Calculus is fun! I just love whenever I solve a cute little problem, much like when you have found the solution to some programming problem.

How do I find the equation of a parabola using the slope of two tangents?

I got this question from a friend of mine who needed help on a Calculus test. At first I thought: "This is just plain easy!", but I must have been a bit rusty in Calculus, since I had to actually read up on it. Embarrassing. Anyway, here is the solution.

The pictures included are from the fine tool called Graphmatica. Remember you do not need a rocket science calculator to do Calculus! ;-) I prefer pencil, paper, a super simple calculator and a tool like Graphmatica. :-)

I will try to explain the solution to the utmost detail. You can solve the problem with a lot less explaining, but if you would like to understand the "why" in the solution, and, believe me, you will, you will have to take these steps.

The problem:

The two tangents T and S touches the parabola in T (30; 12) and S (60; y).
T's angle is 59,53 degrees and S's angle is 26,56 degrees.

Explain why the parabola's function is:

f(x) = -0,02x^2 + 2,9x - 57.

The problem illustrated:

The solution:

In Calculus, I always find it rewarding to approach these types of problems with an example based on letters. This gives me the "recipe" for solving the problem. We cannot use the standard way of finding the function of the parabola, since we only have one and a half point to deal with. So we will have to use the two tangents to find the function. This is just as good, as we know the x where the tangents touches the parabola, the key to solving the problem is to find the slopes of the tangents. Why? Read on!

A tangent is a straight line with a slope. The slope is called m and the straight line is defined by the function:

f(x) = mx +b

The slope m is defined by:

 m = (y - y0 / x - x0) 

The (y - y0 / x - x0) is also called dy / dx and is actually just another way of saying f'(x). You can say dy / dx is the derivative of y with respect to x. It is all the same. Simple huh? :-)

We know that the parabola's function is defined by:

f(x) = ax^2 + bx + c

And we know that the slope m of a tangent at any x on f(x) is the same as f'(x). In our case:

f'(x) = 2ax + b

These two functions will be our recipe for solving the problem.

Let us take a look at the information we have. The best lead on finding the slope must come from the angles provided. Since the angle says something about - well - the slope, we must be able to extract m from the angle. How to do that?

Remember the definition of the slope m?

m = dy / dx

Imagine this as a triangle:

We know that dy / dx is equal to m. But how do we connect the dots? Remember your trigonometry lessons? The tangent function says that the opposite over the adjacent is equal to tangent of the angle A. As the illustration clearly shows dy is the opposite and dx is the adjacent. Meaning that the tangent of the angle A must be equal to dy / dx, which again is equal to f'(x) and finally m. :-)

Now we know that f'(x) is also equivalent to the tangent of the tangent's angle with the x-axis. ;-) And to our luck, this angle has been provided to us.

We now have the following information:

f'(30) = 1.7 = tan(59,53)
 f'(60) = 0,5 = tan(26,56)

We also have one point on the parabola. The point where the tangent T touches the parabola and that is in (30; 12).

Now let us use our recipe functions to solve this problem.

1. x = 30 and f'(30) = 1,7 : 2a(30) + b ==> 60a + b = 1,7
2. x = 60 and f'(60) = 0,5 : 2a(60) + b ==> 120a + b = 0,5
3. The point (30, 12) in the parabola function definition ==> 900a + 30b + c = 12

Great! Now we have three functions with three unknowns.

Subtract function number two from function number one and isolate a:

-60a = 1,2
a = 1,2 / -60 = -0,02

Put this information into function number two and isolate b:

120(-0,02) + b = 0,5
-2,4 + b = 0,5
b = 0,5 + 2,4 = 2,9

Now put this information into function number three and isolate c:

900(-0,02) + 30(2,9) + c = 12
-18 + 87 + c = 12
 c = -57
 

Putting it all together leaves us with the result that the function of the parabola is:

f(x) = -0,02x^2 + 2,9x - 57

Happy programming