Problem 25 - Precalculus: Mathematical for Calculus

Question

Use a graphing device to graph the ellipse. $$\frac{x^{2}}{25}+\frac{y^{2}}{20}=1$$

Step-by-Step Solution

Answer

Graph the ellipse at the origin with semi-major axis 5 and semi-minor axis 4.47.

Step 1: Understand the Equation of an Ellipse

The equation given is \(\frac{x^{2}}{25} + \frac{y^{2}}{20} = 1\), which is in the standard form of an ellipse equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \(a^2 = 25\) and \(b^2 = 20\).

Step 2: Determine the Lengths of the Semi-Axes

Solve for \(a\) and \(b\) by taking the square roots. Thus, \(a = \sqrt{25} = 5\) and \(b = \sqrt{20} \approx 4.47\). These are the lengths of the semi-major and semi-minor axes respectively. Since \(a > b\), the ellipse is horizontally oriented.

Step 3: Identify the Center of the Ellipse

The center of the ellipse is at the origin \((0,0)\) because the equation is not shifted (no \(h\) or \(k\) terms are present).

Step 4: Plot the Axes Extents

On a coordinate plane, plot points at \((5, 0)\), \((-5, 0)\) corresponding to the semi-major axis, and at \((0, 4.47)\), \((0, -4.47)\) corresponding to the semi-minor axis. These points mark the extrema of the ellipse along the x and y directions.

Step 5: Draw the Ellipse

Using the plotted points, sketch the shape of the ellipse. It should be elongated horizontally with the plotted points as the boundaries.

Step 6: Confirm Using a Graphing Device

If using an electronic graphing device, input the original equation \(\frac{x^{2}}{25} + \frac{y^{2}}{20} = 1\) and ensure that the graph aligns with the manually plotted ellipse.

Key Concepts

Semi-Major Axis

The semi-major axis is a crucial aspect of any ellipse. It represents half of the longest diameter of the ellipse. In the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), \(a\) is the semi-major axis if \(a^2 > b^2\). For the problem at hand, the ellipse equation is \(\frac{x^2}{25} + \frac{y^2}{20} = 1\), which implies that \(a^2 = 25\). By taking the square root of \(25\), we find that \(a = 5\).

• The semi-major axis is responsible for the broad stretch of the ellipse along the x-axis.
• It determines the extent of the ellipse in its horizontal direction.

This means the ellipse will extend five units to the right and left of the center at the origin (0,0). The length of the semi-major axis directly affects the overall shape and orientation of the ellipse, helping to define it as more elongated or circular.

Semi-Minor Axis

The semi-minor axis of an ellipse constitutes half of the shortest diameter. When dealing with the standard ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), \(b\) is the semi-minor axis if \(b^2 < a^2\). In the exercise's case, \(b^2 = 20\). Therefore, to find \(b\), we calculate \(b = \sqrt{20} \approx 4.47\).

• The semi-minor axis contributes to the vertical span of the ellipse.
• It measures the compactness of the ellipse along the y-axis.

This axis extends approximately 4.47 units upwards and downwards from the center at (0,0). It holds a key role in balancing the ellipse's dimensions, determining its degree of flatness or roundness.

Graphing Ellipses

Graphing ellipses is all about plotting their defining points and connecting them into a smooth curve. Begin with identifying key attributes such as the semi-major and semi-minor axes, which we previously determined as 5 and approximately 4.47 respectively.

• Start at the center, which is (0,0) when the ellipse equation lacks shifted terms (\(h\) and \(k\)).
• Plot the endpoints of semi-major and semi-minor axes: \((5,0)\), \((-5,0)\), \((0,4.47)\), and \((0,-4.47)\).

After placing these points on a coordinate system, draw a smooth, elongated, oval shape linking the outer tips of these axes. A handy tip when drawing manually is to visualize a rectangle whose corners touch these plotted points, ensuring the ellipse fits snugly inside. When checking this using a digital tool, input the original equation to verify the accuracy of your plotted ellipse. The graphing device should help in confirming that the shape is not only consistent with these axises but also aligned precisely within the determined bounds.