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Chapter 13: Problem 35

Use a graphing calculator to graph each equation. See Using Your Calculator: Graphing Ellipses. $$ \frac{x^{2}}{4}+\frac{(y-1)^{2}}{9}=1 $$

Short Answer

Expert verified
The ellipse is centered at (0, 1) with axes lengths 3 (vertical) and 2 (horizontal).

Step by step solution

01

Identify the Equation Components

The given equation is \( \frac{x^{2}}{4} + \frac{(y-1)^{2}}{9} = 1 \). This represents an ellipse in standard form, centered at \((0, 1)\) since the equation can be rewritten as \((x-h)^2/a^2 + (y-k)^2/b^2 = 1\) where \(h=0\), \(k=1\), \(a^2=4\), and \(b^2=9\).
02

Determine the Values of a and b

The denominators of the equation are the squares of \(a\) and \(b\). Solving these, \(a = \sqrt{4} = 2\) and \(b = \sqrt{9} = 3\). This tells us that the semi-major axis has length 3 (since 9 > 4) and runs along the y-axis, and the semi-minor axis has length 2 and runs along the x-axis.
03

Setup the Graphing Calculator

Power on your graphing calculator and navigate to the graphing mode. To graph the equation \( \frac{x^{2}}{4} + \frac{(y-1)^{2}}{9} = 1 \), you need to isolate \(y\). Rearrange the equation to get \( (y-1)^2 = 9 - \frac{9}{4}x^2 \).
04

Solve for y and Enter the Equations

Solve for \(y\) by taking the square root of both sides: \( y - 1 = \pm \sqrt{9 - \frac{9}{4}x^2} \). Therefore, \( y = 1 \pm \sqrt{9 - \frac{9}{4}x^2} \). Enter these two separate equations in the graphing calculator: \( y_1 = 1 + \sqrt{9 - \frac{9}{4}x^2} \) and \( y_2 = 1 - \sqrt{9 - \frac{9}{4}x^2} \).
05

Graph the Ellipse

After entering the two equations, adjust the window settings to ensure the ellipse is properly visible. Set the x-range from -4 to 4 and the y-range from -2 to 4. This ensures that the entire ellipse will be displayed. Press the graph button to see the graph of the ellipse centered at \((0, 1)\) with a semi-major axis along the y-axis of length 3 and a semi-minor axis along the x-axis of length 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form of Ellipse
The standard form of an ellipse is a specific way of writing the equation of an ellipse to make its characteristics easy to identify. It is written as:
  • \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \)
This form allows you to immediately recognize that the ellipse is centered at the point
  • \((h, k)\).
Here, \(a\) and \(b\) are the distances from the center to the vertices in the x and y directions, known as the semi-axes. In the given equation
  • \( \frac{x^{2}}{4} + \frac{(y-1)^{2}}{9} = 1 \),
  • the center is at \((0, 1)\),
  • \(a\) is \(2\), and \(b\) is \(3\).
Being in standard form helps easily identify the orientation and position of the ellipse just by glancing at the equation.
Graphing Calculator
A graphing calculator is a super useful tool for plotting equations like ellipses. To graph an ellipse such as
  • \( \frac{x^{2}}{4} + \frac{(y-1)^{2}}{9} = 1 \),
you need to solve for \(y\). This involves rearranging the equation into two separate parts because ellipses have two y-values for each x-value, except where they intersect the axes.
  • First, rearrange to \( y = 1 \pm \sqrt{9 - \frac{9}{4}x^2} \)
  • Enter these into your graphing calculator as two separate equations:
  • \( y_1 = 1 + \sqrt{9 - \frac{9}{4}x^2} \)
  • \( y_2 = 1 - \sqrt{9 - \frac{9}{4}x^2} \)
Adjust the view settings on the calculator so that the entire ellipse can be seen, ensuring that the graph spans an appropriate range for both x and y. By entering these equations, the graphing calculator will show both the upper and lower halves of the ellipse on the screen.
Semi-Major and Semi-Minor Axes
Ellipses have two axes: the semi-major and the semi-minor axes. These are crucial because they define the shape and orientation of the ellipse.
  • The semi-major axis is the longest diameter of the ellipse.
  • The semi-minor axis is the shortest diameter.
In the standard form equation \( \frac{x^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), the semi-major and semi-minor axes are determined by \(a\) and \(b\). The larger value between \(a\) and \(b\) indicates which direction the semi-major axis runs. If \(b > a\), like in
  • \( \frac{x^2}{4} + \frac{(y-1)^2}{9} = 1 \),
then the semi-major axis is along the y-axis with length \(3\), while the semi-minor is along the x-axis with length \(2\). Understanding these axes is key to sketching accurate graphs and interpreting the position and proportion of ellipses.

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Most popular questions from this chapter

Write each equation in standard form, if it is not alreacty so, and graph it. If the graph is a circle, give the coordinates of its center and its radius. If the graph is a parabola, give the coordinates of its vertex. $$ x=-\frac{1}{3} y^{2}-2 y $$

Find each product. $$3 x^{-2} y^{2}\left(4 x^{2}+3 y^{-2}\right)$$

Solve each system of equations for real values of x and y. $$ \left\\{\begin{array}{l} x^{2}=4-y \\ y=x^{2}+2 \end{array}\right. $$

Explain the difference between the focus of an ellipse and the vertex of an ellipse.

Solve each system of equations for real values of x and y. $$ \left\\{\begin{array}{l} x-2 y=2 \\ 9 x^{2}-4 y^{2}=36 \end{array}\right. $$
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