Problem 7 - Intermediate Algebra

Question

Sketch the graph of each equation. \(\frac{x^{2}}{4}+\frac{y^{2}}{25}=1\)

Step-by-Step Solution

Answer

The ellipse is centered at (0,0) with its major axis on the y-axis and semi-values of 5 along the y-axis and 2 along the x-axis.

Step 1: Identify the Equation Type

The given equation \(\frac{x^{2}}{4}+\frac{y^{2}}{25}=1\) is a standard form of an ellipse with a horizontal major axis. Recognizing it as an ellipse is crucial for sketching the graph.

Step 2: Determine the Semi-Axes Lengths

In the equation of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), the values \(a^{2}\) and \(b^{2}\) are the denominators. For this equation, \(a^{2}=4\) and \(b^{2}=25\). Thus, \(a=2\) and \(b=5\). These values are the lengths of the semi-major and semi-minor axes, respectively.

Step 3: Identify the Center and Axes Orientation

The center of the ellipse is at the origin \((0,0)\) because there are no \(h\) or \(k\) terms in the equation. The orientation of the axes is horizontal because the larger denominator corresponds to the \(y\)-axis.

Step 4: Sketch the Major and Minor Axes

Plot the center at \((0,0)\). The semi-major axis extends 5 units along the \(y\)-axis in both directions (up and down), and the semi-minor axis extends 2 units along the \(x\)-axis in both directions (left and right). Mark these points: \((0,5), (0,-5), (2,0), (-2,0)\).

Step 5: Draw the Ellipse

Using the points from the major and minor axes, sketch the ellipse. The curve should smoothly connect these points, forming an elongated circle shape, with greater elongation along the \(y\)-axis.

Key Concepts

Ellipse Equation

An ellipse equation in its standard form is typically expressed as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). This formula represents a set of points that, when plotted, form an ellipse. The numbers \(a^2\) and \(b^2\) determine the size of the ellipse. Specifically:

• \(a\) is the distance from the center to the ellipse's furthest edge along the x-axis.
• \(b\) is the distance from the center to the ellipse's furthest edge along the y-axis.

If \(a^2 > b^2\), the ellipse is stretched more along the x-axis, and if \(b^2 > a^2\), the stretching is more along the y-axis. This relationship allows us to determine the orientation of the ellipse. In our example, \(\frac{x^2}{4} + \frac{y^2}{25} = 1\):

• Here, \(a^2 = 4\) and \(b^2 = 25\), thus \(a = 2\) and \(b = 5\).
• The larger \(b^2\) value confirms the stretch along the y-axis making it vertically oriented.

Conic Sections

Ellipses are part of a group of curves known as conic sections. These curves are derived from the intersection of a plane and a double-napped cone. Conic sections include circles, ellipses, parabolas, and hyperbolas. Here are some identifiers of each:

• **Circle**: A special type of ellipse where \(a = b\). Its equation appears as \(\frac{x^2}{r^2} + \frac{y^2}{r^2} = 1\).
• **Ellipse**: Has different lengths for axes \(a\) and \(b\), in its standard equation form as discussed.
• **Parabola**: The result of slicing the cone parallel to its side, giving an open curve.
• **Hyperbola**: Occurs when the plane cuts through both naps of the cone, forming open curves extending in opposite directions.

Understanding where ellipses fit into the bigger picture of conic sections helps to recognize their properties and how they relate to other geometric shapes.

Major and Minor Axes

In the standard ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the terms "major axis" and "minor axis" refer to the principal axes of the ellipse.The major axis is the longest diameter and passes through the foci of the ellipse. It represents the direction in which the ellipse is stretched more. Considering our example, the equation \(\frac{x^2}{4} + \frac{y^2}{25} = 1\) shows:

• The semi-major axis (half the length) is \(b = 5\), oriented along the y-axis because \(b^2 > a^2\).
• The full major axis therefore spans from \((0, -5)\) to \((0, 5)\).

The minor axis is shorter, located perpendicular to the major axis, and in this example, it is along the x-axis. The semi-minor axis has a length of \(a = 2\), spanning from \((-2, 0)\) to \((2, 0)\). Understanding these axes is essential for graphing ellipses since they determine the ellipse's overall shape and size.