Problem 60 - Algebra and Trigonometry

Question

Determine whether the equation defines y as a function of x. (See Example 9.) \(x^{2}+(y-1)^{2}=4\)

Step-by-Step Solution

Answer

The equation does not define \(y\) as a function of \(x\) because it represents a circle, which fails the vertical line test.

Step 1: Understand the equation

The given equation is \( x^2 + (y - 1)^2 = 4 \). This looks like the equation of a circle centered at (0, 1) with a radius of 2.

Step 2: Rearrange terms to identify the form

Rewrite the equation in the standard form of a circle: \( (x - h)^2 + (y - k)^2 = r^2 \). In this equation, \((h, k)\) is the center of the circle and \(r\) is the radius. For this problem, \(h = 0\), \(k = 1\), and \(r = 2\).

Step 3: Identify characteristics of a circle

A circle is a set of points equidistant from a center point. However, the key property of a function is for each input \(x\), there must be exactly one output \(y\).

Step 4: Consider symmetry and vertical line test

For a circle, given any \(x\) (except the extreme points \(x = -2, x = 2\)), there are generally two corresponding \(y\) values (one above the center and one below). This fails the vertical line test, meaning it does not define \(y\) as a function of \(x\).

Step 5: Final conclusion based on properties

Since some \(x\) values map to more than one \(y\) value, the given equation does not define \(y\) as a function of \(x\).

Key Concepts

Function Definition

A function is a type of mathematical relation where each input is associated with exactly one output. It is important to understand this concept when determining if an equation defines a function. To define y as a function of x, for every value of x, there must be only one corresponding value of y. This concept is crucial in many areas of mathematics and is foundational for understanding more complex topics.

For example:

• If for an input of x = 2, there is only one y = 3, it defines a function.
• If x = 2 could result in y = 3 or y = -3, it does not define a function.

Understanding whether an equation forms a function helps in analyzing the behavior of mathematical models and real-life situations.

Vertical Line Test

The vertical line test is a visual way to determine if an equation is a function on a graph. This test involves drawing vertical lines across the graph of an equation.

If any vertical line intersects the graph in more than one point at a time, then the relation is not a function. This is because it shows that for a single x value, there are multiple y values, which contradicts the definition of a function.

In the case of our equation, which represents a circle, if you draw vertical lines through most of the points on the circle, they will intersect at two points. Hence, the equation fails the vertical line test, reaffirming that it is not a function.

Geometry

The geometry of the circle is key in understanding why the given equation does not describe a function. A circle is defined as the set of all points in a plane that are equidistant from a given point, called the center. The given equation is written in the form of a circle, \(x^{2} + (y - 1)^{2} = 4\), which implies:

• The center of the circle is at point (0, 1).
• The radius is 2 units.

A circle's inherent symmetry about its center means that most x values will intersect the circle at two points. This symmetry is what causes circles to fail the vertical line test when determining if they represent functions.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and rewriting equations to better understand their characteristics. For our equation, \(x^{2} + (y - 1)^{2} = 4\), you can see that it is already a form of the circle equation. By recognizing this, one can identify the geometric nature of the problem and apply relevant tests, like the vertical line test, to determine function status.

Rearranging equations might help in:

• Recognizing standard forms that reveal the nature of the graph.
• Simplifying equations for easier calculation or graphing.

By mastering algebraic manipulation, you can often reveal hidden structures and properties within equations, making them easier to interpret and analyze.