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Chapter 9: Problem 48

Use a graphing utility to graph each equation. $$3 x^{2}-6 x y+3 y^{2}+10 x-8 y-2=0$$

Short Answer

Expert verified
The graph will depict a conic section - the exact nature (ellipse, hyperbola or parabola) can be determined by the graphing utility. Refer to the visual output on the graphing tool for the exact shape and location of the graph.

Step by step solution

01

Launch the Graphing Utility

Launch your graphing utility. Graphical Calculation Softwares or online Graphing calculators like Desmos, GeoGebra can be used.
02

Input the equation

Enter the equation \(3 x^{2}-6 x y+3 y^{2}+10 x-8 y-2=0\) into the graphing utility. Make sure to input the equation correctly to get an accurate graph.
03

Plot the graph

After entering the equation, command the calculator to plot the graph. Usually, this is done by pressing a 'Graph' button or a similar command.
04

Interpret the result

Once the graph is plotted, it will appear on the graphing display. This graph represents all the solutions to the equation.

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Key Concepts

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

Graphing Utility
Graphing utilities are digital tools that help us visualize mathematical equations. These tools include software applications like GeoGebra and Desmos, or even graphing calculators available online or in physical form. Here's how they simplify graphing:
  • Accessibility: Most graphing utilities are available online for free, instantly accessible, and often offer step-by-step support to input complex equations.
  • User-Friendly Interface: Modern graphing utilities prioritize user interfaces that guide the student through input and graphing processes, making it easier even for beginners to use.
  • Accuracy: By providing precise calculations and meticulous plotting, graphing utilities eliminate errors often encountered in manual graph drawing.

When working with complex equations, graphing utilities are indispensable; they allow you to visualize what the algebraic expression truly represents in a graphical format. This enables a deeper understanding of mathematical relationships and their geometric representations.
Conic Sections
Conic sections are shapes or curves that result from intersecting a cone with a plane. Depending on the angle and position of the intersection, different conic shapes are formed:
  • Circle: A plane intersects the cone in a manner parallel to its base.
  • Ellipse: Occurs when the plane cuts through the cone at an angle, not parallel to the base or perpendicular to the axis.
  • Parabola: Formed when the intersection is parallel to the slant of the cone.
  • Hyperbola: Occurs when the plane intersects both halves of the double cone.

In our equation, identifying the type of conic section helps in understanding the graph's shape. The given equation is a form of a conic section involving both variables raised to the second power, suggesting that it could be an ellipse based on the coefficients following standard algebraic forms. Conic sections have a wealth of applications ranging from physics to engineering, illustrating the utility of being familiar with these fascinating curves.
Graph Interpretation
Understanding and interpreting a graph can make the numerical data and equations they represent meaningful. When looking at a graph, here's what to focus on:
  • Symmetry: Check if the graph is symmetrical along any axis鈥攁 common feature in conic sections.
  • Intersections: Look for points where the graph intersects the axes, which reveals important solutions to the equation.
  • Shape and Orientation: Analyze the shape of the graph to identify what type of conic section it represents.

For the equation provided, after plotting using a graphing utility, the graph will show a recurved shape that can aid in determining whether it is an ellipse, hyperbola, or another conic. Interpreting these features correctly enhances comprehension of the relationship between algebraic expressions and their visual presentations.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. In each exercise, graph the equation in a rectangular coordinate system. $$\frac{x^{2}}{25}+\frac{y^{2}}{4}=1$$

Find all zeros of \(f(x)=2 x^{3}+x^{2}-13 x+6 . \quad\) (Section 2.5 Example 3)

In Exercises \(78-82,\) use a graphing utility to obtain the plane curve represented by the given parametric equations. Hypocycloid: \(x=4 \cos ^{3} t, y=4 \sin ^{3} t\) \([-8,8,1] \times[-5,5,1], 0 \leq t<2 \pi\)

In Exercises \(21-40\), eliminate the parameter \(t\). Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of \(t .\) (If an interval for \(t\) is not specified, assume that \(-\infty < t < \infty .)\) $$x=2 \sin t, y=2 \cos t ; 0 \leq t < 2 \pi$$

Exercises \(97-99\) will help you prepare for the material covered in the next section. a. Showing all steps, rewrite \(r=\frac{1}{3-3 \cos \theta}\) as \(9 r^{2}=(1+3 r \cos \theta)^{2}\) b. Express \(9 r^{2}=(1+3 r \cos \theta)^{2}\) in rectangular coordinates. Which conic section is represented by the rectangular equation?
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