Problem 11 - A Graphical Approach to College Algebra

Question

Graph each ellipse by hand. Give the domain and range. Give the foci and identify the center. Do not use a calculator. $$\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$$

Step-by-Step Solution

Answer

The center is at (0,0), foci are (\(\pm\sqrt{5}, 0\)), domain is [-3, 3], and range is [-2, 2].

Step 1: Identify the Center and Axes

The given equation is \(\frac{x^2}{9} + \frac{y^2}{4} = 1\). This is the standard form of an ellipse \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\) centered at \((h,k)\). Here, \(h = 0\) and \(k = 0\), so the center of the ellipse is at \((0,0)\). The length of the semi-major axis is \(\sqrt{9} = 3\) (since \(a^2 = 9\)) which is along the \(x\)-axis, and the length of the semi-minor axis is \(\sqrt{4} = 2\) (since \(b^2 = 4\)) which is along the \(y\)-axis.

Step 2: Determine the Vertices

The vertices of the ellipse along the major axis are \((h+a, k)\) and \((h-a, k)\). For this ellipse with center \((0,0)\), the vertices are \((0+3,0) = (3,0)\) and \((0-3,0) = (-3,0)\). The vertices along the minor axis are \((h, k+b)\) and \((h, k-b)\), which are \((0, 0+2) = (0,2)\) and \((0, 0-2) = (0,-2)\).

Step 3: Find the Foci

The foci of an ellipse are along the major axis at \((h \pm c, k)\), where \(c = \sqrt{a^2 - b^2}\). Here, \(c = \sqrt{9 - 4} = \sqrt{5}\). Thus, the foci are \((0 \pm \sqrt{5}, 0)\) or \((\sqrt{5}, 0)\) and \((-\sqrt{5}, 0)\).

Step 4: Determine the Domain and Range

The domain of the ellipse is determined by the extent of the semi-major axis along the \(x\)-axis. Therefore, the domain is \([-3, 3]\). The range is determined by the extent of the semi-minor axis along the \(y\)-axis, giving the range as \([-2, 2]\).

Step 5: Sketch the Ellipse

To sketch the ellipse, mark the center at \((0,0)\), draw the major axis from \(-3\) to \(3\) on the \(x\)-axis, and the minor axis from \(-2\) to \(2\) on the \(y\)-axis. Plot the vertices \((3,0), (-3,0), (0,2), (0,-2)\). Draw a smooth, oval shape connecting these points symmetrically around the center.

Key Concepts

Center of Ellipse

An ellipse is a curve that forms a closed, symmetrical shape, and its defining characteristic is its center. The center of an ellipse is the point around which the shape is symmetrically balanced. In most mathematical problems, the equation of the ellipse is given in the form:

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

In this standard form, \((h, k)\) represents the coordinates of the center of the ellipse. For the example equation \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \), we see that \(h = 0\) and \(k = 0\). Hence, the center of the ellipse is at the origin, \((0,0)\). Understanding where the center is located is crucial as it helps in sketching the ellipse and determining its other properties.

Foci

The foci (singular: focus) of an ellipse are two special points that hold a constant sum of distances to any point on the ellipse's boundary. The foci are key in defining the shape and form of the ellipse. To locate them, use the formula:

• \( c = \sqrt{a^2 - b^2} \)

Where \(a\) is the semi-major axis and \(b\) is the semi-minor axis. For the given ellipse, \(a^2 = 9\) and \(b^2 = 4\), thus \(c = \sqrt{9 - 4} = \sqrt{5} \). The foci are located symmetrically along the major axis, at positions \((h \pm c, k)\). Given the center \((h, k) = (0, 0)\), the foci are \((\sqrt{5}, 0)\) and \((-\sqrt{5}, 0)\). This means they lie on the horizontal (x-axis) at these calculated points.

Domain and Range

A fundamental step in comprehending the dimensions of an ellipse on the coordinate plane is identifying its domain and range. These indicate how far the ellipse stretches along the x-axis and y-axis. The domain represents the horizontal extent of the ellipse. For the provided equation \(\frac{x^2}{9} + \frac{y^2}{4} = 1 \), the domain is determined by the semi-major axis: from \(-a\) to \(a\), which is \([-3, 3]\) since \(a = 3\).

Similarly, the range is the vertical spread, governed by the semi-minor axis: from \(-b\) to \(b\), which is \([-2, 2]\) as \(b = 2\). Understanding these values assists in properly sketching the ellipse and ensures we visualize it within the correct boundaries.

Semi-major and Semi-minor Axes

In an ellipse, the longest diameter is called the major axis and its half is the semi-major axis, while the shortest diameter is the minor axis and its half is the semi-minor axis. These axes define the ellipse's length and width.

• The semi-major axis, in the equation \(\frac{x^2}{9} + \frac{y^2}{4} = 1\), corresponds to the value where the larger denominator is found: \(a^2 = 9\), hence \(a = 3\). It lies along the x-axis in this case.
• The semi-minor axis relates to the smaller denominator: \(b^2 = 4\), so \(b = 2\). It extends along the y-axis.

These axes not only set the dimensions of the ellipse but also play a crucial role in determining the position of the foci. Additionally, knowing the lengths of the axes helps in easily sketching the ellipse, focusing on its symmetry around the center point.