How do you model functions to fit given conditions?

You can choose any value of a you wish and the corresponding c will give you a parabola that passes through those points. For example, if a= 1, c= 3- 1= 2, the parabola is f(x)= x2+ 2. If a= -2, c= 3- (-2)= 5, the parabola is f(x)= -2x2+ 5.
  • #1
pakmingki
93
1
for example, i want to find the coeffiecients of a function f(x) = ax^2 + bx + c for these given conditions:
f(-1) = 3
f(1) = 3

i tried plugging and chugging
for the first condition:
3 = a - b + c
3 = a + b + c

now i tried subtracting the system to get
0 = 2b
so its solved that b = 0

so now we got a step further towards the goal, and we know we can right the function as
f(x) = ax^2 + c
so now we try plugging and chugging again, but then we get
3 = a + c
3 = a + c
since they are the same equation, i don't know how we can solve for a and c.
 
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  • #2
lol you got the equation down to ax^2+c yes? Well for -1 and 1, x^2 is 1!

So now you have it down to a+c=3! No need to solve for, choose any values :D
1 and 2, 3 and 0, -4 and 7, whatever you want :D
 
  • #3
pakmingki said:
for example, i want to find the coeffiecients of a function f(x) = ax^2 + bx + c for these given conditions:
f(-1) = 3
f(1) = 3
You have an obvious problem here- your quadratic, f(x)= ax2+ bx+ c, has three coefficients and you only have two conditions. Given three points, not on a straight line, there exist a unique quadratic passing through them- but there are an infinite number of parabolas passing through two given points.

i tried plugging and chugging
for the first condition:
3 = a - b + c
3 = a + b + c

now i tried subtracting the system to get
0 = 2b
so its solved that b = 0

so now we got a step further towards the goal, and we know we can right the function as
f(x) = ax^2 + c
so now we try plugging and chugging again, but then we get
3 = a + c
3 = a + c
since they are the same equation, i don't know how we can solve for a and c.

Yes, as I said, you don't have enough conditions. You can solve for c in terms of a: c= 3- a. All of the parabolas given by f(x)= ax2+ 3- a will pass through (1, 3) and (-1, 3).
 

1. How do you determine the type of function to use for a given set of conditions?

There are several factors to consider when determining the type of function to use for a given set of conditions. These include the shape of the data, the relationship between the variables, and any known constraints or limitations. It is important to analyze the data and understand the problem before deciding on a specific function.

2. What is the process for creating a mathematical model to fit given conditions?

The process for creating a mathematical model to fit given conditions involves several steps. First, identify the variables and their relationships. Then, choose a suitable function based on the type of data and the desired outcome. Next, gather data points and plot them on a graph. Finally, adjust the parameters of the function to best fit the data points and validate the model using additional data, if available.

3. How do you handle outliers or anomalies when creating a function to fit given conditions?

Outliers or anomalies in data can greatly affect the accuracy of a mathematical model. One approach is to remove these data points and see how it affects the model. If the removed points have a significant impact, it may be necessary to use a more robust model. Another approach is to use a weighted regression where outliers are given less weight in the model. Ultimately, the best approach will depend on the specific data and the goals of the modeling.

4. Can a single function fit all types of data and conditions?

No, a single function cannot fit all types of data and conditions. Different types of data require different types of functions, and a single function may not be flexible enough to fit all types of relationships. It is important to choose the most appropriate function for the specific data and conditions to create an accurate model.

5. How do you validate the accuracy of a function that has been created to fit given conditions?

There are several methods for validating the accuracy of a function created to fit given conditions. One approach is to use a residual analysis, which compares the predicted values from the model to the actual data values. Additionally, cross-validation techniques can be used to test the model on a separate set of data. It is also important to consider the practical implications of the model and if it aligns with the desired outcomes and goals.

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