Understanding Absolute Value Functions

Stretching and compressing absolute value functions can be really interesting! These changes affect how the graph looks and its symmetry. Let’s break this down simply.

What is an Absolute Value Function?

The basic absolute value function is written as $f(x) = |x|$.

When you look at its graph, it has a "V" shape. This "V" is perfectly symmetrical around the $y$-axis.

This means that if you pick a point on the graph, like $(x, y)$, you will also find a matching point on the other side, which is $(-x, y)$.

What Happens When You Stretch the Graph?

If you stretch the graph up, for example, with $f(x) = a|x|$ where $a > 1$, it gets taller.

However, it still stays symmetrical around the $y$-axis.

So, even though the graph becomes higher, the left side still looks like the right side!

What Happens When You Compress the Graph?

On the other hand, if you compress the graph with $0 < a < 1$, like in $f(x) = a|x|$, the shape becomes flatter.

Again, it remains symmetrical! The graph still mirrors itself around the $y$-axis.

Changes in the Horizontal Direction

Now, if you stretch or compress the graph horizontally, that’s a different story.

For instance, if you change it to $f(x) = |bx|$, where $b > 1$, the graph gets narrower.

If $0 < b < 1$, it gets wider.

These horizontal changes can affect how we see symmetry, especially if the point at the top of the "V" (called the vertex) shifts.

Wrapping Up

In summary, vertical changes, like stretching and compressing, keep the symmetry of absolute value functions.

But horizontal changes can mess with that symmetry a bit.

It's really helpful to draw these graphs. When you see the changes visually, it makes everything much clearer!