Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of the given equation. $$ \frac{(x+3)^{2}}{4}+\frac{(y-2)^{2}}{8}=0 $$

Step-by-Step Solution

Answer

The graph of the equation is a single point at \((-3, 2)\).

Step 1: Understand the Equation

Identify the type of equation given by \( \frac{(x+3)^2}{4} + \frac{(y-2)^2}{8} = 0 \). This form displays characteristics of an ellipse equation if the sum on the right-hand side was 1. However, since the sum is 0, set each term individually to zero.

Step 2: Set Each Component to Zero

We have \( \frac{(x+3)^2}{4} = 0 \) and \( \frac{(y-2)^2}{8} = 0 \). Solving \( (x+3)^2 = 0 \) gives \( x = -3 \), and solving \( (y-2)^2 = 0 \) gives \( y = 2 \).

Step 3: Interpret the Solution

Since both \( (x+3)^2 = 0 \) and \( (y-2)^2 = 0 \) must simultaneously hold true, the only solution to the equation is the point \( (x, y) = (-3, 2) \). This means the graph of the equation is a single point.

Step 4: Sketch the Graph

Draw the Cartesian plane. Plot the point \( (-3, 2) \) on this plane. Since this is the only solution, the "graph" is simply this single point.

Key Concepts

Ellipse Equation

An ellipse equation typically resembles the form \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\] Here,

• \( h \) and \( k \) are the coordinates of the center of the ellipse.
• \( a \) is the semi-major axis's length, and \( b \) is the semi-minor axis's length.

This equation is an ellipse when the right-hand side equals 1. However, if we take a look at the given exercise, the equation is \( \frac{(x+3)^2}{4} + \frac{(y-2)^2}{8} = 0 \). Since the equation does not equal 1, it doesn't describe an ellipse in the usual sense. Instead, it implies something different.When each term in the equation equals zero individually (because their sums amount to zero), it simply means there are no additional terms contributing to an ellipse's overall shape. Hence, instead of depicting an extended ellipse with a discernible shape, the solution is pinpointed to the specific point, identifying a condition where both components of the sum are at absolute minimum—and are zero.

Cartesian Plane

The Cartesian plane is a two-dimensional graphing system that allows us to visualize equations like the one in this exercise. It consists of two perpendicular axes—the x-axis (horizontal) and the y-axis (vertical).

• Points on this plane are identified through coordinates \((x, y)\).
• Each axis is a number line extending indefinitely in both directions.

To plot any point, such as the point we derived in the solution \((-3, 2)\), you simply move left or right along the x-axis to the point \(-3\) and then vertically to \(2\) on the y-axis.
The point where these two values meet is where you mark your point. In the case of our exercise, since the graph isn't continuous (like a line or a standard ellipse), this single point is the sole representation, and no further points are plotted.

Solution Interpretation

When interpreting solutions on a Cartesian plane, it's essential to understand what each part of the equation signifies. The given exercise led to the solution \((x, y) = (-3, 2)\), meaning that this point is where both components of the equation reached zero simultaneously.

• This interpretation reveals that despite starting with an equation resembling an ellipse, the actual solution tells a different story.
• In this context, the equation doesn't represent a filled-out shape but instead condenses into a singular location in space.

This exercise's unique outcome demonstrates that even if equations look like they belong to a particular family (like ellipses), their graphical representation can differ widely. Understanding these differences is crucial for accurately depicting and interpreting mathematical expressions in geometry.