Problem 7 - Algebra A Combined Function

Question

Graph each ellipse. $$ 4 x^{2}+25 y^{2}=100 $$

Step-by-Step Solution

Answer

The ellipse's center is at (0,0) with a major axis of 10 along the x-axis and a minor axis of 4 along the y-axis.

Step 1: Identify Standard Form of Ellipse Equation

The equation of the ellipse is given as \( 4x^2 + 25y^2 = 100 \). The standard form of the ellipse equation is \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \((h, k)\) is the center.

Step 2: Divide through by 100

To convert to standard form, divide every term in the equation by 100:\[ \frac{4x^2}{100} + \frac{25y^2}{100} = \frac{100}{100} \] This simplifies to \( \frac{x^2}{25} + \frac{y^2}{4} = 1 \).

Step 3: Recognize ellipse orientation

In the equation \( \frac{x^2}{25} + \frac{y^2}{4} = 1 \), the larger denominator (25) is under \(x^2\). This indicates the ellipse is oriented horizontally.

Step 4: Identify center, major and minor axes

The center \((h,k)\) is \((0,0)\) because there are no \((x-h)^2 \) or \((y-k)^2 \) terms shifting the ellipse. The major axis has length \(2a = 2 \sqrt{25} = 10\) and the minor axis length \(2b = 2 \sqrt{4} = 4\).

Step 5: Plot the ellipse

To graph, plot the center at \((0,0)\), then move 5 units horizontally to the left and right (major axis) and 2 units vertically up and down (minor axis). Connect these four extreme points in a smooth, elliptical shape.

Key Concepts

Standard Form of Ellipse

Ellipses are conic sections that have a particular standard form equation. This equation is used to easily understand and graph ellipses. The standard form for the equation of an ellipse is: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \] Here,

• \((h, k)\) stands for the center of the ellipse.
• \(a\) and \(b\) represent the lengths of the semi-major and semi-minor axes respectively.
• The square terms \((x-h)^2\) and \((y-k)^2\) show distances from the center along the x and y axes.

By comparing a given equation to this standard form, we can determine the position and sizing of the ellipse. If necessary, the equation is rearranged to fit this form before any detailed graphing takes place. This transformation is crucial to simplifying the task of plotting the ellipse effectively.

Ellipse Graphing

Graphing an ellipse starts with converting its equation into the standard form. Once in this form, identifying key features like the center, and the lengths of the major and minor axes becomes much easier. For the equation \[ \frac{x^2}{25} + \frac{y^2}{4} = 1 \], we can see:

• The center is at the origin \((0,0)\).
• The semi-major axis length \(a\) is 5 \((\sqrt{25})\).
• The semi-minor axis length \(b\) is 2 \((\sqrt{4})\).

To graph:

• Start by plotting the center.
• Then, go five units left and right from the center along the x-axis for the major axis.
• Go two units up and down from the center on the y-axis for the minor axis.

Lastly, connect these extremes with a smooth curved line to form the ellipse. It is essential to achieve proportional curves along the specific directions indicated by the semi-axes lengths. Paying attention to symmetry also enhances the accuracy of the graph depiction.

Conic Sections

Ellipses belong to a group of shapes known as conic sections. These shapes are derived from intersections of a plane with a cone. Depending on the angle and position of this intersection, different shapes can result, including circles, ellipses, parabolas, and hyperbolas. Ellipses specifically occur when:

• The cutting plane intersects the cone at an angle relative to its base.
• This angle is neither parallel to the base nor perpendicular to its height, ensuring a closed, oval shape.

Ellipses are characterized by their oval forms and the property that the sum of the distances from any point on the ellipse to its two focal points remains constant. Understanding ellipses as a part of conic sections helps in comprehending their geometric properties and behavior, enriching the ways they can be applied in physical and theoretical contexts.

Ellipse Orientation

The orientation of an ellipse describes its positioning relative to the coordinate axes. It can be either horizontal or vertical:

• If the larger denominator in the standard form equation \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\) is under \((x-h)^2\), the ellipse is horizontal. This means it stretches more along the x-axis.
• Conversely, if the larger denominator is under \((y-k)^2\), the ellipse is vertical and stretches more along the y-axis.

In our case, since \(\frac{x^2}{25} + \frac{y^2}{4} = 1\) has 25 (the larger number) under \(x^2\), the ellipse is horizontally oriented. Recognizing this orientation helps in accurately setting up the ellipse on the graph, which can assist in applications where directionality is important.