Finding an unknown function given the degree, leading coefficient, and two solutions for two given x values...

Question

know the question may seem a little complicated when first read, but I'll explain further. The objective is to write a function based on given information.

For f(x) for example...

If I'm given the type of equation (quadratic, cubic, quartic, etc), the leading coefficient, the constant, and the results of the evaluation of the expression for two x values (for example f(1)=a number, f(2)=another number)

And told to write a function that conforms to those requirements...how would I do this?

I think it involves systems of equations...but I'm not sure about that.

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For further clarification, here's an example of what I mean.

f is a quadratic function
Leading Coefficient = 20
Constant = 20
f(1) = 20
f(2) = 200

Write an equation that conforms

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Keep in mind that that scenario is a strict example; you don't need to solve it (it probably wouldn't work anyway haha). I just want to know how I would go about approaching a problem like this.

Sorry for the lengthy question heh

Answer 1

Hello,

Let's look at a quadratic equation. y = ax^2 + bx + c

Now the leading coefficient is a and the constant is c .
Let c = 20.
So if we know two points say f(1) = 20 and f(-1) = 10 then we put these into the equation giving us: 20 = a(1)^2 + b (1) + 20 Now the second point. 10 = a(-1)^2 + (-10) b = 20

Now you must solve these two equations.

Hope This Helps!!

Answer 2

yes it involves a system of equations.

if the 'type of equations' are polynomes of degree n
then you will need at least n pairs of (x,f(x)).