Problem 36 - Calculus with Concepts in Calculus

Question

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ x^{2}+y^{2}=9 \text { for } x \geq 0 $$

Step-by-Step Solution

Answer

The graph depicts a semicircle (radius 3, centered at the origin) on the right side, which is not a function.

Step 1: Identify the shape of the graph

The equation given is \(x^2 + y^2 = 9\). This is the equation of a circle centered at the origin with a radius of 3.

Step 2: Apply the constraint for \(x\)

The constraint \(x \geq 0\) indicates that only the right half of the circle (including the boundary along the y-axis) should be considered. Thus, the circle is only plotted for \(x \geq 0\).

Step 3: Sketch the graph

To sketch the graph, draw a semicircle of radius 3 centered at the origin that spans the positive x-axis and both the positive and negative y-axes. This semicircle represents the portion of the circle for \(x \geq 0\).

Step 4: Determine if it is a function

A graph represents a function if each \(x\) value corresponds to exactly one \(y\) value. However, for this semicircle, there are values of \(x\) where both positive and negative \(y\) values satisfy the equation. Thus, this graph is not a function.

Key Concepts

Graphing Circles

When it comes to graphing circles, the process starts with understanding the equation that represents the circle. The standard form of a circle's equation is \[(x - h)^2 + (y - k)^2 = r^2 \]where

• \( (h, k) \) is the center of the circle
• \( r \) is the radius of the circle.

In our exercise, the equation is \[x^2 + y^2 = 9\]which simplifies to show a circle centered at the origin \( (0, 0) \) with a radius of \( 3 \) because \( r^2 = 9 \) implies \( r = 3 \). Drawing this circle involves plotting all the points that are exactly 3 units away from the origin, forming a perfect round edge all around.
However, given the constraint \( x \geq 0 \), we only plot the right half of this circle, which is called a semicircle.

Functions and Relations

To better grasp functions and relations, it's important to see the key difference:

• A function is a special type of relation where each input (usually represented by \( x \)) maps to exactly one output (\( y \)).
• A relation is more general - it's a set of ordered pairs without the restriction of mapping one input to just one output.

In the exercise, the relation is represented by the semicircle \( x^2 + y^2 = 9 \); however, this relation fails to qualify as a function because, for some \( x \) values like \( x = 0 \), there are two possible \( y \) values (both positive and negative \( 3 \)).
Thus, while the equation suggests a relation, it does not fulfill the criteria to be a function.

Equation of a Circle

Understanding the equation of a circle is essential when engaging in problems involving the geometric figure. The basic form is \[(x - h)^2 + (y - k)^2 = r^2\]and it precisely defines a circle in terms of its center and radius.

• Here, \( (h, k) \) indicates the center's coordinates.
• \( r \) is the radius.

In our specific case, the circle is centered at the origin \( (0,0) \) with a radius \( 3 \), derived directly from \( x^2 + y^2 = 3^2 \).
Such equations are instrumental in drawing circles graphically, furnishing a clear method to map points equidistant from the center point.

Determining Functions from Graphs

Determining whether a graph represents a function involves a handy test called the vertical line test. This test helps quickly assess the function status:

• If any vertical line drawn through the graph intersects it at more than one point, the graph is not a function.
• If a vertical line only crosses the graph once no matter where it's drawn, then it represents a valid function.

In the given exercise, using this test on the semicircle graph from the equation \[x^2 + y^2 = 9 \]makes clear that there are places where a vertical line will touch the graph twice. This confirms that the graph is indeed not a function.
Such an evaluation is crucial as it informs us about how inputs relate to outputs and the uniqueness of this correspondence in mathematical functions.