Unlock the Vertex: Absolute Value Functions Explained!

The absolute value function, often visualized using Desmos, presents a unique challenge: understanding its turning point, the vertex of absolute value. This point reveals crucial information about the function’s behavior. Many students first encounter the complexities of finding this vertex of absolute value during pre-calculus studies, a foundational course often guided by resources from organizations like the National Council of Teachers of Mathematics (NCTM). Algebra provides the tools needed to manipulate and analyze these functions, revealing that the vertex of absolute value determines the minimum or maximum output. Recognizing the vertex of absolute value in equations and graphs equips one with a fundamental understanding of function transformations and applications in fields such as engineering.

Image taken from the YouTube channel The Organic Chemistry Tutor , from the video titled How To Find The Vertex and Axis of Symmetry of Absolute Value Functions – Algebra .

Understanding the Vertex of Absolute Value Functions

This guide explains absolute value functions with a specific focus on identifying and understanding their vertex. We will cover the definition of absolute value, how it impacts the function’s graph, and provide methods for locating the vertex.

What is an Absolute Value Function?

An absolute value function is a function that contains an algebraic expression within absolute value symbols. Remember, the absolute value of a number is its distance from zero on the number line. Therefore, it’s always non-negative (zero or positive).

The Basic Absolute Value Function

The simplest absolute value function is f(x) = |x|. This function takes any input, ‘x’, and returns its absolute value.

• For example:
  • f(3) = |3| = 3
  • f(-3) = |-3| = 3

Graphing the Basic Function

The graph of f(x) = |x| creates a V-shaped graph, with the point of the V at the origin (0, 0). This point is crucial and is, in fact, the vertex of the absolute value.

What is the Vertex of Absolute Value?

The vertex of absolute value is the point where the absolute value function changes direction. On the graph, it is the sharp corner of the V-shape. It’s either the minimum point (when the V opens upwards) or the maximum point (when the V opens downwards) of the function.

The General Form of an Absolute Value Function

The general form helps in easily identifying the vertex. It’s given by:

f(x) = a|x – h| + k

Where:

• ‘a’ determines the direction and steepness of the V-shape.
  • If ‘a’ is positive, the V opens upwards.
  • If ‘a’ is negative, the V opens downwards.
  • The larger the absolute value of ‘a’, the steeper the sides of the V.
• ‘(h, k)’ represents the coordinates of the vertex of absolute value. ‘h’ shifts the graph horizontally, and ‘k’ shifts it vertically.

Understanding the Parameters ‘h’ and ‘k’

• ‘h’ – Horizontal Shift: The value of ‘h’ shifts the graph left or right. Note that inside the absolute value, it appears as (x – h).
  • If ‘h’ is positive, the graph shifts right by ‘h’ units.
  • If ‘h’ is negative, the graph shifts left by the absolute value of ‘h’ units. (For example, if it’s (x – (-2)) which simplifies to (x + 2), it shifts left by 2 units).
• ‘k’ – Vertical Shift: The value of ‘k’ shifts the graph up or down.
  • If ‘k’ is positive, the graph shifts up by ‘k’ units.
  • If ‘k’ is negative, the graph shifts down by the absolute value of ‘k’ units.

How to Find the Vertex of Absolute Value

There are a few ways to find the vertex:

1. Directly from the General Form: As mentioned before, in the general form f(x) = a|x – h| + k, the vertex is (h, k).

2. Finding the Zero of the Expression Inside the Absolute Value:

   a. Set the expression inside the absolute value equal to zero and solve for ‘x’. This ‘x’ value is the ‘h’ coordinate of the vertex.

   b. Substitute this ‘x’ value back into the entire function f(x) to find the corresponding ‘y’ value, which is the ‘k’ coordinate of the vertex.

3. Using a Table of Values (less efficient): You can create a table of x and f(x) values. Look for the x-value where the f(x) values start increasing or decreasing symmetrically. This is generally more time-consuming.

Examples of Finding the Vertex

Example 1: Using the General Form

Let’s find the vertex of absolute value for the function f(x) = 2|x – 3| + 1.

• Comparing this to the general form, we have a = 2, h = 3, and k = 1.
• Therefore, the vertex is (h, k) = (3, 1).

Example 2: Finding the Zero of the Expression

Consider the function g(x) = -|x + 2| – 4.

• Expression inside the absolute value: x + 2
• Set it to zero: x + 2 = 0
• Solve for x: x = -2
• Substitute x = -2 into the function: g(-2) = -|-2 + 2| – 4 = -|0| – 4 = -4
• Therefore, the vertex is (-2, -4).

How the ‘a’ Value Affects the Graph and Vertex

The ‘a’ value in the general form f(x) = a|x – h| + k plays a crucial role in determining the shape of the graph and whether the vertex of absolute value represents a minimum or maximum point.

• a > 0 (a is positive): The V-shape opens upwards. The vertex (h, k) is the minimum point on the graph. The function has a minimum value of ‘k’ at x = h.
• a < 0 (a is negative): The V-shape opens downwards. The vertex (h, k) is the maximum point on the graph. The function has a maximum value of ‘k’ at x = h.
• |a| > 1 (The absolute value of ‘a’ is greater than 1): The graph is steeper than the basic absolute value function f(x) = |x|.
• 0 < |a| < 1 (The absolute value of ‘a’ is between 0 and 1): The graph is less steep (wider) than the basic absolute value function f(x) = |x|.

The following table summarizes these effects:

And that’s a wrap on understanding the vertex of absolute value! Hopefully, you now feel more confident tackling these types of problems. Go forth and conquer those equations!