Parabola equations review

In summary, the equation for the parabola with a vertex at (0,0), an axis along the x-axis, and passing through the point (-2,-4) is y^2 = 4py. This is because the parabola is symmetrical with respect to the y-axis, so the equation should be in terms of y rather than x.
  • #1
fitz_calc
41
0

Homework Statement


Write the equation for the parabola. Vertex (0,0), axis along x-axis, passes thru (-2,-4).


The Attempt at a Solution



I thought since the parabola resides on the x-axis that I was supposed to use x^2=4py, with a parabola looking similar to this:

Untitled-9.png


However, the solution is actually in the form of y^2=4py, looking like this:

Untitled2-1.png


I thought if it was along the x-axis it would look similar to the first pic? (This problem is just a refresher for the technical calc course I'm enrolled in, sorry for the crappy mspaint pics)
 
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  • #2
It seems that you have confused the plots of the parabolas with respect to their equations.
Assuming you have your axes right (x+ point to the right, y+ point up) parabolas of the form "x^2 = 4py" correspond to the 2nd image you attached, while those of the form "y^2 = 4px" correspond to the 1st one.

This is easy to see, as -in the "x^2 = 4py" case- for each y you should have too solutions for x (since x is squared), therefore the symmetry must be with respect to Y, thus the 2nd image.
 
  • #3
I think I'm confused because the problem says axis along x-axis; in this case, I thought 'along the x axis' meant the equation x^2=4py would be used. The book, however, says y^2=4py is to be used...
 
  • #4
anybody? This seems to be a very simple but I can't figure out why I cannot visualize this problem.
 
  • #5
fitz_calc said:
I think I'm confused because the problem says axis along x-axis; in this case, I thought 'along the x axis' meant the equation x^2=4py would be used. The book, however, says y^2=4py is to be used...
"axis" means "axis of the parabola". I believe the wording would have been clearer if your book said, "axis of the parabola... parallel to the x axis". "Along", as used was intended to mean "in the same direction as..."; my opinion is that it is a little more difficult to interpret.
 

Related to Parabola equations review

1. What is a parabola equation?

A parabola equation is a mathematical representation of a parabola, which is a symmetrical curve that can be found in many natural and man-made objects, such as bridges, satellite dishes, and projectiles. It is expressed in the form y = ax^2 + bx + c, where a, b, and c are constants that determine the shape, position, and orientation of the parabola.

2. How do you graph a parabola equation?

To graph a parabola equation, you can follow these steps:

  • 1. Plot the y-intercept, which is the value of c in the equation.
  • 2. Find the x-intercepts by setting y = 0 and solving for x. These points are also known as the roots or solutions of the equation.
  • 3. Use the axis of symmetry formula, x = -b/2a, to find the x-coordinate of the vertex.
  • 4. Substitute the x-coordinate of the vertex into the equation to find the y-coordinate.
  • 5. Plot the vertex and connect the points to create the parabola curve.

3. What is the significance of the coefficient a in a parabola equation?

The coefficient a determines the direction and width of the parabola. If a is positive, the parabola opens upwards and is narrow. If a is negative, the parabola opens downwards and is wide. The absolute value of a also represents the distance between the vertex and the focus of the parabola.

4. Can a parabola have more than one x-intercept?

Yes, a parabola can have zero, one, or two x-intercepts, depending on the values of the coefficients a, b, and c. If the discriminant, b^2-4ac, is positive, the parabola will have two x-intercepts. If the discriminant is zero, the parabola will have one x-intercept at the vertex. If the discriminant is negative, the parabola will have no x-intercepts.

5. How are parabola equations used in real life?

Parabola equations have many applications in real life, such as in physics, engineering, and economics. They can be used to model the trajectory of a projectile, the shape of a satellite dish, or the profit function of a business. They are also used in optics to design reflective surfaces that focus parallel light rays at a single point, such as in telescopes and solar cookers.

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