Graphical Representation of Functions

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Interpret a function’s behavior from its graph.
  • Identify key features such as intercepts and intervals of increase/decrease.
  • Connect graphs to tables, formulas, and function notation.

Key Ideas

A graph represents all the input–output pairs of a function. Every point on the graph has the form \((x, f(x))\).

Important features to look for:

  • \(x\)-intercepts (where the graph crosses the \(x\)-axis)
  • \(y\)-intercept (where it crosses the \(y\)-axis)
  • Increasing/decreasing intervals
  • Maximum/minimum points
  • End behavior (how the graph behaves as \(x\) becomes large or small)

Common Problem Types

1. Intercepts

Read where the graph crosses an axis.

2. Increasing/Decreasing Behavior

Identify where the function rises or falls as \(x\) increases.

3. Reading Function Values

Locate a point with the given \(x\)-value.

4. Connecting Graphs to Rules/Tables

Interpret graph shape relative to formula or table data.

Strategies

  • To find the \(y\)-intercept, look at where the graph hits the vertical axis (\(x = 0\)).
  • To find \(x\)-intercepts, look for where the graph crosses the \(x\)-axis (output = 0).
  • A function is increasing when the graph rises as you move right; decreasing when it falls.
  • Use endpoints, turning points, and slope direction to determine behavior.
  • Connect visual features to notation: if the graph hits \((3, -2)\), then \(f(3) = -2\).

Worked Examples

Example 1 — Finding a \(y\)-Intercept

If the graph passes through \((0, 4)\), then: \[ f(0) = 4 \]

Example 2 — Increasing Interval

If the graph rises as \(x\) moves from \(1\) to \(4\), the function is increasing on \((1, 4)\).

Example 3 — Reading an \(x\)-Intercept

If the graph crosses the \(x\)-axis at \(x = -3\), then: \[ f(-3) = 0 \]

WarningCommon Mistakes
  • Mixing up \(x\)-intercepts and \(y\)-intercepts.
  • Reversing intervals when describing increasing/decreasing behavior.
  • Assuming the graph is linear between plotted points.

Practice Problems

  1. A graph shows \(f(0) = -2\). What is the \(y\)-intercept?
  2. A graph crosses the \(x\)-axis at \(x = 5\). What does this mean?
  3. If a graph decreases on \((0, 4)\), describe what happens to \(f(x)\).
  4. A point \((3, 7)\) lies on the graph. What is \(f(3)\)?
  5. If the graph increases on \((-\infty, -1)\), describe the behavior.

1.
\(y\)-intercept is \((0, -2)\).

2.
The function satisfies \(f(5) = 0\).

3.
As \(x\) increases from 0 to 4, \(f(x)\) decreases.

4.
From the point \((3, 7)\) → \(f(3) = 7\).

5.
As \(x\) moves right toward \(-1\), the function rises.

Summary

  • A graph displays all \((x, f(x))\) pairs of a function.
  • Key features include intercepts, increasing/decreasing intervals, maxima/minima, and end behavior.
  • Use the graph to read outputs directly: find \(x\), then locate the corresponding \(y\).
  • Increasing means “rising to the right”; decreasing means “falling to the right.”
  • Intercepts reveal where the function touches or crosses the axes.
  • Always check the axes first—intercepts give quick insight.
  • Use left-to-right movement to identify increasing/decreasing behavior.
  • Read values carefully: graphs show approximate values unless labeled.
  • Turning points mark changes between rising and falling intervals.