# What is the angle of rotation for a parabola given its vertex and equation?

• chaoseverlasting
In summary: This is the same as the angle of rotation of the original parabola. If you can follow what I have done then you should be able to solve the problem. In summary, the conversation discusses finding the equation of the axis of a parabola with a vertex at (-3,1) and touching the x-axis at the origin. Various methods and approaches are suggested, including using the properties of parabolas and solving geometric equations. Ultimately, it is concluded that the equation of the axis is y = -1/9(x+3)^2 + 1, taking into account the tilt of the parabola.
chaoseverlasting
If someone could tell me how to go about it, I'd really appreciate it. Just point me in the right direction please. Kinda stuck.

The vertex of a parabola is (-3,1) and it touches the x-axis at the origin. Find the equation of the axis of the parabola.

chaoseverlasting said:
If someone could tell me how to go about it, I'd really appreciate it. Just point me in the right direction please. Kinda stuck.

The vertex of a parabola is (-3,1) and it touches the x-axis at the origin. Find the equation of the axis of the parabola.

Did you mean, the parabola intersects the x-axis at the origin? Or am I missing something?

No, the x-axis is the tangent to the parabola at the origin. I don't know what to do with it.

I don't think that possible.

Couldnt it be a tilted parabola of the type $$y^2=4ax$$ so that its lower arm touches the x-axis at the origin?

yes it could. In that case, it would be opening to the right.

So how would you find its axis?

it would simply be y = 1.

Why would it be y=1?

Since the axis has to go through the parabola's vertex, and must be parallell to the x-axis.

Why would the axis of the parabola have to parallel to the x-axis? We are told that the x-axis is tangent to the parabola at (0,0). No line parallel to the axis of a parabola is ever tangent to it.

THe x-axis is tangent to the parabola at the origin.

The symmetry axis of the parabola is parallell to the x-axis, because the the tangent to the vertex is parallell to the y-axis.
The symmetry axis of a parabola is perpendicular to the tangent at the vertex.

It doesn't say the tangent at the vertex is parallel to the y-axis. You don't know the equation so you cannot assume that. the parabola could be tiled so that the axis of the parabola makes some angle with the x-axis and one of the arms just touches the x-axis at the origin.

arildno said:
THe x-axis is tangent to the parabola at the origin.
Yes, that is given.

The symmetry axis of the parabola is parallell to the x-axis, because the the tangent to the vertex is parallell to the y-axis.
Where was that said?

The symmetry axis of a parabola is perpendicular to the tangent at the vertex.

Where do you begin on something like this?

I'm sorry, I was wrong.

chaoseverlasting said:
If someone could tell me how to go about it, I'd really appreciate it. Just point me in the right direction please. Kinda stuck.

The vertex of a parabola is (-3,1) and it touches the x-axis at the origin. Find the equation of the axis of the parabola.

Well, if it is a parabola, the equation is of the form y = f(x) = ax^2+bx+c

Now, f(-3)=1 and

f(0)=0

Finally, f'(0)=0

Also, since f(x)= ax^2+bx+c and f(0)=0 c must also equal zero.

EDIT: I think I'm wrong again (as usual). I guess the equation of a parabola does not have to be of the form ax^2+bx+c=f(x)

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Could it be that the question is incomplete?

I think the equation should be y=a(x-h)^2+k
Since (-3,1) is the vertex h=-3 and k=1
we get y=a(x+3)^2+1
If x=0 and y=0
9a+1=0 a=-1/9 y=1-((x+3)^2)/9

Yes, that would be true if the axis were parallel to the x-axis, but it isn't so. The parabola is tilted.

accidental double post.

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You can construct a geometric solution by using the property of a parabola that the gradient (tan of angle) of the "vertex to point" chord is equal to one half of the gradient of the tangent at the point.

See if you can follow through on this and after a bit of basic geometry you should be able to get an equation (based on the tan of sum/difference formula) of something like :

$$\frac{\tan(\theta)}{1 + 2 \tan^2(\theta)} = \frac{1}{3}$$

Actually that is the solution for the angle whose tan is one half that of the required angle of rotation, but it's going to be pretty easy if you can get to this point.

Note that the above is a quadratic so there are actually two solutions. One solution has smaller rotation (measured from vertical) and a longer focal length whereas the second solution has greater rotation (axis is closer to the x-axis) and a shorter focal length.

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Sorry, you lost me. The slope of the chord from a point to the vertex is equal to half the slope of the tangent at that point?

chaoseverlasting said:
Sorry, you lost me. The slope of the chord from a point to the vertex is equal to half the slope of the tangent at that point?

Yeah that's correct for a parabola of the usual form (x^2=4ay). Say you denote the vertex point "O" and some general point on the parabola as point "P", then the gradient of the straight line joining "O" and "P" is one half of the gradient of the tangent at "P". This a well known property of a parabola that's very easily proven.

Take a look at the attachment. This diagram shows the parabola after its been rotated back to a vertical axis. Your goal is to determine the angle PRQ though which we have rotated it to make it vertical. This of course will be the original angle of the axis (measured from vertical) of the original given parabola.

In this diagram we know the following facts.
1. tan(PRQ) = 2 tan(POR)
2. PS = 3
3. OS = 1
4. OSP is a right angle.

I used the above information and some geometry to solve for tan(POR) and hence to find angle PRQ.

#### Attachments

• parabola_dxf.jpg
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## 1. What is the axis of symmetry of a parabola?

The axis of symmetry of a parabola is a vertical line that divides the parabola into two equal halves. It is also the line where the parabola is symmetrical or mirror-like.

## 2. How do you find the axis of a parabola?

The axis of a parabola can be found by using the formula x = -b/2a, where a and b are the coefficients of the quadratic equation. This formula gives the x-coordinate of the vertex, which is also the axis of symmetry.

## 3. What is the significance of the axis of symmetry in a parabola?

The axis of symmetry is significant because it helps us identify the vertex of the parabola, which is the highest or lowest point on the curve. It also helps us graph the parabola accurately.

## 4. Can the axis of symmetry be a horizontal line?

No, the axis of symmetry of a parabola is always a vertical line. This is because the parabola is symmetrical along a vertical line, and its equation is in the form of y = ax^2 + bx + c.

## 5. How does the axis of symmetry affect the roots of a parabola?

The axis of symmetry has no direct effect on the roots of a parabola. However, it does help us determine the x-coordinate of the vertex, which is also the x-coordinate of the roots if the parabola intersects the x-axis. Additionally, the roots are equidistant from the axis of symmetry.

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