Graphing Functions

13 min read

Quick Reference

The Cartesian plane, plotting ordered pairs, and reading a curve are the picture side of a function. They prepare the formal function-notation work and every later graphing skill in MATH141.

Try This First

Plotting points by hand for two minutes builds the instinct the definitions below make precise. A wrong turn while counting is common and shows exactly which coordinate to watch. There is no need to rush this.

A Function as a Process You Can Picture

A function is a process that takes an input and gives back exactly one output. The graph is the picture of that process: for every input $x$ that the function accepts, you compute the output $f(x)$, and you mark the single point $(x, f(x))$. Sweep through every allowed input and the marks fill in a curve.

Two plain-language observations carry most of the meaning, and it helps to hold both at once:

• As the input moves left to right, the output is whatever height the curve sits at. Reading the picture from left to right is reading the process from its first input to its last.
• One input, one height. Because the process returns exactly one output for each input, a curve that is a function never stacks two points directly above the same input.

These two ideas appear again as the vertical line test and as the meaning of increasing and decreasing, so it is worth seeing them first in words.

Prerequisite Hub

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    subgraph BuildsOn["Builds On"]
        A["Algebraic<br/>Simplification"]
        B["Solving<br/>Inequalities"]
    end

    subgraph ThisSkill["This Skill"]
        C["Graphing<br/>Functions"]
    end

    subgraph Unlocks
        D["Transformations<br/>of Graphs"]
        E["Trig<br/>Identities"]
        F["Functions and<br/>Function Notation"]
    end

    A --> C
    B --> C
    C --> D
    C --> E
    C --> F

    style C fill:#d1fae5,stroke:#a565f0,stroke-width:3px

Builds on (have these ready before starting):

Unlocks (these become reachable after this node):

No cross-course prerequisites apply to this node.

Prerequisite Check

Official Definition

Cartesian (rectangular) coordinate system. A grid system based on a horizontal axis (the $x$-axis) and a vertical axis (the $y$-axis) that intersect at the origin, used to plot ordered pairs $(x, y)$.

Source: OpenStax College Algebra 2e, Section 2.1 The Rectangular Coordinate Systems and Graphs.

A few terms travel with this definition and are worth pinning down in the same breath:

• The origin is the intersection point of the two axes, the ordered pair $(0, 0)$.
• An ordered pair $(x, y)$ lists the horizontal coordinate first and the vertical coordinate second. Order matters: $(3, 5)$ and $(5, 3)$ are different locations.
• The two axes cut the plane into four quadrants, numbered counterclockwise from the upper right.
• The graph of a function $f$ is the set of all points $(x, f(x))$ for inputs $x$ that the function accepts.

The vertical line test below is a direct visual consequence of the one-output-per-input meaning of a function, not a separate result to memorize.

Two Representations on One Screen

Every height on a graph can be read three ways, and translating between them is the skill: a worded rule, a table of input-output pairs, and the plotted curve. Take the rule “the output is the input plus one.”

Worded rule: the output is one more than the input.

Table:

Graph: plotting those five points and connecting them gives a straight line that rises one unit of height for every one unit to the right, crossing the vertical axis at height $1$.

Translate in both directions to lock the idea in: pick a sixth input, say $5$, find its height from the rule (which is $6$), and predict where the point lands before plotting it. Then go the other way: read the height at input $-3$ off the line and confirm the rule gives the same number.

Worked Example 1: Plot, Then Read

Build the graph of the rule “the output is the input squared,” written $f(x) = x^2$, then read three features off it.

Predict first. The square of any real number is never negative, so before plotting, predict that no part of the curve dips below the horizontal axis.

Step 1: Build a table. Choose inputs spread on both sides of zero.

Step 2: Plot and connect. The five points form a U-shaped curve with its lowest point at the origin, opening upward.

Step 3: Read three features.

• Lowest height: the curve bottoms out at $(0, 0)$, so the smallest output is $0$, matching the prediction.
• Symmetry: the input $-2$ and the input $2$ both give height $4$. The left half is a mirror image of the right half across the vertical axis.
• Where the height equals 4: there are two inputs, $-2$ and $2$. One height paired with two inputs is allowed for a function. Two heights for one input is what is forbidden.

Check against the prediction. No plotted point sits below the horizontal axis, and the curve confirms it: the lowest height is $0$.

Worked Example 2: The Vertical Line Test

Decide whether a given curve is the graph of a function, and explain the decision in terms of the one-output-per-input meaning.

Curve A: the straight line through the plotted points $(-1, -1)$, $(0, 1)$, and $(1, 3)$.

Curve B: the full circle of radius $2$ centered at the origin, passing through $(2, 0)$, $(0, 2)$, $(-2, 0)$, and $(0, -2)$.

Predict first. A function assigns one height per input. Predict which curve might stack two heights over a single input.

Test Curve A. Slide an imaginary vertical line across the plane. At every input, the line crosses the straight line exactly once. One input, one height, everywhere. Curve A is a function.

Test Curve B. Slide the same vertical line to the input $x = 0$. It crosses the circle at $(0, 2)$ and at $(0, -2)$, two heights over the single input $0$. The same double crossing happens at $x = 1$, where the circle passes through about $(1, 1.73)$ and $(1, -1.73)$. One input, two heights, breaks the promise. Curve B is not a function.

Check against the prediction. The circle is the curve that stacks two heights over one input, exactly where a closed loop must.

The vertical line test is just this slide-a-line check made into a habit. A vertical line that ever meets a curve more than once exposes an input with more than one output.

Common Misconceptions

Practice Problems

Mastery Checklist

Novice (Level 1-2):

• I can plot an ordered pair and name its quadrant by reading the signs of its coordinates.
• I can build a table of points from a worded rule or an algebraic rule.
• I can state, in plain words, that a function returns exactly one output for each input.

Competent (Level 3-4):

• I can apply the vertical line test and justify it from the one-output-per-input meaning.
• I can read crossings of an axis and the lowest or highest height off a built graph.
• I can translate between a rule, a table, and a graph in both directions.

Proficient (Level 5):

• I can interpret a context graph by naming what each axis measures before describing the curve.
• I can spot and correct an iconic-graph misreading.
• I can explain why a circle fails to be a function while still being a valid set of points.

Connections

Looking back:

• Algebraic simplification supplies the evaluation step behind every plotted point.
• Solving inequalities supplies the interval language for describing where a curve sits relative to an axis.

Looking ahead:

• Transformations of graphs (transformations-of-graphs): shifting, stretching, and reflecting a known curve, which Worked Example 1 previewed when the square curve slid down to $f(x) = x^2 - 4$.
• Functions and function notation (m141-functions-and-function-notation): the formal $f(x)$ machinery, now grounded in the picture of plotted heights.
• Trig identities (trig-identities): the graphical instinct here supports recognizing periodic curves later in the course.

Real-world connections:

• A distance-over-time graph reports motion, not the shape of the path.
• A temperature-over-day graph shows how one quantity rises and falls as another advances, the same input-to-output reading practiced here.

Resources

Page numbers for the textbook sections are pending faculty verification.

Last updated: 2026-06-16