Problem 16 - Beginning and Intermediate Algebra

Question

Identify the center of each ellipse and graph the equation. $$\frac{x^{2}}{9}+y^{2}=1$$

Step-by-Step Solution

Answer

The center of the ellipse is at the origin (0,0) with a semi-major axis of 3 units and a semi-minor axis of 1 unit. To graph the ellipse, mark the center (0,0) and draw points at (3,0), (-3,0), (0,1), and (0,-1). Then, connect these points with a smooth curve to form the ellipse.

Step 1: Identify center and axes

\( \frac{x^2}{9} + \frac{y^2}{1} = 1 \). Center: \( (0,0) \). Semi-major axis \( a = 3 \) (along x-axis), semi-minor axis \( b = 1 \) (along y-axis).

Step 2: Graph

Plot points \( (\pm 3, 0) \) and \( (0, \pm 1) \) and draw the ellipse through them.

Key Concepts

Graphing Ellipses

Graphing an ellipse involves understanding its equation and translating it onto a coordinate plane. The standard form of an ellipse's equation is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\). Here, \((h, k)\) represents the center, while \(a\) and \(b\) denote the lengths of the semi-major and semi-minor axes, respectively.
To graph an ellipse:

• Identify the center, \((h, k)\) from the equation.
• Determine the values of \(a\) and \(b\) by taking the square roots of \(a^2\) and \(b^2\).
• Use these values to plot points along the major and minor axes.
• Connect these points smoothly to form the ellipse.

Remember, the major axis is along the direction of the larger number between \(a\) and \(b\).

Semi-Major Axis

The semi-major axis is the longest radius of an ellipse, extending from the center to the ellipse's edge. It lies along the axis where the denominator of the equation is larger.
For example, in the equation \(\frac{x^{2}}{9}+\frac{y^{2}}{1}=1\), since \(a^2 = 9\), the semi-major axis is 3, which is longer than the semi-minor axis. Key points:

• It determines the direction and length of the ellipse's longest stretch.
• The value \(a\) is calculated by \(a = \sqrt{a^2}\).
• On the graph, the total length is \(2a\), as it stretches both ways from the center.

This axis is crucial for accurately drawing the ellipse's longest dimensions.

Semi-Minor Axis

The semi-minor axis is the shortest radius of an ellipse, perpendicular to the semi-major axis. It helps define how "squished" the ellipse appears.
In the equation \(\frac{x^{2}}{9}+\frac{y^{2}}{1}=1\), with \(b^2=1\), the semi-minor axis is 1.Important aspects:

• It provides the shortest diameter of the ellipse.
• Its length is found using \(b = \sqrt{b^2}\).
• On the graph, the total length is \(2b\).

Understanding both semi-major and semi-minor axes is essential in graphing ellipses effectively.

Conic Sections

Ellipses are a type of conic section, which are curves obtained by intersecting a plane with a cone. The four basic types of conic sections are circles, ellipses, parabolas, and hyperbolas.
Ellipses specifically occur when the intersecting plane cuts through both nappes of the cone but not as steep as parabolas or hyperbolas. Characteristics of ellipses:

• They have two focal points that describe their shape.
• Ellipses can become circles if the semi-major and semi-minor axes are equal.
• Their unique properties and shapes have vital applications in astronomy, engineering, and design.

Studying conic sections helps in understanding the complex relationships and shapes within mathematics.