Problem 28 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

Graph the equations \(y = 1 + \sqrt{1-x^2}\) and \(y = 1 - \sqrt{1-x^2}\).

Step 1: Identify the Circle Equation

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

Step 2: Solve for y

We need to solve for \(y\) in terms of \(x\). Start by isolating \((y-1)^2\) by subtracting \(x^2\) from both sides: \((y-1)^2 = 1 - x^2\).

Step 3: Take the Square Root

To solve for \(y\), take the square root of both sides: \(y - 1 = \pm\sqrt{1-x^2}\).

Step 4: Isolate y

Add \(1\) to both sides to isolate \(y\): \(y = 1 \pm \sqrt{1-x^2}\). This gives us two equations: \(y_1 = 1 + \sqrt{1-x^2}\) and \(y_2 = 1 - \sqrt{1-x^2}\).

Step 5: Graph the Equations

Graph the equations \(y_1 = 1 + \sqrt{1-x^2}\) and \(y_2 = 1 - \sqrt{1-x^2}\). These two graphs together form the circle originally given by the equation \((y-1)^2 + x^2 = 1\).

Key Concepts

Equation of a Circle

The equation of a circle is a mathematical representation that defines all the points that form a circle on a plane. A standard form of a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\), where \(h, k\) is the center of the circle and \r\ is the radius. In this case, the equation \( (y-1)^2 + x^2 = 1\) specifies a circle with the center at \(0, 1\) and a radius of \1\. Knowing how to identify these components in the equation is important:

• The term \(x^2\) means the circle's center is along the vertical line at \x=0\.
• The expression \((y-1)^2\) means the center is one unit up from the origin, at \y=1\.
• The \1\ on the right-hand side represents the squared radius, so the radius is \sqrt{1} = 1\.

Understanding the equation helps you to quickly find these crucial properties of the circle.

Solving for y

To graph the given circle equation, we first need to solve for \(y\). This means rearranging the equation so that \(y\) is isolated. Starting with \( (y-1)^2 + x^2 = 1\), we subtract \(x^2\) from both sides: \( (y-1)^2 = 1 - x^2\). This step effectively isolates the \( (y-1)^2\) term. Once we have reached this point, our next step is to simplify further by taking the square root of both sides. The solution involves two parts:

• Subtract \(x^2\) from \1\ to isolate \( (y-1)^2\).
• Remember that solving a squared equation by taking the square root introduces two potential solutions, as the square root can have both a positive and negative value.

Thus, the resulting expressions are: \( y-1 = \pm\sqrt{1-x^2}\). Solving for \(y\) further requires adding \(1\) to both sides of the equation.

Square Root Method

The square root method is a useful way to solve equations where a variable is squared, as it helps to find all possible values the variable can take. In our circle equation, after isolating \( (y-1)^2\), we apply this method to solve for \(y\). By taking the square root of both sides, we determine:

• The expression becomes \( y - 1 = \pm\sqrt{1 - x^2}\), highlighting that there are two values for \(y\) for each value of \(x\).
• Adding \(1\) to both sides finally gives us \(y = 1 + \sqrt{1-x^2}\) and \(yy = 1 - \sqrt{1-x^2}\).

These two equations represent the upper and lower halves of the circle. This method ensures that we have accounted for all possible values and graphing both expressions together will accurately display the entire circle.

Graphing Techniques

Graphing the identified equations \(y_1 = 1 + \sqrt{1-x^2}\) and \(y_2 = 1 - \sqrt{1-x^2}\) requires careful application of your graphing skills. Here’s how you can effectively plot them to form a circle:

• Start by setting up a coordinate plane, ensuring that you have a clear scale and both positive and negative axes.
• Plot a few points for \(y_1\) where \(x\) is from \(-1\) to \(1\), completing the top half of the circle by calculating corresponding \(y\) values using \(y_1 = 1 + \sqrt{1-x^2}\).
• Do the same for \(y_2\), which is \(y = 1 - \sqrt{1-x^2}\), to identify the lower half of the circle.
• Connect these points smoothly to see the perfect circular shape.

This technique not only brings the mathematical expression to life on a graph but also strengthens your understanding of how algebraic formulas translate into geometric figures. Remember, precision in plotting and connecting points will yield the most accurate representation of the circle.