Problem 31 - Intermediate Algebra

Question

Graph each equation. \(16(x-2)^{2}+4(y+4)^{2}=256\)

Step-by-Step Solution

Answer

The ellipse is centered at (2, -4) with a semi-major axis of 8 along the y-axis and a semi-minor axis of 4 along the x-axis.

Step 1: Identify the Equation Form

The equation \(16(x-2)^{2}+4(y+4)^{2}=256\) represents an ellipse in its standard form, which is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\). We need to rewrite the given equation to match this form.

Step 2: Divide by the Constant

First, divide the entire equation by 256 to rewrite it in the standard form of an ellipse: \[ \frac{16(x-2)^2}{256} + \frac{4(y+4)^2}{256} = \frac{256}{256} \]This simplifies to:\[ \frac{(x-2)^2}{16} + \frac{(y+4)^2}{64} = 1 \]

Step 3: Identify Center and Lengths of Axes

The rewritten equation \( \frac{(x-2)^2}{16} + \frac{(y+4)^2}{64} = 1 \) represents an ellipse centered at \((h, k) = (2, -4)\). The denominators beneath \((x-2)^2\) and \((y+4)^2\) represent \(a^2 = 16\) and \(b^2 = 64\). Therefore, \(a = 4\) and \(b = 8\), indicating the semi-major axis is 8 (along the y-axis) and the semi-minor axis is 4 (along the x-axis).

Step 4: Graph the Ellipse

To graph the ellipse, start by plotting its center at \((2, -4)\). From the center, move 4 units left and right for the semi-minor axis along the x-axis, and 8 units up and down for the semi-major axis along the y-axis. This will give the boundary of the ellipse.

Key Concepts

Graphing Equations

Graphing equations, especially for ellipses, can be a fun way to visualize mathematical concepts. When graphing an ellipse, it's crucial to translate the algebraic form of the equation into a visual format. Start by determining the center of the ellipse. For our example, this is located at (2, -4). Next, identify the lengths of the semi-major and semi-minor axes to determine the width and height of the ellipse.

Once you have this information, you can draw the ellipse on a coordinate plane. Begin by plotting the center, then use the axes lengths to pinpoint the major bounds of the ellipse. These points will help in sketching the elliptical shape.

Remember, the ellipse is symmetric about its center, so ensure to balance your points on both sides along each axis.

Ellipse Standard Form

The standard form of an ellipse equation is an application of the concept of conic sections. It shows the relation between the x and y coordinates to form an ellipse. The typical standard form is:

• \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]

Breaking Down the Equation

In this equation:

• \((x-h)^2\) and \((y-k)^2\) are squared terms showing distances from the center \((h, k)\).
• \(a^2\) and \(b^2\) are the squares of the semi-major and semi-minor axes lengths.
• The value 1 on the right signifies that the terms on the left are normalized, a necessary form for straightforward graphing.

Having the equation in this format simplifies identifying the ellipse's center and axes lengths.

Axes Lengths

Understanding the axes lengths of an ellipse is crucial in graphing and visualizing it. These lengths define the shape and orientation of the ellipse:

• The **semi-major axis** is the longest radius, stretching from the center to the farthest edge of the ellipse.
• The **semi-minor axis** is the shortest radius, extending from the center to the closest edge of the ellipse.

Calculating Axes Lengths

In our case, the ellipse has semi-major and semi-minor axes lengths derived from the denominators in the standard form equation:

• The semi-minor axis: \(a = 4\) (from \(a^2 = 16\))
• The semi-major axis: \(b = 8\) (from \(b^2 = 64\))

With our center at (2, -4), these axes determine the spread of the ellipse on the coordinate plane. A longer semi-major axis means the ellipse stretches further along that direction, giving it an oval shape.

Equation Transformation

Equation transformation is a vital step in simplifying and solving complex equations. When transforming equations for ellipses, the goal is to convert them into a standard form that is easy to work with.

In the original exercise, we began with the equation:

• \[ 16(x-2)^2 + 4(y+4)^2 = 256 \]

Steps in Transformation

• **Divide through by 256** to normalize the equation:\[\frac{16(x-2)^2}{256} + \frac{4(y+4)^2}{256} = 1\]
• **Simplify** the fractions to rewrite the equation in standard form:\[\frac{(x-2)^2}{16} + \frac{(y+4)^2}{64} = 1\]

This transformation makes it easier to identify elements such as the center and axes lengths, aiding in graphing the ellipse accurately.