Identifying Equations: Parabolas, Circles, & More

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In summary, when trying to identify an equation as a parabola, hyperbola, ellipse, circle, straight line, or none of the above, it is important to pay attention to the powers of the variables and their arrangement. It can be difficult to deduce the type of conic section from a graph, especially if there has been a linear transformation. However, there are general forms for each type of conic section that can serve as a guide. It is important to compare the given equation to these general forms and analyze any differences. It is also helpful to write down all the general forms and analyze them in order to find a pattern. It is not recommended to rely solely on a calculator, as it is important to understand the
  • #1
oray
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Homework Statement



when asked to identify an equation as a parabola, hyperbolas, ellipses, circles, straight lines, or none of the above, how can i deduce which is which?

the problems given are:

2x^2+2y^2=9 (im pretty sure this one's a circle, just by graphing it, but id like confirmation)
x^2=16-4y^2
x^2/16 + y/25 =1
3x^2 =7 + 3y^2
x/16 + y/25 = 1
x^2 = 16 - (y-3)^2
 
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  • #2
That can be difficult, especially if the graph is plotted on axes which have been translated, rotated or have undergone some linear transformation. You have to express it in the form of a matrix, find some eigenvalues and eigenvectors and then re-express the entire matrix equation in terms of the new coordinate axes. It's difficult to explain here. What do your notes say?

On the other hand, if they are plotted on the standard coordinate axes, then use this as a rough guide:

Ellipse (a circle is a special type of ellipse):

[tex]\frac{x^2}{\alpha^2} + \frac{y^2}{\beta^2} = 1[/tex]

Hyperbola:

[tex]\frac{x^2}{\alpha^2} - \frac{y^2}{\beta^2} = 1 \ \mbox{or} \ -\frac{x^2}{\alpha^2} + \frac{y^2}{\beta^2} = 1 [/tex]

Parabola:

[tex]x^2 = \alpha y \ \mbox{or} \ y^2 = \alpha x[/tex]

[tex] \alpha[/tex] and [tex]\beta[/tex] are arbitrary non-zero real numbers.
 
  • #3
Lines can have the form

y=mx+b or Ax+By+C=0

For circles: coefficients are 1

Best thing is to go thru ur book: Write down all the general forms you can find, and analyze the differences in each one. If you have all of them in front of you, it will be easier to find a pattern.
 
  • #4
ok here's what i came up with...
2x^2+2y^2=9 CIRCLE
x^2=16-4y^2 CIRLCE
x^2/16 + y/25 =1 ELLIPSE
3x^2 =7 + 3y^2 ELLIPSE
x/16 + y/25 = 1 PARABOLA.

did i get them right? i really need confirmation asap thanks for all the help guys!
 
  • #5
ALL WRONG but 1.

1st one is correct, rest is wrong.

Pay attention to whether your variable is LINEAR or QUADRATIC!
 
  • #6
rocomath said:
ALL WRONG but 1.

1st one is correct, rest is wrong.

Pay attention to whether your variable is LINEAR or QUADRATIC!

im confused. can someone walk me through this step by step?

i can keep guessing but i don't have much time and i need to figure this out. so can someone offer some guidance?

thanks :)

EDIT after looking through my textbook I've changed my answers

2x^2+2y^2=9 CIRCLE
x^2=16-4y^2 PARABOLA
x^2/16 + y/25 =1 PARABOLA
3x^2 =7 + 3y^2 HYPERBOLA
x/16 + y/25 = 1 LINE
x^2 = 16 - (y-3)^2 CIRCLE
 
Last edited:
  • #7
Did you type them in correctly? If so, then you only got the 1st one correct.

Go back and read Defender's post! Compare your answers to the general forms he gave you.
 
  • #8
i did, and edited my answers. one thing i wasnt sure about was something like this:

3x^2 =7 + 3y^2

are we assuming u subtract 3y^2 from both sides so u get:

3x^2 - 3y^2 =7

which looks like the formula for a hyperbola? anyway, someone please check my answers. thanks!
 
Last edited:
  • #9
ONCE AGAIN, read Defender's post and compare it to yours.

Before I sleep, "pay close attention" to the powers of your variables, this APPLIES to both x and y.

If x and y are both linear, then it's a line. If x and y are both quadratic, then it's either a circle, ellipse, or hyperbola.

If x or y is either linear or quadratic, vice versa, then it's a parabola.

... read your book and quit using your lame calculator, gnite and goodluck
 
Last edited:
  • #10
AH got it.

2x^2+2y^2=9
x^2=16-4y^2
x^2/16 + y/25 =1
3x^2 =7 + 3y^2
x/16 + y/25 = 1
x^2 = 16 - (y-3)^2
circle.
circle.
parabola.
hyperbola.
line.
circle.
brilliant!
 
  • #11
2nd one still wrong, rest good.
 
  • #12
rocomath said:
2nd one still wrong, rest good.

how? it has 2 exponents meaning its a hyperbola, circle, or ellipse, and the denominators are the same making it a circle? right?
 
  • #13
[tex]x^2 = 16 - 4y^2[/tex]

The denominators are not the same. Yes, it has two square terms.

Move all the terms to the LHS, and try to make the RHS 1. Now look at the possible conic section equations given above. Which type is your equation?
 

1. What is the difference between a parabola and a circle?

A parabola is a curved shape that is created by graphing a quadratic equation, while a circle is a closed shape that is created by graphing an equation in which both the x and y variables are squared.

2. How can I determine the shape of an equation?

To determine the shape of an equation, you can look at the exponents of the variables. For example, an equation with an exponent of 2 will create a parabola, while an equation with both x and y variables squared will create a circle.

3. What are some real-world applications of parabolas and circles?

Parabolas are commonly used in physics to describe the trajectory of a projectile, while circles are used in many engineering and design applications, such as creating circular structures or calculating the circumference of a circle.

4. How can I identify the vertex of a parabola?

The vertex of a parabola is the point where the parabola changes direction. This point can be found by using the formula (-b/2a, f(-b/2a)), where a and b are the coefficients of the quadratic equation.

5. Can an equation have both a parabola and a circle?

No, an equation can only represent one shape at a time. If an equation contains both squared variables and non-squared variables, it cannot be graphed as a parabola or a circle.

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