Problem 3 - Algebra 2

Question

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. $$ x^{2}+y^{2}=x+2 $$

Step-by-Step Solution

Answer

The equation is a circle centered at \((\frac{1}{2}, 0)\) with radius \(\frac{3}{2}\).

Step 1: Identify the Equation Type

The given equation is \(x^2 + y^2 = x + 2\). Notice that both \(x^2\) and \(y^2\) terms have the same coefficient, which indicates it is a circle.

Step 2: Move Linear Terms

To convert the equation into the standard form of a circle, move \(x\) and the constant 2 to the left side: \(x^2 + y^2 - x = 2\).

Step 3: Complete the Square for x

To complete the square, we need to transform \(x^2 - x\). Add and subtract \(\left( \frac{1}{2} \right)^2 = \frac{1}{4}\) inside the equation: \((x^2 - x + \frac{1}{4}) + y^2 = 2 + \frac{1}{4}\).

Step 4: Rewrite Completed Square

Rewrite \(x^2 - x + \frac{1}{4}\) as \((x - \frac{1}{2})^2\). The equation now is \((x - \frac{1}{2})^2 + y^2 = \frac{9}{4}\).

Step 5: Recognize the Standard Form

The equation \((x - h)^2 + (y - k)^2 = r^2\) represents a circle, where \(h\) and \(k\) are the circle's center, and \(r\) is the radius. In this case, center \((\frac{1}{2}, 0)\) and radius \(\frac{3}{2}\).

Step 6: State the Graph Type and Draw

The graph is a circle centered at \((\frac{1}{2}, 0)\) with a radius of \(\frac{3}{2}\). Plot this circle on a coordinate plane by marking the center and outlining the circle using the radius.

Key Concepts

Equation of a Circle

In geometry, understanding the equation of a circle is crucial. A standard circle equation in the Cartesian plane has the form \((x - h)^2 + (y - k)^2 = r^2\). Here,

• \((h, k)\) represents the circle's center coordinates.
• \(r\) is the radius of the circle.

This standard form is tidy and makes it easy to identify the circle's key features just by looking at it.
To convert any circle equation into this format, you need to correct each variable's expression by completing the square. This ensures the equation clearly details the center and radius.

Completing the Square

Completing the square is a method used to simplify equations, especially those of circles and parabolas. It involves a few algebraic steps:

• Arrange all terms on one side of the equation.
• Identify and group the terms to complete the square, usually in expressions involving a quadratic term like \(x^2\).
• For an expression like \(x^2 - x\), add and subtract \(\left(\frac{1}{2}\right)^2\) to create a perfect square trinomial \((x - \frac{1}{2})^2\).

This method refines the equation, so it aligns with the standard form of a circle, displaying the center and radius neatly.
It's especially handy for manipulating circles in problems where graphical representation is needed.

Graphing Equations

Graphing equations allows us to visually interpret what an equation represents. For circles, the following steps are key:

• Identify the center of the circle from the equation, \((h, k)\), through its standard form \((x - h)^2 + (y - k)^2 = r^2\).
• Determine the radius \(r\), which is the distance from the center to any point on the circle.
• On a coordinate graph, mark the center, then use the radius to plot points around the center.
• Finally, draw a smooth curve connecting these points to form a circle.

Graphing is a tangible way to see the relationship represented by the equation, making abstract math concepts easier to grasp.
It's an essential skill often required in geometry to thoroughly understand the properties of geometric shapes.

Geometric Shapes

Geometric shapes are the backbone of geometry and understanding their algebraic representation is crucial. Common shapes include:

• Circle: Defined by \((x - h)^2 + (y - k)^2 = r^2\).
• Parabolas: Often expressed as \(y = ax^2 + bx + c\) or \(x = ay^2 + by + c\).
• Ellipses: Formed by \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\).
• Hyperbolas: With the equation \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\).

Recognizing these forms and translating them into standard form helps in visualizing them through graphs.
Each shape has unique properties and characteristics, which influence how equations are transformed and interpreted in geometric terms.