# Piecewise Function: Graph

How to graph a piecewise function on a coordinate plane: 3 examples and their solutions.

## Example

y = 2x + 3

(x < 0)

So draw y = 2x + 3

on the left side of x = 0.

Slope-Intercept Form

x < 0 does not include x = 0.

So draw an empty circle at x = 0.

y = x - 1

(x ≥ 0)

So draw y = x - 1

on the right side of x = 0.

x ≥ 0 does include x = 0.

So draw a full circle at x = 0.

This is the graph of the given piecewise function.

## Example

y = x^{2}

(x < 1)

So draw y = x^{2}

on the left side of x = 1.

Quadratic Function: Vertex Form

x < 1 does not include x = 1.

So draw an empty circle at x = 1.

y = x

(x ≥ 1)

So draw y = x

on the right side of x = 1.

x ≥ 1 does include x = 1.

So draw a full circle at x = 1.

Both graphs meet at the same point

when x = 1.

So remove the circle

and connect the graphs.

This is the graph of the given piecewise function.

## Example

See (x + 1)(x - 3)/(x - 3).

When x ≠ 3,

x - 3 ≠ 0.

So you can cancel (x - 3) factors.

Then

(x + 1)(x - 3)/(x - 3) = x + 1.

Simplify a Rational Expression

Rewrite the given piecewise function.

Change

y = (x + 1)(x - 3)/(x - 3)

to

y = x + 1.

And write

y = -2

(x = 3).

Graph this piecewise function.

y = x + 1

(x ≠ 3)

So draw y = x + 1.

And draw an empty circle at x = 3.

The empty circle means

the y value is not on the line y = x + 1.

y = -2

(x = 3)

So draw a point on (3, -2).

This is the graph of the given piecewise function.