Understanding Absolute Value Functions

Absolute value functions are a special type of math function. They have unique shapes when you draw them on a graph. In Grade 9 Pre-Calculus, it's important to learn how these functions change the way a graph looks. Let's take a closer look at these functions and what makes them special.

What Are Absolute Value Functions?

An absolute value function is written like this:

$f(x) = |x|$

This means that it makes any negative numbers positive. So, if you put in a negative number, the function gives you the positive version of that number.

A more general way to write absolute value functions is:

$f(x) = a|bx + c| + d$

Here’s what those letters mean:

• $a$: Affects how stretchy or squished the graph is, and whether it goes up or down.
• $b$: Changes how wide or thin the graph looks.
• $c$: Moves the graph left or right.
• $d$: Moves the graph up or down.

Key Features of the Graph

1. Vertex:

   • The vertex is the peak point or the lowest point of the graph. You can find it using the formula $(-\frac{c}{b}, d)$. This point is important because it shows where the graph turns.

2. Intercepts:

   • Y-intercept: This is where the graph crosses the vertical line (y-axis). For the function $f(x) = |x|$, the y-intercept is at (0, 0).
   • X-intercept(s): This is where the graph crosses the horizontal line (x-axis). For $f(x) = |x|$, the x-intercept is also at (0, 0). If $d$ is not zero, you have to solve the equation to find x-intercepts.

3. Symmetry:

   • Absolute value functions are symmetrical around the line $x = 0$. This means if there is a point on the graph at $(x, f(x))$, there will also be a matching point at $(-x, f(x))$.

4. End Behavior:

   • How the graph behaves as you move far left or far right depends on the $a$ value:
     • If $a > 0$, as you go to either positive or negative infinity, $f(x)$ will go up to positive infinity: $\lim_{x \to \pm \infty} f(x) = \infty$
     • If $a < 0$, as you go to either infinity, $f(x)$ will go down to negative infinity: $\lim_{x \to \pm \infty} f(x) = -\infty$
   • For $f(x) = |x|$, the ends go up forever.

5. Asymptotes:

   • Absolute value functions don’t have flat lines that they get closer to (asymptotes) because they keep going up or down forever.

Quick Review of Key Features

• Vertex: The high or low point of the graph.
• Intercepts: Where the graph crosses the x and y axes.
• Symmetry: The graph looks the same on both sides of the y-axis.
• End Behavior: Depends on $a$ to see if it goes up or down forever.
• Asymptotes: None because the graph can rise or fall indefinitely.

By learning about these features, students can see how absolute value functions are different from linear ones. They also help in understanding more complicated math ideas in the future.