Problem 37 - Calculus with Concepts in Calculus

Question

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ x^{2}+y^{2}=4 \text { for } y \leq 0 $$

Step-by-Step Solution

Answer

The graph is the lower semi-circle and it is not a function.

Step 1: Recognize the Equation Type

Observe the equation \(x^2 + y^2 = 4\). This is the equation of a circle centered at the origin (0, 0) with a radius of 2.

Step 2: Apply the Condition for Hemisphere

Notice the condition \(y \leq 0\). This condition limits the graph to the lower half of the circle, or the lower semi-circle, since \(y\) includes only values less than or equal to 0.

Step 3: Sketch the Graph of the Lower Semi-Circle

Draw a circle centered at the origin with a radius of 2. Then, only keep the lower half of this circle, where \(y \leq 0\). This lower semi-circle is what will remain as our graph.

Step 4: Determine if the Graph is a Function

To determine if the graph is a function, use the Vertical Line Test. If any vertical line crosses the graph more than once, it is not a function. In this case, any vertical line through the circle from \(x = -2\) to \(x = 2\) will cross the lower semi-circle twice, confirming it is not a function.

Key Concepts

Circle Equations

Circle equations are mathematical representations of circles on a coordinate plane. The standard form of a circle's equation is \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle. A circle centered at the origin, or point (0, 0), is described by this equation. In this type of equation:

• \(x\) and \(y\) are the coordinates of any point on the circle.
• \(r\) is the radius, which determines the size of the circle.

This form allows you to easily identify the center of the circle (the origin) and its radius. In our original exercise, the equation \(x^2 + y^2 = 4\) indicates a circle with a radius of 2 centered at the origin. This sets the stage for sketching the graph and applying other necessary tests to the equation.

Vertical Line Test

When determining whether a graph represents a function, the vertical line test is a straightforward method to apply. It involves:

• Imagining drawing vertical lines (parallel to the y-axis) across the graph.
• Observing how many times these lines intersect the graph.

A graph is a function if and only if no vertical line intersects the graph more than once. This is because functions have only one output \(y\) for every input \(x\). In the case of our semi-circle graph from the exercise, any vertical line in the range \(-2 \leq x \leq 2\) will intersect the semi-circle twice. This definitively shows that the graph is not a function, as it fails the vertical line test.

Semi-Circle Graph

A semi-circle graph arises when an additional condition limits half of the circle from being part of the graph. In our exercise, the condition \(y \leq 0\) specifies we only depict the lower part of the circle. Here’s how it works:

• The original circle, \(x^2 + y^2 = 4\), spans the entire xy-plane.
• The condition \(y \leq 0\) restricts us to only include the lower half, resulting in a semi-circle.

To draw it, first, sketch the full circle centered at the origin with a radius of 2. Then, erase or ignore the top half (where \(y > 0\)). This leaves a shape that resembles a half-bowl sitting along the x-axis and serves as the desired semi-circle graph.