Problem 100 - A Graphical Approach to Precalculus with Limits

Question

Graph by hand the equation of the circle or the parabola with a horizontal axis. $$(x+1)^{2}+y^{2}=9$$

Step-by-Step Solution

Answer

The circle is centered at (-1, 0) with a radius of 3.

Step 1: Identify the Equation Type

The given equation \[(x+1)^2 + y^2 = 9\]is the equation of a circle. This can be deduced because it follows the general form of a circle's equation: \[(x-h)^2 + (y-k)^2 = r^2\]where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2: Find the Center of the Circle

By comparing \[(x+1)^2 + y^2 = 9\]with \[(x-h)^2 + (y-k)^2 = r^2\],we identify that:- \(h = -1\), \(k = 0\).Thus, the center of the circle is \((-1, 0)\).

Step 3: Determine the Radius

From the equation \[(x+1)^2 + y^2 = 9\],we see that this is equal to \[r^2 = 9\].Therefore, the radius \(r\) is the square root of \(9\), which gives us \[r = 3\].

Step 4: Draw the Circle

To graph the circle:1. Plot the center of the circle at \((-1, 0)\).2. From the center, measure and plot the radius \(r = 3\) units up, down, left, and right.3. Connect these points smoothly to form a circle.

Key Concepts

Graph by Hand

When graphing a circle by hand, start with identifying the essential elements given in the equation. In this case, the equation \((x+1)^2 + y^2 = 9\) clearly presents a circle. The key to drawing any circle accurately by hand is:

• Understanding the center of the circle, which guides where you place it on the graph, and
• The radius, which determines the size of the circle plotted.

Begin by marking the center point on your graph. Using a compass or any straight-edge tool, measure the radius from the center and make even marks in all cardinal directions on the grid.

Once you have these guidelines, connect the points with a smooth curve, ensuring to maintain a constant distance from the center to form a perfect circle.

Center of the Circle

The center of a circle in its equation is represented by the point \((h, k)\) in the standard form \((x-h)^2 + (y-k)^2 = r^2\). For this exercise, the equation is \((x+1)^2 + y^2 = 9\).

Let's break it down to find the center:

• Compare it to \((x-h)^2 + (y-k)^2 = r^2\).
• Here, \(x+1\) becomes \((x-(-1))\), so \(h = -1\).
• The \(y\) term does not have a \(k\), which means \(k = 0\).

Therefore, the center of the circle is \((-1, 0)\). Understanding this position is crucial because it anchors the circle on the graph paper knowing it’s your main reference point for constructing accurate points around to sketch the circle.

Radius Calculation

Determining the radius of the circle is a straightforward process once you've pinned down the equation's format. For the circle \((x+1)^2 + y^2 = 9\), the term on the right of the equality, \(9\), represents \(r^2\), which is the square of the radius.

Here's how you find the radius:

• The equation \((x+1)^2 + y^2 = 9\) is akin to \(r^2 = 9\).
• To find \(r\), solve for the square root of \(9\).
• The square root of \(9\) is \(3\), thus \(r = 3\).

Knowing the radius is vital as it defines the boundary of the circle from its center, ensuring that all graph points are equidistant from the center point \((-1, 0)\) when you draw it.