Problem 65 - Algebra 2

Question

Graph each pair of equations. Identify the conic section represented by the graph. Then write the equation of the conic section. $$ \begin{array}{l}{y=3+\sqrt{16-(x-4)^{2}}} \\\ {y=3-\sqrt{16-(x-4)^{2}}}\end{array} $$

Step-by-Step Solution

Answer

The given equations represent a circle with its center at (4, 3) and a radius of 4. The equation of the conic section is \((x - 4)^2 + (y - 3)^2 = 16\)

Step 1: Plotting the Equations

The first thing to note is that there are two equations: one is the upper half and the other is the lower half of the plot. Graphing these two equations, produces a shape that is symmetric about the line \(y = 3\). Specifically, for any value of \(x\), the value of \(y\) is either \(3 + \sqrt{16 - (x - 4)^2}\) or \(3 - \sqrt{16 - (x - 4)^2}\). This implies that for any given \(x\), y extends \(\sqrt{16 - (x - 4)^2}\) above and below 3. This pattern suggests a circle or an ellipse.

Step 2: Identify the Conic Section

To identify whether the conic section is a circle or an ellipse, we need to observe the shape of the graph. Given the roots and how the \(y\) extends above and below \(3\), we find that it forms a circle.

Step 3: Write the Equation of the Conic Section

To write the equation of this conic section, we first need to know the general form of a circle's equation, which is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) are the center of the circle and \(r\) is the radius. Substituting our values, we get \((x - 4)^2 + (y - 3)^2 = 16\)

Key Concepts

Circle Equations

Understanding circle equations is key to graphing them correctly. A circle is a set of points in a plane that are a fixed distance, called the radius, from a given point known as the center. The standard form of a circle's equation is

• \((x - h)^2 + (y - k)^2 = r^2\)

where

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

The equation given in the original exercise, \((x - 4)^2 + (y - 3)^2 = 16\), clearly follows this standard form. Here the center of the circle is at \((4, 3)\), and the radius is 4, since \(16 = 4^2\). Recognizing this pattern allows us to quickly identify and write down the equation of a circle from its graph.

Graphing Equations

Graphing equations is a vital skill for visualizing mathematical relationships. Let's focus on how to graph a circle from its equation. Using the equation \((x - 4)^2 + (y - 3)^2 = 16\), follow these steps:

• First, identify the center of the circle from the equation. Here, it's \((4, 3)\).
• Next, determine the radius. With \(r^2 = 16\), the radius \(r = 4\).
• Plot the center on a graph. From this point, mark four equally spaced points 4 units away at the top, bottom, left, and right. These will give you initial points of reference for the circle.
• Then, using these points, sketch a smooth curve to form a complete circle. Ensure that all parts of your circle are equidistant from the center.

Graphing conic sections like circles helps in visualizing and solving problems involving these equations.

Symmetry in Graphs

Symmetry is a useful property for graphing and understanding equations of conic sections. When a graph is symmetric, it looks the same on both sides of a given line or point.
In this exercise, the two equations

• \(y = 3 + \sqrt{16 - (x-4)^2}\)
• \(y = 3 - \sqrt{16 - (x-4)^2}\)

create a graph that is symmetric about the line \(y = 3\). This horizontal line serves as an axis of symmetry.
Why is symmetry important? It allows us to predict missing points on a graph. If you know half the circle, symmetry helps you complete the other half without recalculating more points.
Thus, symmetry not only simplifies graphing but also aids in recognizing the nature of conic sections, leading directly to identifying the circle here.