Problem 57 - Precalculus: Functions and Graphs

Question

Exer. \(51-58:\) Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse. $$y=2-7 \sqrt{1-\frac{(x+1)^{2}}{9}}$$

Step-by-Step Solution

Answer

Lower half of the ellipse; full equation is \( \frac{(x+1)^2}{9} + \frac{(y-2)^2}{49} = 1 \).

Step 1: Identify the Square Root Expression

The given equation is \( y = 2 - 7 \sqrt{1 - \frac{(x+1)^2}{9}} \). Notice the term under the square root, \( 1 - \frac{(x+1)^2}{9} \). This expression resembles the form used in the equation of an ellipse.

Step 2: Match the Form of the Ellipse Equation

Recall the standard form of the equation of an ellipse centered at \( (h, k) \) with axes aligned to the coordinate axes:\[ (x-h)^2 / a^2 + (y-k)^2 / b^2 = 1 \]For a horizontal ellipse, the equation is rearranged as \[ y = k \pm b \sqrt{1 - \frac{(x-h)^2}{a^2}} \].In the given expression, \( h = -1 \), \( a^2 = 9 \), \( b = 7 \), and \( k = 2 \). This matches the form of a horizontal (left or right half) ellipse.

Step 3: Determine Which Half is Represented

In the equation \( y = k - b \sqrt{1 - \frac{(x-h)^2}{a^2}} \), the negative sign before the square root means the equation represents the lower half of the ellipse, not the upper half, because \( b \sqrt{1 - \frac{(x-h)^2}{a^2}} \) is subtracted from \( k \).

Step 4: Formulate the Full Equation of the Ellipse

Based on the standard form rearrangement and the values for \( h \), \( k \), \( a^2 \), and \( b \), the full equation of the ellipse is:\[ \frac{(x+1)^2}{9} + \frac{(y-2)^2}{49} = 1 \].

Key Concepts

Equation of an Ellipse

An ellipse is a curve on a plane that surrounds two focal points, such that the sum of the distances to the two focal points is constant for every point on the curve. Its equation helps in understanding the relationship between these points and how they form an ellipse.
To formulate the equation of an ellipse, identify two axes: **major** and **minor**. The major axis is the longest diameter, while the minor axis is the shortest. These axes are crucial in determining the shape and size of the ellipse. The typical equation structure involves the use of these axes.
For an ellipse centered at \( (h,k) \), the standard algebraic form involves two fractions: one for the x-axis and one for the y-axis. They're bounded by a constant value of 1 as follows: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \] where \( a \) and \( b \) are the semi-major and semi-minor axes respectively.

Standard Form of Ellipse

The standard form of an ellipse's equation provides the precise mathematical representation of an ellipse. It helps in understanding both vertical and horizontal ellipses and interpreting the graph.
The general form, as mentioned previously, changes slightly based on the orientation of the ellipse:

• Horizontal Ellipse: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \)
• Vertical Ellipse: \( \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \)

Here, \( a \) is the semi-major axis and \( b \) is the semi-minor axis. The center is noted as \( (h, k) \), which is a key point in determining how the ellipse fits on a coordinate plane. Recognizing this standard form is fundamental for graphing and analyzing ellipses within coordinate systems.

Half of an Ellipse

In mathematical exercises, equations can sometimes describe only half of an ellipse. Recognizing which half is represented involves a careful analysis of its equation.
Consider the equation: \( y = k \pm b \sqrt{1-\frac{(x-h)^2}{a^2}} \). Here,

• The upper half is denoted by the "+" sign, resulting in values > \( k \).
• The lower half is depicted by the "−" sign, creating points < \( k \).

Understanding these relationships allows us to interpret which portion of the ellipse a given equation represents. This is crucial for solving problems or graphing half-ellipses effectively, as it lends insight into the inherent symmetry of ellipses.

Graphing Ellipses

Graphing an ellipse requires knowing its center, and lengths of the semi-major and semi-minor axes. These components help determine its shape and position.
Follow these steps to graph an ellipse:

• Identify the center \((h, k)\) from the equation.
• Draw the semi-major axis along its respective direction depending on ellipse orientation (horizontal or vertical).
• Plot the semi-minor axis in the perpendicular direction.
• Sketch the curve, ensuring it smoothly connects through intercepted axes points.

Ellipses can be asymmetrical, but they always retain this elliptical shape. Visualization of an ellipse on a graph is not only useful in mathematics but applicable in physics and astronomy for real-world applications.