Graphing a Rotated Conic on a Graphing Calculator

In summary, to graph the conic x^2+xy+y^2=10 using a graphing utility, you need to solve for y by rewriting the equation as a quadratic equation and using the quadratic formula. The angle of rotation for the axis is 45 degrees and the equation in terms of x' and y' is (x')^2/(20/3)+(y')^2/20=1. However, to graph the rotated function on a calculator, you need to solve for y and use the quadratic formula to get a real answer.
  • #1
themadhatter1
140
0

Homework Statement


Use a graphing utility to graph the conic. Determine the angle through which the axis are rotated.

[tex]x^2+xy+y^2=10[/tex]

Homework Equations


[tex]\cot2\theta=\frac{A-C}{B}[/tex]
[tex]x=x'\cos\theta-y'\sin\theta[/tex]
[tex]y=x'\sin\theta+y'\cos\theta[/tex]

The Attempt at a Solution



I can find the angle of rotation to be 45 degrees and I know the equation in terms of x' y' is.

[tex]\frac{(x')^2}{\frac{20}{3}}+\frac{(y')^2}{20}=1[/tex]

However I don't know how to graph the rotated function. My graphing calculator can not graph implicit functions. I can get function in terms of x',y' in terms of y' and graph that, but that won't be on an x',y' axis on my calculator, only an x,y axis. How can I graph the rotated function on a calculator?
 
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  • #2
To graph the conic using a graphing utility, you need to solve the conic for y. Rewrite the equation as a quadratic equation to y (the x's will be part of the coefficients):
[tex]\begin{aligned}
x^2 + xy + y^2 &= 10 \\
y^2 + (x)y + (x^2 - 10) &= 0 \\
\end{aligned}[/tex]
Use the quadratic formula with [tex]a = 1[/tex], [tex]b = x[/tex], and [tex]c = (x^2 - 10)[/tex].69
 
  • #3
eumyang said:
To graph the conic using a graphing utility, you need to solve the conic for y. Rewrite the equation as a quadratic equation to y (the x's will be part of the coefficients):
[tex]\begin{aligned}
x^2 + xy + y^2 &= 10 \\
y^2 + (x)y + (x^2 - 10) &= 0 \\
\end{aligned}[/tex]
Use the quadratic formula with [tex]a = 1[/tex], [tex]b = x[/tex], and [tex]c = (x^2 - 10)[/tex].


69

I did that and I wind up getting an imaginary answer. My calculator comes up with errors.

[tex]x=\frac{-x\pm\sqrt{-3x^2-40}}{2}[/tex]
 
  • #4
themadhatter1 said:
I did that and I wind up getting an imaginary answer. My calculator comes up with errors.

[tex]x=\frac{-x\pm\sqrt{-3x^2-40}}{2}[/tex]

(I assume you meant to type "y" on the left side.) Should be + 40, not - 40. Underneath the square root you should have set up
[tex]\sqrt{x^2 - 4(1)(x^2 - 10)}[/tex]
69
 
Last edited:
  • #5
eumyang said:
(I assume you meant to type "y" on the left side.) Should be + 40, not - 40. Underneath the square root you should have set up
[tex]\sqrt{x^2 - 4(1)(x^2 - 10)}[/tex]



69

Thanks for all your help. It works fine now.
 

1. How do I graph a rotated conic on a graphing calculator?

To graph a rotated conic on a graphing calculator, you will need to use the "rotate" function, which can usually be found in the "transform" or "edit" menu. You will need to enter the center of rotation and the angle of rotation, and then graph the original conic as usual. The graph should automatically rotate to the desired position.

2. What are the advantages of graphing a rotated conic on a graphing calculator?

Graphing a rotated conic on a graphing calculator allows you to easily visualize and analyze the properties of the conic. It also makes it easier to identify key points, such as the vertex or focus, and to find the equation of the conic in its rotated position.

3. Can I graph any type of conic on a graphing calculator?

Yes, most graphing calculators have the ability to graph various types of conics, including circles, ellipses, parabolas, and hyperbolas. However, depending on the model and software, some calculators may have limitations on the size and complexity of the conic that can be graphed.

4. How do I know if I have graphed the rotated conic correctly?

To ensure that you have graphed the rotated conic correctly, you can check the equation of the conic in its rotated position. You can also use the trace function on the graphing calculator to check if the points on the graph match the expected coordinates of the conic.

5. Are there any tips for graphing a rotated conic on a graphing calculator?

One useful tip is to use the zoom and window settings on the calculator to adjust the scale and view of the graph. This can help you see the rotated conic more clearly and make any necessary adjustments to the center or angle of rotation. Additionally, it is helpful to check your work by graphing the original conic and comparing it to the rotated conic to ensure accuracy.

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