Translating Piecewise Functions

Translating Piecewise Functions
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  • 0:05 Understanding…
  • 1:19 Unknown Piecewise Functions
  • 1:59 Stretching,…
  • 3:22 Transformations on the Graph
  • 6:12 Lesson Summary
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Instructor: Tyler Cantway

Tyler has tutored math at two universities and has a master's degree in engineering.

Piecewise functions are two or more functions that only use part of a domain. If these functions need to be changed, we can translate them without even having to know the equation.

Understanding Piecewise Functions

Have you ever put together a puzzle? I've put together lots of puzzles. One time, I started putting a puzzle together on my desk and before I knew it, I had run out of room. I didn't want the puzzle to fall apart, but I wanted to finish it. I just carefully pushed the puzzle up a few inches on the desk. Then, I pushed it to the left a few more inches. What was interesting is that to move the whole puzzle, all I had to do was move the pieces that were already snapped together. It didn't matter how many pieces there were. It didn't matter what the picture was on the puzzle. I just carefully moved it to a better place.

Piecewise functions are when we take small bits and pieces of graphs and fit them together. It can be pieces of two simple functions or many pieces of complicated functions. But, just like the puzzle, sometimes we will have to move a piecewise function to a better location on the graph. Moving and changing a graph is called translation. A graph can be translated up, down, left, right, reflected, or stretched. There are special ways to translate piecewise functions so they can be put in the right places on a graph.

Unknown Piecewise Functions

When I moved that puzzle on the desk, it didn't matter the size, shape, or even what picture was on the puzzle. I just moved it. If we were given a piecewise graph that looks like the one below, we might not know the pieces of the functions that created it. That's okay, because we can just call them f(x) and g(x). In fact, we can use any letter to name each piece of a piecewise function. If there were more pieces, we could use more letters. Just know that we can translate even the most complicated piecewise functions by making a few changes to the equation. Let's take a look at how to translate them.

You can use any letter to identify a piece of a piecewise function.
graph example of piecewise function

Stretching, Reflecting, Shifting

First, let's take a look at the basic formula. When we don't know what a function is, we write it as f(x). Here is that function with the changes we can potentially make to it:

A*f(x+B) + C

You will notice that A and C are outside the parentheses, but B is inside the parentheses. That means B will act differently from A and C.

Let's start with A and C. They change the y values of the function, which means they make vertical translations. A is multiplied by the function, which causes stretching. Multiplying A times the y values stretches it in the vertical direction. Remember, if A is negative, it will change the sign of the y values, which reflects it about the x axis.

C controls shifting up or down. If C is added, we shift the graph up. If C is subtracted, we shift the graph down.

Let's go back to B. B is inside the parentheses, which means it changes the x value and controls translations left or right. It also means that B acts opposite of what you would think. If B is added, you actually shift the graph to the left. If B is subtracted, you shift the graph right.

Transformations on the Graph

Now that we know how to translate the graph by using A, B, and C, let's do it on a piecewise function. If we had the graph below, we can tell it has two pieces. We don't have the piecewise function written, so we call the left side f(x) and the right side g(x).

Consider the left side f(x) and the right side g(x).
example graph with two pieces

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