How Do You Correctly Derive the Equation of a Parabola with a Horizontal Axis?

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In summary, the conversation discusses the process of finding the equation of a parabola with a vertex at (h, k) and an axis parallel to the x-axis. The individual suggests shifting the origin to (h, k) and using the equation y^2 = 4ax in the new coordinate system. However, the other person points out that this will only change the location of the parabola, not its orientation. They suggest rotating the coordinate system instead. The conversation ends with a question about the desired form of the new parabola.
  • #1
Ashu2912
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Hi! I am trying to find the equation of a parabola with vertex as (h,k) and axis parallel to the x-axis. However, I am not able to derive the correct result.
(1) I shift the origin to the point (h,k).
(2) Now the equation of the parabola in the new system becomes y[itex]^{2}[/itex] = 4ax.
(3) Now, we know that when we shift the origin without rotation of axes to a point (h,k) (wrt the old system), the locus in the old system is changed by replacing x by x-h and y by y-k. This gives us the equation of the locus in the new system.
Now, since the equation of the parabola in the new system is y[itex]^{2}[/itex]=4ax, then the equation of the parabola in the old system must be (y+k)[itex]^{2}[/itex] = 4a(x+h), as on replacing x by x-h and y by y-k in this equation, we get the equation of the parabola in the new system, which is y[itex]^{2}[/itex] = 4ax.
Please help in finding the fault in this derivation. Thanks!
 
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  • #2
Do you want to transform a parabola whose axis is along the y-axis ( f(x) = 4a x^2 ) to one whose axis is parallel to the new x-axis? You will have to rotate the coordinate system, not simply move the origin. What form do you want the new parabola to take?
 

Related to How Do You Correctly Derive the Equation of a Parabola with a Horizontal Axis?

1. What is the equation of a parabola?

The equation of a parabola is a quadratic equation in the form y = ax^2 + bx + c, where a, b, and c are constants and x is the variable. It represents a U-shaped curve on a graph.

2. How do you graph a parabola using its equation?

To graph a parabola using its equation, you can plot points by substituting different values for x and solving for y. Another method is to find the vertex, axis of symmetry, and y-intercept using the formula x = -b/2a and then plotting these points on the graph.

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

The coefficient a determines the shape and direction of the parabola. If a is positive, the parabola will open upwards, and if a is negative, the parabola will open downwards. The absolute value of a also affects the steepness of the curve.

4. How do you find the vertex of a parabola from its equation?

The vertex of a parabola can be found by using the formula x = -b/2a to find the x-coordinate, and then substituting the x-coordinate into the equation to find the y-coordinate. The vertex is represented as (x,y).

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

Yes, a parabola can have up to two x-intercepts, depending on the value of the discriminant (b^2 - 4ac). If the discriminant is positive, the parabola will have two x-intercepts. If the discriminant is zero, the parabola will have one x-intercept (the vertex). If the discriminant is negative, the parabola will have no x-intercepts and will not intersect the x-axis.

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