Problem 56 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve for \(y\) and use a graphing utility to graph each of the resulting equations in the same viewing window. (Adjust the viewing window so that the circle appears circular.) \(x^{2}+y^{2}=36\)

Step-by-Step Solution

Answer

The solution for \(y\) in the equation \(x^{2} + y^{2} = 36\) results in two equations: \(y = \sqrt{36 - x^{2}}\) and \(y = -\sqrt{36 - x^{2}}\). These equations, when graphed, form a circle with radius 6 units.

Step 1: Isolate \(y\)

The given equation is \(x^{2} + y^{2} = 36\). In order to isolate \(y\), subtract \(x^{2}\) from both sides of the equation to get \(y^{2} = 36 - x^{2}\).

Step 2: Solve for \(y\)

Taking the square root of both sides to solve for \(y\), this equation actually yields two solutions: \(y = \sqrt{36 - x^{2}}\) and \(y = -\sqrt{36 - x^{2}}\), because the square root of a number is always positive and negative. In geometrical terms, these two solutions represent the upper and lower halves of the circle.

Step 3: Graph the equations

Using a graphing utility, graph both equations (\(y = \sqrt{36 - x^{2}}\) and \(y = -\sqrt{36 - x^{2}}\)) in the same viewing window. Make sure to adjust the window so that the graph appears as a circle rather than an ellipse due to scaling issues. Each equation will create a semi-circle, and together they will form a full circle.

Key Concepts

Graphing Utility

A graphing utility is a helpful tool that allows you to draw mathematical graphs quickly and accurately. It can be a software program, an app, or a physical device like a graphing calculator. These tools are particularly useful for visualizing complex equations, like those of circles or other geometric shapes.

When tasked with graphing equations such as ones derived from a circle equation, a graphing utility simplifies the task by automating the plotting process. Here are a few steps in using a graphing utility:

• Input the equation: Enter the mathematical expressions into the graphing utility. This can typically be done by typing the equation directly.
• Set the viewing window: Adjust the scale and range for both the x-axis and y-axis so that the entire graph is visible and correctly proportioned. This is crucial when graphing circles since incorrect scaling can make circles appear elongated.
• Analyze the graph: Once the equations are entered and the viewing window set, the utility will display the graph. Check if the circle appears as it should.

A graphing utility is ideal for handling equations with multiple solutions, as seen in the circle equation. It allows students to see both the positive and negative solutions with ease, helping them understand the symmetry and shape of circles.

Solving for y

In the equation of a circle, like the one given: \(x^2 + y^2 = 36\), solving for \(y\) involves isolating the variable. This is a common algebraic technique used to rearrange an equation to find the value of one variable in terms of others.

When solving for \(y\), follow these steps:

• Move other terms: Subtract \(x^2\) from both sides to focus on \(y\).
• Resulting equation: You end up with \(y^2 = 36 - x^2\).
• Square root: Take the square root of both sides to solve for \(y\).

This process gives two expressions: \(y = \sqrt{36 - x^2}\) and \(y = -\sqrt{36 - x^2}\). Both equations represent valid solutions in the graph, corresponding to different parts of the circle. Solving for \(y\) in this manner is critical in separating the upper and lower segments of a circle.

Square Root

Understanding square roots is essential when solving equations involving squares like \(y^2 = 36 - x^2\). A square root is a number that, when multiplied by itself, gives the original number.

To find the square root:

• Recognize solutions: Square roots produce both positive and negative solutions. This results in \(y = \sqrt{36 - x^2}\) and \(y = -\sqrt{36 - x^2}\).
• Include all solutions: Both square roots cover different halves of the full circle, the positive for the top half, the negative for the bottom half.

Visualizing with square roots helps to understand how parts of a graph are connected. It's important to remember that the sign before the square root can affect which part of a circle is shown on a graph.

Graphing a Circle

Circles are unique shapes defined by their radius and center. In the equation \(x^2 + y^2 = 36\), 36 is the square of the radius. To properly graph a circle, ensure both components \(x^2\) and \(y^2\) provide full symmetry.

Graphing a circle involves:

• Understanding the equation: Recognize \(x^2 + y^2 = r^2\) format, where \(r\) is the circle's radius.
• Using both equations: Employ both \(y = \sqrt{36 - x^2}\) and \(y = -\sqrt{36 - x^2}\) for full circular coverage.
• Adjusting the plot: Set the graph to ensure a circular appearance, as any discrepancy can look like an ellipse due to aspect ratio issues.

Making these adjustments provides a true representation of the geometric properties of a circle, making it a fundamental exercise in graphing equations.