Vertical Line Test - October 10, 2013

Welcome back to Professor Baker's Math Class! Today, we delved into the Vertical Line Test. This test is a visual method used to determine whether a relation defined on a graph is a function. Let's break it down:

What is a Function?

Before we dive into the test, let's quickly recap what a function is. A function is a special type of relation where each input (usually denoted as $x$) has only one output (usually denoted as $y$). In other words, for every $x$-value, there can be only one corresponding $y$-value.

The Vertical Line Test Explained

The Vertical Line Test states that if you can draw any vertical line that intersects the graph of a relation at more than one point, then the relation is not a function. Conversely, if every vertical line intersects the graph at at most one point, then the relation is a function.

Examples

Let's consider some examples:

  • Example 1: Imagine a set of points on a graph. If a vertical line can pass through two or more of those points simultaneously, this implies that a single $x$-value has multiple $y$-values. Hence, it's not a function.
  • Example 2: Consider a circle. If you draw a vertical line through the circle, it will intersect the circle at two points (except at the extreme left and right points). This means for one $x$ value there are two possible $y$ values and therefore a circle is not a function.
  • Example 3: Consider a line where for every $x$ there is only one $y$. No vertical line can pass through more than one point on that line and therefore a line is a function.

Why Does It Work?

The Vertical Line Test is based on the definition of a function. If a vertical line intersects a graph at two points, say $(x, y_1)$ and $(x, y_2)$, then we have the same $x$-value ($x$) mapped to two different $y$-values ($y_1$ and $y_2$). This violates the definition of a function.

Class Notes & Homework

Refer to the class notes and worksheet for more examples and practice problems. Remember, practice makes perfect!

  • Class Notes: (Attached PDF)
  • Worksheet: Page 1 & Page 2 (Check attachments or course website)

Homework: Complete problems #22-32 on pages 71 and 72 in your textbook. Don't hesitate to ask questions if you get stuck! Good luck, and remember, you can do it!