Problem 18 - Intermediate Algebra

Question

Sketch the graph of each equation. \(\frac{(x+3)^{2}}{16}+\frac{(y+2)^{2}}{4}=1\)

Step-by-Step Solution

Answer

Sketch an ellipse centered at \((-3, -2)\) with a major axis of 4 and a minor axis of 2.

Step 1: Recognizing the Equation Type

The given equation \(\frac{(x+3)^{2}}{16}+\frac{(y+2)^{2}}{4}=1\) is the standard form of an ellipse. In general form, an ellipse is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), where \((h, k)\) is the center and \(a\) and \(b\) are the semi-major and semi-minor axes respectively.

Step 2: Identifying Parameters of the Ellipse

Compare the equation \(\frac{(x+3)^{2}}{16}+\frac{(y+2)^{2}}{4}=1\) with the standard form to identify the parameters:- \(h = -3\), \(k = -2\), the center of the ellipse is \((-3, -2)\).- \(a^2 = 16\), thus \(a = 4\).- \(b^2 = 4\), thus \(b = 2\).This means the semi-major axis is 4 and the semi-minor axis is 2.

Step 3: Locating the Center

Plot the point \((-3, -2)\) on the coordinate plane, which is the center of the ellipse.

Step 4: Drawing the Axes

From the center \((-3, -2)\), move 4 units left and right to draw the horizontal major axis and 2 units up and down to draw the vertical minor axis.

Step 5: Sketching the Ellipse

Connect the ends of the axes with a smooth, oval-like curve to complete the sketch of the ellipse. The result should be an ellipse centered at \((-3, -2)\) with its major axis parallel to the x-axis and minor axis parallel to the y-axis.

Key Concepts

Equation of an Ellipse

An ellipse is a geometric shape that looks like an elongated circle. The standard equation for an ellipse with a horizontal major axis is \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \] where

• \((h, k)\) is the center of the ellipse
• \(a\) is the semi-major axis
• \(b\) is the semi-minor axis.

When comparing any given ellipse equation to its standard form, key parameters such as the center and the lengths of the axes can be identified.
In the equation \( \frac{(x+3)^{2}}{16} + \frac{(y+2)^{2}}{4} = 1 \), the center of the ellipse is located at \((-3, -2)\). Knowing how to interpret these parameters will help you draw the ellipse accurately on a graph.

Graphing Conic Sections

Graphing conic sections involves plotting shapes like ellipses, circles, parabolas, and hyperbolas. Each has a distinct equation form.
An ellipse is part of these conic sections and is characterized by its oval shape. Graphing an ellipse starts by identifying its center, semi-major axis, and semi-minor axis. Once these are known, you plot the center and draw the axes to guide the shape.
For our ellipse equation example, you begin by plotting the center point \((-3, -2)\). This forms the reference point from which you'll extend both axes: horizontal for the major and vertical for the minor.
Graphing requires careful measurement: move equal distances from the center, and smoothly connect these points to form the curve. This meticulous approach ensures an accurate representation of the ellipse, capturing its precise geometry.

Semi-major Axis

The semi-major axis is the longest radius of an ellipse, extending from the center to the farthest edge. It's crucial in defining the shape and size of the ellipse.
In the standard form equation \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), the value of \(a\) determines the length of this axis.
For the given equation \(\frac{(x+3)^{2}}{16} + \frac{(y+2)^{2}}{4} = 1 \), \(a^2 = 16\), which means \(a = 4\). This indicates that the semi-major axis extends 4 units from the center.

• Move 4 units left and right from the center point \((-3, -2)\), parallel to the x-axis.

This axis is critical in determining how stretched the ellipse appears horizontally. Always consider the semi-major axis first when sketching or analyzing an ellipse.

Semi-minor Axis

The semi-minor axis is the shorter radius of an ellipse, stretching from the center to the closest edge. Although shorter than the semi-major axis, it equally influences the shape's appearance.
In the ellipse's standard equation, the semi-minor axis is represented by \(b\). It is derived from \(b^2\), which is the denominator of the second term.
For the equation \( \frac{(x+3)^{2}}{16} + \frac{(y+2)^{2}}{4} = 1 \), \(b^2 = 4\), resulting in \(b = 2\). This indicates the semi-minor axis is 2 units long.
To visualize, from the center \((-3, -2)\), move 2 units up and 2 units down, parallel to the y-axis.

• Together with the semi-major axis, the semi-minor axis helps "round out" the ellipse.

Understanding these components allows for accurate sketching and analysis, providing insight into the ellipse's overall structure.