Problem 28 - Algebra and Trigonometry

Question

Graph the circle \((y-1)^{2}+x^{2}=1\) by solving for \(y\) and graphing two equations as in Example \(3 .\)

Step-by-Step Solution

Answer

Graph \(y = 1 \pm \sqrt{1 - x^2}\) for \(-1 \leq x \leq 1\).

Step 1: Identify the Circle Equation

The given equation is \((y - 1)^2 + x^2 = 1\). This represents a circle centered at \((0, 1)\) with a radius of 1.

Step 2: Solve for y

To express the equation in terms of \(y\), isolate \((y-1)^2\):1. Rewrite the equation as \((y - 1)^2 = 1 - x^2\).2. Take the square root of both sides to get \(y - 1 = \pm \sqrt{1 - x^2}\).3. Solve for \(y\) by adding 1: \(y = 1 \pm \sqrt{1 - x^2}\).

Step 3: Determine Domain of x

Since \(\sqrt{1 - x^2}\) is defined when the expression inside the square root is non-negative, \(1 - x^2 \geq 0\). This gives the domain: \(-1 \leq x \leq 1\).

Step 4: Graph the Equations

Graph the two equations solved from Step 2:1. \(y = 1 + \sqrt{1 - x^2}\) represents the upper semi-circle.2. \(y = 1 - \sqrt{1 - x^2}\) represents the lower semi-circle.Consider the domain \(-1 \leq x \leq 1\) for both functions and plot them to form the complete circle.

Key Concepts

Circle Equation

The equation \( (y - 1)^2 + x^2 = 1 \) is a classic example of a circle equation. A circle in mathematics is defined as the set of all points in a plane that are equidistant from a fixed point called the center. The general form of a circle equation is \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center of the circle and \( r \) is the radius.
This particular equation shows a circle centered at \( (0, 1) \), since the equation compares to the form \( (0 - x)^2 + (y - 1)^2 = 1^2 \).
The radius, \( r \), is the square root of the constant on the right-hand side, which is 1. Hence, the radius is also 1.
This tells us that every point on the circle is 1 unit away from the center \( (0, 1) \). Understanding this form is crucial when graphing circles, as it allows us to quickly identify the center and radius.

Solving for y

In order to graph the circle described by the equation, solving for \( y \) requires rewriting the equation in terms of \( y \).
First, isolate \( (y - 1)^2 \) by moving \( x^2 \) to the other side of the equation:

\( (y - 1)^2 = 1 - x^2 \).
Next, take the square root of both sides to eliminate the square on \( y - 1 \):

• \( y - 1 = \sqrt{1 - x^2} \) and \( y - 1 = -\sqrt{1 - x^2} \).

Now solve for \( y \) by adding 1 to each part:

• \( y = 1 + \sqrt{1 - x^2} \)
• \( y = 1 - \sqrt{1 - x^2} \)

These equations represent the top and bottom halves of the circle (semi-circles). The "+" solution represents the upper half, and the "−" solution represents the lower half.

Domain of x

The domain of \( x \) is critical when graphing, as it shows the interval of \( x \) values that keep the circle equation defined and real.
For this particular equation, consider the expression inside the square root, \( \sqrt{1 - x^2} \).
The square root is only defined for non-negative numbers, so you set the expression under the square root to be greater than or equal to zero:

• \( 1 - x^2 \geq 0 \)

This inequality can be transformed to find the range of \( x \):

• \( x^2 \leq 1 \)

This means that \( -1 \leq x \leq 1 \). So, the domain for \( x \) in this circle is \( x \) values between -1 and 1, inclusive.
This domain ensures the circle is graphed correctly and remains a real, visual representation of the equation.

Semi-Circle

When the circle equation \( (y - 1)^2 + x^2 = 1 \) is broken into two separate equations of \( y \), it results in the equation describing two semi-circles.

• The equation \( y = 1 + \sqrt{1 - x^2} \) describes the upper semi-circle.
• The equation \( y = 1 - \sqrt{1 - x^2} \) describes the lower semi-circle.

Each of these semi-circles half the original circle, which allows graphers to ensure both the top and bottom are included.
This also simplifies the process of dividing and visualizing the circle.
By graphing each equation separately within the domain of \( x \), \( -1 \leq x \leq 1 \), you will see the familiar round shape of a circle materialize in two halves.
Remembering that each half is a separate equation allows graphers flexibility and accuracy when rendering circle graphs.