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Chapter 0: Problem 69

(i) Use a graphing utility to graph the equation in the first quadrant. [Note: To do this you will have to solve the equation for \(y\) in terms of \(x .]\) (ii) Use symmetry to make a hand-drawn sketch of the entire graph. (iii) Confirm your work by generating the graph of the equation in the remaining three quadrants. $$ 4 x^{2}+16 y^{2}=16 $$

Short Answer

Expert verified
The graph is a complete ellipse centered at the origin, extending from \(-2\) to \(2\) on the x-axis and from \(-1\) to \(1\) on the y-axis.

Step by step solution

01

Solve for y in terms of x

To solve the equation \( 4x^2 + 16y^2 = 16 \) for \( y \), first divide every term by 16 to simplify. This yields \( \frac{x^2}{4} + y^2 = 1 \). Then, solve for \( y \): \( y = \pm \sqrt{1 - \frac{x^2}{4}} \). This represents an ellipse.
02

Use a Graphing Utility in the First Quadrant

Graph the equation \( y = \sqrt{1 - \frac{x^2}{4}} \) for \( x \geq 0 \) in the first quadrant using a graphing utility. This will show the upper half of the ellipse from \( x = 0 \) to \( x = 2 \).
03

Reflect the Graph Across the x-axis

Since the given equation \( 4x^2 + 16y^2 = 16 \) is symmetric with respect to the x-axis, the part of the graph in the first quadrant can be reflected across the x-axis to cover the fourth quadrant. This gives the lower half of the ellipse from \( x = 0 \) to \( x = 2 \).
04

Reflect the Graph Across the y-axis

Similarly, the equation is symmetric with respect to the y-axis. Reflect the existing parts (in the first and fourth quadrants) over the y-axis to cover the second and third quadrants, thereby completing the ellipse.
05

Confirm with Full Ellipse Equation

To confirm, use a graphing utility to plot the full ellipse by keeping \( y = \pm \sqrt{1 - \frac{x^2}{4}} \) for the range of \( x \) from \( -2 \) to \( 2 \), which confirms the hand-drawn symmetry-based sketch.

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Key Concepts

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

Graphing Utility
A graphing utility is a powerful tool that helps visualize mathematical equations, making abstract concepts more tangible. When dealing with an ellipse like the one given by the equation \(4x^2 + 16y^2 = 16\), a graphing utility simplifies plotting by providing a clear visual representation in specific quadrants. To use a graphing utility for this equation:
  • Firstly, solve the equation for \(y\) in terms of \(x\). This means getting \(y\) by itself on one side of the equation, resulting in \(y = \pm \sqrt{1 - \frac{x^2}{4}}\).
  • Next, to focus on the first quadrant where both \(x\) and \(y\) are positive, use the equation \(y = \sqrt{1 - \frac{x^2}{4}}\) and input this into the graphing utility.
  • This process will produce a graph that shows the upper portion of the ellipse only in the first quadrant, from \(x = 0\) to \(x = 2\).
Graphing utilities can manage complex calculations and equations, allowing you to concentrate on understanding the shape and behavior of the graph, rather than getting bogged down in the arithmetic.
Symmetry
Symmetry in geometry provides a fascinating way to understand shapes, reflecting them across specific lines or axes without changing their inherent properties. Symmetry is critical for sketching graphs by hand, particularly with ellipses. Consider our equation \(4x^2 + 16y^2 = 16\), which describes an ellipse.
  • The equation is symmetric with respect to the x-axis. This means the graph above and below the x-axis are mirror images. By reflecting the first quadrant's portion of the graph across the x-axis, we obtain the graph in the fourth quadrant. This forms the lower half of the ellipse.
  • It's also symmetric about the y-axis. That means the sections on the left and right of the y-axis mirror each other. By reflecting the parts in the first and fourth quadrants across the y-axis, we capture the ellipse in the second and third quadrants, completing its entire shape.
Understanding geometric symmetry allows you to predict or confirm other sections of the graph without detailed calculations. It's an efficient and visual method for expanding portions of a graph across its natural symmetric properties.
Quadrants
The Cartesian coordinate system, divided into four quadrants, helps us describe the position of any point in the plane. An ellipse drawn in this system often spans different quadrants, as is the case for the ellipse \(4x^2 + 16y^2 = 16\). Quadrants are numbered counterclockwise:
  • The first quadrant, where both \(x\) and \(y\) are positive, shows the beginnings of our ellipse.
  • The fourth quadrant reflects the portion where \(x\) is positive but \(y\) is negative.
  • Quadrant two, where \(x\) is negative and \(y\) is positive, is reached through reflection across the y-axis.
  • Finally, quadrant three, where both \(x\) and \(y\) are negative, completes the ellipse.
When using quadrants, you can easily map and reflect sections of a graph from one part of the plane to others. Understanding how each part of the graph aligns with different quadrants helps in visualizing the entire shape, especially when using symmetry to complete graphs beyond just calculations.

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Most popular questions from this chapter

The function \(\cot ^{-1} x\) is defined to be the inverse of the restricted cotangent function $$ \cot x, \quad 00 \\ \pi+\tan ^{-1}(1 / x), & \text { if } x<0\end{array}\right.\) (b) \(\sec ^{-1} x=\cos ^{-1} \frac{1}{x}, \quad\) if \(|x| \geq 1\) (c) \(\csc ^{-1} x=\sin ^{-1} \frac{1}{x}, \quad\) if \(|x| \geq 1\).

Sketch the graph of the equation by making appropriate transformations to the graph of a basic power function. If you have a graphing utility, use it to check your work. (a) \(y=2(x+1)^{2}\) (b) \(y=-3(x-2)^{3}\) (c) \(y=\frac{-3}{(x+1)^{2}}\) (d) \(y=\frac{1}{(x-3)^{5}}\)

A soccer player kicks a ball with an initial speed of 14 \(\mathrm{m} / \mathrm{s}\) at an angle \(\theta\) with the horizontal (see the accompanying figure). The ball lands \(18 \mathrm{~m}\) down the field. If air resistance is neglected, then the ball will have a parabolic trajectory and the horizontal range \(R\) will be given by $$ R=\frac{v^{2}}{g} \sin 2 \theta $$ where \(v\) is the initial speed of the ball and \(g\) is the acceleration due to gravity. Using \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\), approximate two values of \(\theta\), to the nearest degree, at which the ball could have been kicked. Which angle results in the shorter time of flight? Why?

The perceived loudness \(\beta\) of a sound in decibels \((\mathrm{dB})\) is related to its intensity \(I\) in watts per square meter \(\left(\mathrm{W} / \mathrm{m}^{2}\right)\) by the equation $$ \beta=10 \log \left(I / I_{0}\right) $$ where \(I_{0}=10^{-12} \mathrm{~W} / \mathrm{m}^{2}\). Damage to the average ear occurs at \(90 \mathrm{~dB}\) or greater. Find the decibel level of each of the following sounds and state whether it will cause ear damage. $$ \begin{array}{ll} \hline \text { SOUND } & {I} \\ \hline \text { (a) Jet aircraft (from 50 ft) } & 1.0 \times 10^{2} \mathrm{~W} / \mathrm{m}^{2} \\ \text { (b) Amplified rock music } & 1.0 \mathrm{~W} / \mathrm{m}^{2} \\ \text { (c) Garbage disposal } & 1.0 \times 10^{-4} \mathrm{~W} / \mathrm{m}^{2} \\ \text { (d) TV (mid volume from 10 ft) } & 3.2 \times 10^{-5} \mathrm{~W} / \mathrm{m}^{2} \\ \hline \end{array} $$

True-False Determine whether the statement is true or false. Explain your answer. $$ \begin{aligned} &\text { Each curve in the family } y=2 x+b \text { is parallel to the line }\\\ &y=2 x \end{aligned} $$
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