Problem 3 - Algebra A Combined Function

Question

Graph each ellipse. $$ \frac{x^{2}}{9}+y^{2}=1 $$

Step-by-Step Solution

Answer

The ellipse is horizontally oriented with vertices (-3,0), (3,0), (0,-1), and (0,1).

Step 1: Identify the Equation Type

The given equation is \(\frac{x^2}{9} + y^2 = 1\). This is the standard form of an ellipse centered at the origin with the major and minor axes aligned with the coordinate axes.

Step 2: Determine the Axes Lengths

The standard equation for an ellipse is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, \(a^2 = 9\) so \(a = 3\), and \(b^2 = 1\) so \(b = 1\). Thus, the lengths of the axes are 2a = 6 for the x-axis and 2b = 2 for the y-axis.

Step 3: Identify the Orientation

Since \(a > b\), the ellipse is horizontally oriented with the major axis along the x-axis and the minor axis along the y-axis.

Step 4: Find the Vertices

The vertices of the ellipse are located at the endpoints of the major and minor axes. For the x-axis, vertices are at (-3, 0) and (3, 0). For the y-axis, vertices are at (0, -1) and (0, 1).

Step 5: Sketch the Ellipse

Draw the ellipse by plotting the vertices: (-3,0), (3,0), (0,-1), and (0,1). Connect these points smoothly forming an ellipse shape elongating horizontally.

Key Concepts

Standard Form of an Ellipse

The equation \(\frac{x^2}{9} + y^2 = 1\) follows the standard form of an ellipse. In general, the standard form of an ellipse is given as:

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

This form helps us easily identify the properties of an ellipse, such as its size and orientation. In our equation, the terms \(\frac{x^2}{9}\) and \(y^2\) indicate that the ellipse's center is at the origin, (0,0), since there are no terms added or subtracted from the \(x\) or \(y\) variables. By understanding this form, we can proceed to uncover the ellipse's axis lengths and other geometric features.

Axes Lengths of an Ellipse

To find the lengths of the axes, we examine the denominators in the equation \(\frac{x^2}{9} + y^2 = 1\). For an ellipse in the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), \(a\) and \(b\) represent the semi-major and semi-minor axes, respectively.

• In our equation, \(a^2 = 9\), which means \(a = 3\). The length of the major axis is therefore \(2a = 6\).
• Since \(b^2 = 1\), \(b = 1\). Thus, the length of the minor axis is \(2b = 2\).

These values allow us to graphically represent the ellipse with the correct proportions.

Ellipse Orientation

The orientation of an ellipse describes whether it is stretched along the x-axis or the y-axis. This is determined by comparing the values of \(a\) and \(b\). When \(a > b\), as in our equation \(\frac{x^2}{9} + y^2 = 1\), the ellipse is horizontally oriented.

• The major axis is the longer axis, aligned along the x-axis, forming a horizontal ellipse in this case.
• The minor axis is shorter and aligned along the y-axis.

Understanding the orientation helps visualize how the ellipse stretches and guides the accurate sketch of its shape.

Ellipse Vertices

Vertices are crucial points that define the extremities of an ellipse along its axes. They play an integral role in graphing the ellipse accurately. For a horizontally oriented ellipse, the vertices are determined as follows:

• Major axis vertices are at \((-a,0)\) and \((a,0)\), which for \(a = 3\) are \((-3,0)\) and \((3,0)\).
• Minor axis vertices are at \((0,-b)\) and \((0,b)\), with \(b = 1\) resulting in vertices at \((0,-1)\) and \((0,1)\).

These vertices outline the outer shape of the ellipse, and plotting them is fundamental when drawing the complete ellipse.