Transforming Absolute Value

Objective

• Review over what absolute value is
• Review over how to translate the absolute value graph right/left
• Review over how to translate the absolute value graph up and down
• Go over how to reflect the absolute value graph
• Go over how to stretch/compress the absolute value graph
• Do some examples
• Homework

Absolute Value

So that’s a piecewise function, but this would be a poor power point if it didn’t have the absolute value function in it as well.

So who remembers the absolute value of something?

For those that don’t, the absolute value of a number is categorized as it’s distance from 0, and is written as |some number|

So, as an example, the absolute value of -2 would be written as:

|-2|

And categorized as the distance -2 is from 0.

So if we were to look at a number line:

-2

-1

0

1

2

-2

0

So, from this we know that

|-2| = 2

The graph of f(x) = |x|

That looks familiar though…..

Seems an awful lot like a piece wise function, right?

That’s because it is.

So why do we do this?

It make it easier to see how a bunch of relatable functions mash together to form a big one.

To be completely honest, this is what you will use more in life than a regular function, mainly because life is not a nice, even equation.

It’s filled with all sorts of bumps and variables, and making piecewise functions is a great way to filter through all of the issues and see what’s really happening.

So, how to we transform absolute value left and right?

Well, to be honest, we transform the graph the exact same way we transform a quadratic equation.

Remember how we transformed quadratics?

Where r is how far right we want to move the graph and

U is how far up we want to move the graph

Well, when it comes to absolute value functions, it’s not really much different.

Let’s start with an example:

Now let’s say we want to move this graph 3 to the right

Then our new equation would look like:

So then, how do we move the graph up/down?

Exactly the same way that we do it for a quadratic equation.

So, as a reminder, here is the quadratic equation to move a graph

Where r is how far right we want to move the graph (again) and

U is how far up we want to move the graph

Let’s start with the same example:

Now let’s say we want to move this graph up 4

Then our new equation would look like:

Now let’s put them together!

So now let’s put them together:

So, again, let’s start with the same first example:

Now let’s say we want to move this graph up 4

And to the left 5

Then our new equation would look like:

So, what we can say is, just like our old equation:

For absolute value, our equation would be:

Something important to point out:

R and u are the vertex of the graph, or in other words, the very bottom point of the absolute value graph.

So, in our example, the vertex of this graph would be:

(-5, 4)

Now, let’s see some examples!

REFLECTING THE GRAPH ACROSS THE X-AXIS

So, just like we did with the quadratics graph, the way we’re going to reflect the graph of: f(x) = |x| is by multiplying |x| by a negative number.

See, when we do that, each value for f(x) becomes negative, thus making all of the points go below the x-axis instead of above.

So here’s an example:

Reflecting f(x) = |x|

So first, let’s start with our original function:

As we can see, all of the points are above the x-axis,

And

But, we want f(x) to be negative instead.

So, what we do is multiply the function by -1.

So our equation becomes:

And our graph becomes:

And our points become:

SO THAT’S IT?

That’s it!

If you want to reflect the function (or more importantly, if you are tested and need to reflect the function) all you need to do is multiply the function by a negative number.

But it doesn’t have to necessarily be negative 1, it could be any other negative.

But before we get into that, let’s go over how to stretch/compress the graph as well.

STRETCHING/COMPRESSING THE GRAPH

Again, to stretch or compress the graph, we’re going to do the same thing that we did to the quadratics graph.

So, the way we are going to stretch the graph of: f(x) = |x| is by multiplying |x| by an integer number.

If, instead, we want to compress the graph of: f(x) = |x|, we want to multiply |x| by a fraction.

See, again, to make the graph stretch, we want the y values to increase drastically, while the x values increase at the same rate.

However, if we want to make the graph compress, we want the y values to increase much slower, while the x values increase at the same rate.

So here’s an example:

Stretching f(x) = |x|

So first, let’s start with our original function:

As we can see, all of the points are above the x-axis,

And

But, we want f(x) to be stretched instead.

So, let’s try multiplying the function by 2.

So our equation becomes:

And our graph becomes:

And our points become:

Compressing f(x) = |x|

So first, let’s start with our original function:

As we can see, all of the points are above the x-axis,

And

But, we want f(x) to be compressed instead.

So, let’s try multiplying the function by 1/2.

So our equation becomes:

And our graph becomes:

And our points become:

NOW LET’S PUT THEM ALTOGETHER!

So now let’s put them everything we know together:

So, again, let’s start with the same first example:

Now let’s say we want to move this graph up 4

To the left 5

Compress by ¼

And reflect about the x-axis

Then our new equation would look like:

So, what we can say is, just like our old equation:

For absolute value, our equation would be:

Something important to point out:

R and u are still the vertex of the graph, or in other words, the very bottom point of the absolute value graph.

So, in our example, the vertex would still be:

(-5, 4)

Now, let’s see some examples!

Example 1:

Give the equation for the following absolute value function graph and identify the vertex:

Well, as we can see, the graph has shifted 2 units to the left

2 units down

It seems to be stretched by 2

And it’s reflected.

So, we can safely say:

R = -2

U = -2

S = -2

So, our equation would look like:

And, since r = -2, and u = -2, we can see that the vertex is:

(-2, -2)

Example 2:

Give the equation for the following absolute value function graph and identify the vertex:

Well, as we can see, the graph has shifted

2 units to the left

1 unit down

Stretched by 1/3

And it’s been reflected.

So, we can safely say:

R = -2

U = -1

S = -1/3

So, our equation would look like:

And, since r = -2, and u = -1, we can see that the vertex is:

(-2, -1)

Example 3:

Graph the following equation and identify the vertex:

​

We can see, when we compare this equation to our discovered equation:

​

That:

R = 2

U = 3

S = 4

Which means we are moving to left 2 and up 3.

So, our graph would look like:

We can also see that since r = 2 and u = 3 that the vertex of this graph will be: (2, 3)