Problem 1 - Algebra A Combined Function

Question

Graph each ellipse. $$ \frac{x^{2}}{4}+\frac{y^{2}}{25}=1 $$

Step-by-Step Solution

Answer

The ellipse is centered at (0, 0), with a semi-major axis of length 5 along the y-axis and a semi-minor axis of length 2 along the x-axis.

Step 1: Identify the Equation of the Ellipse

The given equation is \( \frac{x^2}{4} + \frac{y^2}{25} = 1 \). This is in the standard form of an ellipse, \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where \( a^2 \) and \( b^2 \) are the denominators. Here, \( a^2 = 4 \) and \( b^2 = 25 \). Since \( b^2 > a^2 \), this is a vertical ellipse.

Step 2: Calculate the Semi-Major and Semi-Minor Axes

For an ellipse, the semi-major and semi-minor axes are given by \( a = \sqrt{a^2} \) and \( b = \sqrt{b^2} \). With \( a^2 = 4 \), \( a = \sqrt{4} = 2 \). With \( b^2 = 25 \), \( b = \sqrt{25} = 5 \). Thus, the semi-major axis is 5 units and the semi-minor axis is 2 units.

Step 3: Determine the Center of the Ellipse

The center of the ellipse in this form is at the origin, \((0, 0)\). This comes directly from the standard form of the ellipse equation, as there are no \( x \) or \( y \) constant terms added.

Step 4: Plot the Axes

Starting from the center \((0, 0)\), plot the semi-major axis vertically because \( b \) is greater than \( a \). Move 5 units up to \((0, 5)\) and 5 units down to \((0, -5)\). For the semi-minor axis, move 2 units right to \((2, 0)\) and 2 units left to \((-2, 0)\).

Step 5: Sketch the Ellipse

Use the plotted points to sketch the shape of the ellipse. Connect the points such that the ellipse is elongated along the y-axis, ensuring to create a smooth, symmetric shape.

Key Concepts

Graphing ellipses

Graphing an ellipse involves plotting key points that outline its shape. Each ellipse is symmetric and oval-shaped, characterized by its axes.
To begin graphing an ellipse, you first identify its center, which serves as the reference point for plotting other points.
Next, the lengths of the semi-major and semi-minor axes are determined, representing the ellipse's radius along its longest and shortest dimensions, respectively.
Align these axes along the coordinate plane based on which is longer (major) or shorter (minor), extending from the center.

• For a vertical ellipse, the major axis stretches vertically.
• For a horizontal ellipse, it stretches horizontally.

Plot the major and minor points, then smoothly sketch the curved, symmetrical outline connecting these points.

Standard form of an ellipse

The standard form of an ellipse is the mathematical formula to describe its shape and position on a graph. It looks like this:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]This formula indicates the relationships between the coordinates of points on the ellipse. Here, \(a\) and \(b\) represent the distances from the center of the ellipse to its edge along the x-axis and y-axis, respectively.
In the equation, the value under each squared term (\(a^2\) and \(b^2\)) denotes how much the ellipse is stretched in that direction.

• If \(b^2 > a^2\), the ellipse is stretched vertically.
• If \(a^2 > b^2\), it is pushed out horizontally.

Understanding this form helps decide the ellipse's orientation and size.

Semi-major and semi-minor axes

The semi-major and semi-minor axes are crucial in defining an ellipse's shape. They are the two main lengths extending from the center to the ellipse's edge.
The semi-major axis is the longer of the two, while the semi-minor axis is the shorter.

• The semi-major axis has the length \(b = \sqrt{b^2}\).
• The semi-minor axis has the length \(a = \sqrt{a^2}\).

In our problem, the calculations reveal that the semi-major axis is 5 units and the semi-minor axis is 2 units.
These axes help in plotting as they define the fireworks spread of the ellipse along its major and minor orientations on the graph.

Center of an ellipse

The center of an ellipse is like its anchor point from which everything else spreads out. In the standard equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), when there are no added constants, it means the center is at the origin, \((0, 0)\).
From this central point:

• The semi-major axis extends in one direction, and its opposite.
• The semi-minor axis stretches perpendicular to the semi-major axis, also in both directions.

By knowing the center, you can position the rest of the ellipse more easily on the graph, ensuring it's symmetric and neatly aligned with its axes.