Problem 35 - Calculus with Concepts in Calculus

Question

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ x^{2}+y^{2}=9 $$

Step-by-Step Solution

Answer

The graph is a circle with center (0,0) and radius 3, and it is not a function.

Step 1: Identify the Equation Type

The equation given is \( x^2 + y^2 = 9 \), which resembles the standard form of a circle \( x^2 + y^2 = r^2 \). Here, \( r^2 = 9 \) thus \( r=3 \). This indicates the graph is a circle with radius 3.

Step 2: Determine the Center

The standard form of a circle \( (x-h)^2 + (y-k)^2 = r^2 \) provides the center at \((h, k)\). In this equation, \( x^2 + y^2 = 9 \), both \( h \) and \( k \) are zero, so the center of the circle is at the origin (0,0).

Step 3: Sketch the Graph

Draw a circle on a coordinate plane with center at (0,0) and a radius of 3. The circle will pass through the points (3,0), (-3,0), (0,3), and (0,-3) on the axes.

Step 4: Check Function Criteria

To determine if the graph is that of a function, apply the vertical line test. If any vertical line crosses the graph more than once, it is not a function. A circle fails this test, as a vertical line through the center will intersect the circle at two points.

Key Concepts

Equation of a Circle

In mathematics, the equation of a circle is expressed in its standard form as \((x-h)^2 + (y-k)^2 = r^2\). It describes a circle on a coordinate plane, where:

• \( (h, k) \) is the center of the circle.
• \( r \) is the radius of the circle, which determines its size.

For the given equation \(x^2 + y^2 = 9\) in the exercise, we can match it with the standard form to identify its specific features:

• The center of the circle is at the origin \((0, 0)\) because \(h\) and \(k\) both equal zero in the equation.
• The radius \(r\) is 3, as calculated from \(r^2 = 9\).

With these parameters, the circle can be sketched on the graph with center at \((0, 0)\) and touching the axes at points \((3,0)\), \((-3,0)\), \((0,3)\), and \((0,-3)\). Understanding these components helps in visualizing the circle and recognizing how equations translate into geometrical shapes.

Vertical Line Test

The vertical line test is an important tool in mathematics used to determine if a graph represents a function. The concept is simple:

• If a vertical line crosses a graph in more than one place at any given x-coordinate, the graph does not represent a function.

In the context of our exercise, the graph in question is a circle, defined by the equation \(x^2 + y^2 = 9\). When you draw a vertical line through this circle, you will often find that it intersects the circle at two points except at the edge. This tells us:

• A circle fails the vertical line test because each vertical line through its center crosses the circle twice.

Therefore, in mathematical terms, the graph of a circle is not a function. Recognizing the outcomes of the vertical line test is key to distinguishing between functions and non-functions.

Functions in Mathematics

Functions are a fundamental concept in mathematics, frequently used to describe relationships between two sets of numbers. By definition, a function assigns each element in a domain (input value \(x\)) to exactly one element in a range (output value \(y\)).

• A function is often described using an equation like \(y = f(x)\).
• Functions can be visualized as curves or lines on a graph where no vertical line intersects the graph more than once.

Returning to our original exercise, we attempt to determine if the graph of \(x^2 + y^2 = 9\) is a function using the vertical line test. Since this graph is a circle, and vertical lines intersect the circle more than once, it does not meet the criteria for a function.

• This means that while every function has a graph, not every graph is the graph of a function.

Understanding the definition and properties of functions allows for a deeper comprehension of how different mathematical graphs behave and relate to each other.