# Finding unknown coefficients of a quadratic equation describing a PDF

• RawlinsCross
In summary, the student is trying to find a, b, and c for a function that has a quadratic equation as its graph. They know that three unknowns require three equations and that one is integrating f(x) from 0 to 30. They found a, b, and c by solving the first equation and then using the second equation to determine that when x=30 f(x) = 0. They also found a third equation by differentiating the continuous probability function.
RawlinsCross

## Homework Statement

I have a probability distribution function (PDF) going fromi 0 to a on the y-axis and 0 to 30 on the x-axis. The function describing the PDF is the quadratic equation f(x) = a+bx+cx^2
I have to find a, b, and c.

## The Attempt at a Solution

I know that three unknown require 3 equations.

1. By definition the area under a PDF = 1 so one equation is intergrating f(x) from 0 to 30 and have that equal to 1. When that's worked out you get 30a+450b+9000C=1

2. I know that when x = 30 f(x) = 0. Therefore we get a+30b+900 = 0

but what is the third equation? I know that when x = 0 f(x) = a but that's only the y-intercept

Any ideas?

You expect $a$ to be the maximum value of $f(x)$ correct? doesn't that mean that the vertex of your quadratic is at $(0,a)$?...what does that make $b$?

Attached is a picture of the function. We have to determine a, b, c as I described above. I know I have two of the equations write. Based on the schematic can we be certain that the vertex is at (0,a)?

If it is and if the equation is written in the form y = a+bx+cx^2 then I can come up with a third equation.

a = -b/2c (what do you think?)

#### Attachments

• Capture1.JPG
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I won't be able to see the picture you've attached until it is approved by site admin (which sometime takes days!)

...Was that sketch given along with the problem statement, or was that just how you interpreted the question?

I would have assumed that it would look more like this:
pic

In which case, the fact that the maximum value is $a$ and it occurs at $x=0$, tells you that is where the vertex is.

Last edited:
I think I figured it out... as I tried the vertex but got lost in calculations. So I just thought some more and I think I answered it:

so we have f(x) = a+bx+cx^2 (where x goes from 0 to 30) and are asked to determine a, b, and c. The key here is we know that the function is a probability distribution function which has key characteristics:

1. The area under the curve is 1 (this gives you the first equation by integration)
2. The value of the pdf is 0 at x=30 (this is just a simple subst of x into the main equation)
3. This is the one I couldn't get. But I recalled that when you differentiate a continuous probability function you get the cumulative distribution function which always reaches 1 when the at maximum x value (30). Therefore we differentiate

F(x) (capital F denotes cdf, small f denotes pdf) = b + 2cx
F(30) = 1 = b+60x

Then I found a 3-unknown, 3 equation calculator on the web and got a = 15.1 b= -2.01 and c=0.050

Thoughts?

## 1. How can I find the unknown coefficients of a quadratic equation?

To find the unknown coefficients of a quadratic equation, you will need to have at least three points on the curve of the equation. These points can be obtained from the graph or from the given data. Once you have the points, you can set up a system of equations and solve for the unknown coefficients using algebraic methods.

## 2. Can I use calculus to find the unknown coefficients?

Yes, you can use calculus to find the unknown coefficients of a quadratic equation. You can use the derivative of the equation to find the slope of the curve at a given point, and then use the values of the points to set up a system of equations and solve for the unknown coefficients.

## 3. Is there a specific formula for finding the unknown coefficients?

Yes, there is a formula for finding the unknown coefficients of a quadratic equation. It is called the quadratic formula, which is (-b ± √(b^2-4ac)) / 2a. This formula can be used to solve for the unknown coefficients given the values of a, b, and c in the quadratic equation.

## 4. Can I use a computer program to find the unknown coefficients?

Yes, there are many computer programs and online tools available that can help you find the unknown coefficients of a quadratic equation. These programs use algorithms and numerical methods to solve for the coefficients, making the process faster and more accurate.

## 5. Are there any assumptions or limitations when finding the unknown coefficients?

Yes, there are some assumptions and limitations when finding the unknown coefficients of a quadratic equation. One of the main assumptions is that the equation accurately describes the given data or graph. Additionally, the equation must be in the standard form of ax^2 + bx + c in order for the methods to work properly.

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