Graphing Linear Functions

Q: What's the difference between $f(x)$ and $y$?

On a graph, nothing — $f(x)$ is the $y$-coordinate. Function notation just makes it easier to talk about specific inputs, like $f(3)$ meaning the $y$-value at $x = 3$.

The graph of a linear function is a straight line, and every point on that line is a solution of the equation. That second part is the key idea: a graph is not decoration — it is a picture of every input-output pair the function produces.

Because two points determine a line, graphing a linear function only ever takes three moves: find two points, plot them, and draw the line through them. Everything in this article is a strategy for finding those two points quickly.

Function notation is just $y$

When you see $f(x) = 2x - 1$ , read $f(x)$ as the $y$ -value. The equation $y = 2x - 1$ and the function $f(x) = 2x - 1$ have exactly the same graph. Writing $f(3)$ just means the $y$ -value when $x = 3$ .

To find a point, pick any $x$ , substitute, and compute. For $f(x) = 2x - 1$ : $f(0) = -1$ gives the point $(0, -1)$ , and $f(2) = 3$ gives the point $(2, 3)$ . Plot both and connect them — the graph below shows the result.

Choosing smart points

Any two $x$ -values work, but some make the arithmetic easier. $x = 0$ is almost always a good first pick, because it gives the $y$ -intercept immediately. For an equation like $x + 2y = 4$ that is not solved for $y$ , use the intercepts: set $x = 0$ to find where the line crosses the $y$ -axis, then set $y = 0$ to find where it crosses the $x$ -axis.

If the function has a fraction slope, pick $x$ -values that clear the fraction. For $f(x) = \dfrac{2}{3}x + 1$ , choosing $x = 0$ and $x = 3$ keeps every coordinate a whole number.

Is a point on the graph?

A point is on the graph exactly when substituting its $x$ -value into the function produces its $y$ -value. To test whether $(2, 5)$ is on the graph of $f(x) = 4x - 3$ , compute $f(2) = 4(2) - 3 = 5$ . It matches, so the point is on the line. If the output had been anything other than $5$ , the point would miss the line.

Worked examples

Example 1: graph from function notation

Graph $f(x) = 2x - 1$ .

Evaluate at

x = 0

f(0) = 2(0) - 1 = -1

Evaluate at

x = 2

f(2) = 2(2) - 1 = 3

Plot

(0, -1)

and

(2, 3)

, then draw the line through them

Answer: The line through $(0, -1)$ and $(2, 3)$ .

Example 2: graph using intercepts

Graph $x + 2y = 4$ .

Set

x = 0

and solve

2y = 4 \;\Rightarrow\; y = 2

Set

y = 0

and solve

x = 4

Plot

(0, 2)

and

(4, 0)

, then draw the line through them

Answer: The line through $(0, 2)$ and $(4, 0)$ .

Example 3: test whether a point is on the graph

Is $(2, 5)$ on the graph of $f(x) = 4x - 3$ ?

Substitute the

x

-value

f(2) = 4(2) - 3

Simplify

f(2) = 5

The output equals the point's

y

-value

Answer: Yes — $(2, 5)$ is on the graph.

Try one yourself

Common questions

Why do I only need two points?

Two points determine exactly one line. A third point is still worth plotting as a check — if it doesn't land on the same line, one of your points has an arithmetic mistake.

What's the difference between

f(x)

and

y

On a graph, nothing — $f(x)$ is the $y$ -coordinate. Function notation just makes it easier to talk about specific inputs, like $f(3)$ meaning the $y$ -value at $x = 3$ .

Which two points should I pick?

Whichever make the arithmetic clean. $x = 0$ is usually the easiest start. If the slope is a fraction, pick an $x$ -value that's a multiple of the denominator.

How do I graph an equation like

x + 2y = 4

that isn't solved for

y

Use the intercepts: plug in $x = 0$ to get one point, then $y = 0$ to get another. Or solve the equation for $y$ first and graph it like any other function.

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Graphing Linear Functions

Function notation is just $y$

Choosing smart points

Is a point on the graph?

Worked examples

Example 1: graph from function notation

Example 2: graph using intercepts

Example 3: test whether a point is on the graph

Try one yourself

Common questions

Keep learning

Rate of Change and Slope

Slope from a Table

Vertical Translations of Linear Functions

Piecewise and Step Functions

Graphing Linear Functions

Function notation is just yyy

Choosing smart points

Is a point on the graph?

Worked examples

Example 1: graph from function notation

Example 2: graph using intercepts

Example 3: test whether a point is on the graph

Try one yourself

Common questions

Keep learning

Rate of Change and Slope

Slope from a Table

Vertical Translations of Linear Functions

Piecewise and Step Functions

Function notation is just $y$