2-4 A Variety of Graphs

Piecewise Functions

What are Piecewise Functions?

Piecewise functions are defined for specific domains. The most basic example of a piecewise function is the absolute value function . Why?

That’s right. If x > 0 the graph looks one way, and if x < 0 the graph looks different.

y x

Huh? What did she say?

This makes sense, right?

So, to the right of zero, what is the slope of the line? To the left of zero, what is the slope?

if x 0

if x

x y x

x 0

(If this doesn’t make sense, then just plot points and see what you get)

What about ?

It follows that

What does it look like? Trends?

y 2x 4

x 2x 4 if

y 2x 42x 4 f xi 2

2

Will we graph this way?

Nope. At least I won’t. Instead, I prefer to find the vertex of the graph and then plot 2 points on either side of the vertex.

Why find the vertex?Why 2 points instead of just 1 on each side?

But, how do I find the vertex?

Lets look at a couple of graphs online to see what happens to the absolute value and see if we can figure out how to find the vertex.

Graphing Absolute Value

1. Set the inside of the absolute value = 0; solve for x. Plug this x back in and solve for y. You now have the coordinates of the vertex.

2. Plug in 2 “x” values on each side of the vertex (equidistant works well) and make sure the corresponding y values are equal. (IF they aren’t, what have you just figured out?

Examples

Graph the following

1.

2.

y x 3 2

y 2x 4

The next kind of piecewise function

The form of this function is similar to this:

This looks worse than it is. Essentially the function is split into multiple functions based on particular domains. (you might only see 2 functions, you could see 4..)

1

2

3

fn if x a

f(x) fn if b<x a

fn if x b

Think of it as “sub” graphs and make charts.

This may seem rote, but it is probably the easiest way to do this problem. Make a table for each domain; pick 3 numbers “a and above”, 3 between b and a, and then 3 numbers “ below b”.

Then plot those points. Just be careful of the endpoints.

Example

3. Graph the following

x if x 2

f(x) 3x 1 if 0 x 2

2x if x 0