Problem 11 - College Algebra and Calculus: An Applied Approach

Question

Use the Vertical Line Test to decide whether \(y\) is a function of \(x\). \(x^{2}+y^{2}=9\)

Step-by-Step Solution

Answer

By applying the Vertical Line Test on the graph of the equation, it is determined that \(y\) is not a function of \(x\).

Step 1: Understanding the Equation

The given equation is \(x^{2}+y^{2}=9\), which is in the standard form of a circle. The centre of this circle is at the origin (0,0) and the radius is \(\sqrt{9}\) which is 3.

Step 2: Graphing the Equation

On graphing this equation, a circle with a centre at the origin and a radius of 3 is obtained. The circle is symmetric about x-axis and y-axis.

Step 3: Applying the Vertical Line Test

On drawing vertical lines across the graph, it is observed that each line crosses the circle at more than one point. Therefore, by the Vertical Line Test, \(y\) is not purely a function of \(x\).

Key Concepts

Functions and the Vertical Line Test

A function is a special relationship between two sets where each element in the first set, typically called the domain, is paired with exactly one element in the second set, known as the range. In simpler terms, a function is like a machine that gives you one output for any input you provide.

To determine whether a graph represents a function, we often use the *Vertical Line Test*. This is a visual way to decide if a curve is a graph of a function of a variable. The rule is simple:

• If a vertical line crosses the graph in more than one place, then the graph does not represent a function.
• Conversely, if a vertical line crosses the graph at exactly one point everywhere, then it is a function.

This test is especially helpful when dealing with curves, as it quickly shows us whether each x-value (input) corresponds to exactly one y-value (output).

Understanding Circle Equations

The standard form of a circle equation is given by \[x^2 + y^2 = r^2\]where \((x, y)\) are the coordinates of a point on the circle and \(r\) is the radius.

In the equation \(x^2 + y^2 = 9\), we can recognize it as the circle's equation centered at the origin \((0, 0)\) with a radius of \(3\), since \(r^2 = 9\) implies \(r = \sqrt{9} = 3\).

Circles are unique in their geometric properties:

• They are perfectly symmetrical about both axes.
• The distance from the circle's center to any point on the circle is the same, known as the radius.

This symmetry is what causes vertical lines drawn through the circle to touch more than one point, which leads into the application of the Vertical Line Test.

Graphing Techniques for Circles and the Vertical Line Test

Graphing circles requires identifying both the center and the radius from the equation. For the equation \(x^2 + y^2 = 9\), the center is at \((0, 0)\), and the radius is \(3\).

To graph this circle:

• Start by plotting the center point at the origin.
• From the center, move outwards in all directions to a distance of 3 units and plot several points.
• These create the outline of the circle, which can then be smoothly connected to form the circle.

**Using the Vertical Line Test**
Once the circle is graphed, apply the Vertical Line Test by drawing vertical lines over the curve. If any line intersects the circle at more than one point, it's clear the graph is not a function. Circles often fail this test because, for many x-values, there are two corresponding y-values, particularly on the upper and lower parts of the circle.

This emphasizes why a circle cannot be expressed as \(y\) being a function of \(x\), highlighting the utility of graphing and this simple test.