Problem 51 - Intermediate Algebra

Question

Use a graphing calculator to graph each equation. $$ x^{2}+y^{2}=7 $$

Step-by-Step Solution

Answer

Graph the circle centered at (0,0) with a radius of approximately 2.65.

Step 1: Identify the Equation Type

The given equation is in the form of \[ x^2 + y^2 = r^2 \]which is the standard form of a circle centered at the origin \((0,0)\) with radius \(r\).

Step 2: Calculate the Radius

From the equation \[ x^2 + y^2 = 7 \]we see that\( r^2 = 7 \).Thus, the radius \( r \) is \( \sqrt{7} \approx 2.65 \).

Step 3: Input Equation into Graphing Calculator

Access the graphing function on your graphing calculator. Input the equation \[ x^2 + y^2 = 7 \]which represents a circle.

Step 4: Graph the Equation

Plot the equation on the graphing calculator. You should see a circle centered at the origin \((0,0)\) with a radius of approximately 2.65.

Key Concepts

Equation of a Circle

When you're dealing with the equation of a circle, it's important to recognize its specific form. A circle's equation is typically written as \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) represents the center of the circle and \(r\) is the circle's radius.
In the special case where the circle is centered at the origin, which means \((h, k) = (0, 0)\), the equation simplifies to \(x^2 + y^2 = r^2\).
This simplified form is what we have here: \(x^2 + y^2 = 7\).

• The expression \(x^2\) involves the horizontal distance from the center.
• The expression \(y^2\) involves the vertical distance from the center.

The sum of these squared terms equals \(r^2\), the squared radius of the circle. This concise format helps us easily identify the circle's properties and graph it using tools like graphing calculators.

Radius Calculation

Understanding how to calculate the radius from the circle's equation is crucial.
To find the radius, look at \(r^2\) in the equation \(x^2 + y^2 = r^2\).
From \(x^2 + y^2 = 7\), we can see that \(r^2 = 7\).

To find \(r\), simply take the square root of both sides ---- \(r = \sqrt{7}\).
This process reveals that the radius of our circle is \(\sqrt{7} \approx 2.65\).

• The square root transformation is key to translating \(r^2\) into \(r\).
• It's important to note that the radius is always a positive value as it represents a distance.

Calculating the radius accurately allows you to graph the circle correctly and interpret its size compared to other circles or geometric figures.

Standard Form

The standard form of a circle's equation is a powerful tool in geometry and algebra.
It provides a straightforward way to understand and graph circles.
For example, the standard form \( (x - h)^2 + (y - k)^2 = r^2 \) directly tells us about the circle's center \((h, k)\) and radius \(r\).

• If \(h = 0\) and \(k = 0\), as in \(x^2 + y^2 = 7\), the circle is centered at the origin.
• The equation \((x - h)^2 + (y - k)^2 = r^2\) provides all the necessary information to plot the circle on a graph.

This form highlights the relationship between algebra and geometry, simplifying how we visualize and analyze circle-related problems.
Understanding the standard form also makes it easier to translate between different mathematical representations, such as shifting a circle from one location to another on a coordinate plane.