Problem 98 - A Graphical Approach to College Algebra

Question

In Exercises \(97-108,\) graph by hand the equation of the circle or the parabola with a horizontal axis. $$x^{2}+y^{2}=25$$

Step-by-Step Solution

Answer

The graph is a circle centered at the origin with radius 5.

Step 1: Identify the Equation Type

The given equation is \(x^2 + y^2 = 25\). This is the standard form of a circle's equation, \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle.

Step 2: Find the Radius

To find the radius, compare the equation \(x^2 + y^2 = 25\) with \(x^2 + y^2 = r^2\). Here, \(r^2 = 25\), so the radius \(r\) is \(\sqrt{25} = 5\).

Step 3: Find the Center

The standard form \(x^2 + y^2 = r^2\) indicates that the circle is centered at the origin \((0, 0)\) because there are no \(x\) or \(y\) terms present to move the center.

Step 4: Plot the Circle on the Graph

Draw a graph with the x-axis and y-axis. Mark the center of the circle at the origin \((0, 0)\). From the origin, mark points 5 units away in all directions: up, down, left, and right. Connect these points in a smooth circular arc.

Key Concepts

Radius Calculation

When dealing with the equation of a circle, one of the primary tasks is to determine the radius. The standard form of a circle's equation is given as \(x^2 + y^2 = r^2\), where \(r\) symbolizes the radius.
The equation you are given might look a bit like a puzzle initially, but the key is to identify its structure. In our case, we have \(x^2 + y^2 = 25\). Recognize that this matches the format \(x^2 + y^2 = r^2\), but instead of an unknown \(r\), we have 25. This means we equate \(r^2\) to 25.
To find \(r\), you simply calculate the square root of 25, which gives us \(r = 5\).
Now you understand that the circle we are dealing with has a radius of 5 units. The radius directly influences how large the circle will appear when graphed. In the context of graphing, it indicates how far from the center in any direction the circle extends.

Origin Center

Understanding the center of a circle is critical when graphing. In the standard form equation \(x^2 + y^2 = r^2\), if there are no terms such as \(x - h\) or \(y - k\), then the circle's center is at the origin, which is \((0, 0)\).
It's important to remember:

• If your equation adds terms like \((x - h)^2 + (y - k)^2 = r^2\), then the center shifts to \((h, k)\).
• In our current scenario of \(x^2 + y^2 = 25\), the absence of additional \(h\) or \(k\) in the equation solidly places the circle's center at (0, 0).

Knowing the center is at the origin greatly simplifies the drawing and visualization of the circle, as you can use the graph's axes as natural guides. Grasping this concept supports accurate graph representations and deeper comprehension of circle equations.

Equation Identification

Identifying the type of equation you're facing paves the way for a smoother graphing process. The equation \(x^2 + y^2 = 25\) must be recognized as that of a circle. This recognition stems from the equation's distinct similarity to \(x^2 + y^2 = r^2\), the classic circle equation form.
Here's how to effectively recognize circle equations:

• Look for both \(x^2\) and \(y^2\) terms, each with a coefficient of 1.
• Ensure the equation equates to a constant, indicating the square of the radius.

These hints solidify an equation's identity as a circle.
Understanding the nature of the equation before graphing can save time and prevent errors. It enables you to prepare appropriately, such as determining the radius and center intuitively, ensuring you are ready to draw accurate and correctly scaled graphs.