Problem 24 - Intermediate Algebra

Question

Graph each equation. \(4 x^{2}+9 y^{2}=36\)

Step-by-Step Solution

Answer

The graph is an ellipse centered at the origin with vertices at (-3, 0), (3, 0), (0, -2), and (0, 2).

Step 1 - Identify the Equation Type

The given equation \(4x^2 + 9y^2 = 36\) is in the standard form of an ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) are the semi-axes lengths.

Step 2 - Transform into Standard Form

Divide the entire equation by 36 to transform it into the standard form: \(\frac{4x^2}{36} + \frac{9y^2}{36} = 1\), simplifying to \(\frac{x^2}{9} + \frac{y^2}{4} = 1\).

Step 3 - Identify the Semi-Axes Lengths

From the equation \(\frac{x^2}{9} + \frac{y^2}{4} = 1\), identify \(a^2 = 9\) and \(b^2 = 4\). Thus, \(a = 3\) and \(b = 2\).

Step 4 - Sketch the Ellipse

Since \(a = 3\) and \(b = 2\), sketch the ellipse centered at the origin (0,0). The ellipse will intersect the x-axis at (-3,0) and (3,0), and the y-axis at (0,-2) and (0,2). Connect these points in an oval shape to complete the ellipse.

Key Concepts

Standard Form of an Ellipse

Understanding the standard form of an ellipse is pivotal in graphing and analyzing ellipses. An ellipse can be generally represented in the standard form as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, \(a\) and \(b\) are the lengths of the semi-axes. By rewriting an ellipse equation into this format, it becomes easier to visualize and sketch the ellipse.

In the given equation from the exercise, \(4x^2 + 9y^2 = 36\), we transform it into the standard form by dividing every term by 36 to get \(\frac{x^2}{9} + \frac{y^2}{4} = 1\). This new form reveals the relationship of the ellipse's axes directly from the denominators in the fractions.

Semi-Axes Lengths

The semi-axes lengths \(a\) and \(b\) give significant insight into the shape and orientation of an ellipse. They are derived from the standard form, \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), for a horizontal ellipse, or \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\), for a vertical one.

The values of \(a\) and \(b\) are the square roots of the denominators: \(a = \sqrt{9} = 3\) and \(b = \sqrt{4} = 2\). This reflects that the ellipse stretches 3 units along the x-axis and 2 units along the y-axis starting from the center of the ellipse. These measurements not only help in sketching the ellipse but also in understanding its dimensions and proportions.

Graphing Ellipses

Graphing an ellipse involves plotting points and connecting them smoothly to represent its oval shape. The semi-axes values \(a\) and \(b\) inform where the ellipse will touch both axes. For our ellipse, centered at the origin (0,0), it intersects the x-axis at the points \((-3, 0)\) and \((3, 0)\) and the y-axis at \((0, -2)\) and \((0, 2)\).

Once these major and minor axis points are plotted, they provide a guide for drafting the ellipse's oval shape. Drawing symmetric curves around these points ensures an accurate representation. Always remember to maintain the proportions indicated by \(a\) and \(b\) to correctly represent the ellipse's shape.