decreasing intervals, and even/odd symmetry from a function's graph or equation.

Intercepts

The y-intercept is where the graph crosses the y-axis — find it by evaluating f(0). The x-intercepts (also called zeros or roots) are where the graph crosses the x-axis — find them by solving f(x) = 0.

Increasing and decreasing intervals

A function is increasing on an interval if its graph rises as you move left to right across it, and decreasing if it falls. These intervals are always described using x-values only, written in interval notation.

Even and odd symmetry

Some functions have a symmetry you can check algebraically, without graphing anything:

Even: f(−x) = f(x) for every x. The graph is a mirror image across the y-axis.
Odd: f(−x) = −f(x) for every x. The graph has 180° rotational symmetry about the origin.
Neither: most functions satisfy neither condition, and that's fine — symmetry is a bonus feature, not a requirement.

f(x) = x³ − 3x — increasing on (−∞, −1) and (1, ∞); decreasing on (−1, 1)

Worked Example 1 · Finding intercepts

ProblemFind the intercepts of f(x) = x² − 4.

1y-intercept: f(0) = 0² − 4 = −4, so the point is (0, −4).

2x-intercepts: solve x² − 4 = 0 → x² = 4 → x = ±2.

y-intercept: (0, −4) · x-intercepts: (−2, 0) and (2, 0)

Worked Example 2 · Reading increasing/decreasing from a graph

ProblemUsing the diagram above of f(x) = x³ − 3x, describe where the graph rises and falls.

1Moving left to right, the curve rises until the local max at x = −1, then falls until the local min at x = 1, then rises again.

Increasing: (−∞, −1) and (1, ∞) · Decreasing: (−1, 1)

Worked Example 3 · Testing even and odd

ProblemTest f(x) = x&sup4; − 3x² for symmetry. Replace x with −x: f(−x) = (−x)&sup4; − 3(−x)² = x&sup4; − 3x².

1Since f(−x) = f(x), f is even.

2Now test g(x) = x³ − x. g(−x) = (−x)³ − (−x) = −x³ + x = −(x³ − x) = −g(x).

f is even · g is odd

Guided practice

1.Find the x- and y-intercepts of f(x) = 2x − 6.
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y-intercept: f(0) = −6, point (0, −6). x-intercept: 2x − 6 = 0 → x = 3, point (3, 0). (0, −6) and (3, 0)
2.A graph rises from (−∞, 0) up to a peak at x = 2, then falls forever after. State the increasing and decreasing intervals.
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Increasing: (−∞, 2) · Decreasing: (2, ∞)
3.Determine whether f(x) = x² + 5 is even, odd, or neither.
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f(−x) = (−x)² + 5 = x² + 5 = f(x). f is even.
4.Determine whether g(x) = 2x³ + x is even, odd, or neither.
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g(−x) = 2(−x)³ + (−x) = −2x³ − x = −(2x³ + x) = −g(x). g is odd.
5.Determine whether h(x) = x² + x is even, odd, or neither.
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h(−x) = (−x)² + (−x) = x² − x. This matches neither h(x) = x²+x nor −h(x) = −x²−x. h is neither even nor odd.

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