Write the equation of a polynomial function that passes through A, B, C, and D.
f(x) =
Answer (2)
Assume the polynomial is a cubic function: f ( x ) = a x 3 + b x 2 + c x + d .
Substitute the coordinates of points A, B, C, and D into the cubic function to create a system of four equations.
Solve the system of equations to find the coefficients a, b, c, and d.
Since the coordinates are not provided, the general form of the polynomial is: f ( x ) = a x 3 + b x 2 + c x + d .
Explanation
Understanding the Problem The problem asks for the equation of a polynomial function that passes through four points: A, B, C, and D. However, the coordinates of these points are not provided. To find a unique polynomial, we need to make some assumptions. A common approach is to assume that we are looking for a cubic polynomial that passes through these points, assuming that the x-coordinates of the points are distinct.
Setting up the Cubic Polynomial Let's assume the coordinates of the points are A ( x 1 , y 1 ) , B ( x 2 , y 2 ) , C ( x 3 , y 3 ) , and D ( x 4 , y 4 ) . We want to find a cubic polynomial of the form f ( x ) = a x 3 + b x 2 + c x + d that passes through these points. This means that when we plug in the x-coordinate of each point into the polynomial, we should get the corresponding y-coordinate.
Creating a System of Equations We can set up a system of four linear equations with four unknowns (a, b, c, d) by substituting the coordinates of the points into the polynomial equation:
y 1 = a x 1 3 + b x 1 2 + c x 1 + d y 2 = a x 2 3 + b x 2 2 + c x 2 + d y 3 = a x 3 3 + b x 3 2 + c x 3 + d y 4 = a x 4 3 + b x 4 2 + c x 4 + d
Solving this system of equations will give us the values of a, b, c, and d, which will define our cubic polynomial.
General Form of the Polynomial Since the coordinates of the points A, B, C, and D are not given, we cannot solve for the specific values of a, b, c, and d. Therefore, the general form of the polynomial function is:
f ( x ) = a x 3 + b x 2 + c x + d
where a, b, c, and d would be determined by solving the system of equations if the coordinates were provided.
Examples
Polynomial functions are used in various fields, such as physics, engineering, and economics, to model curves and relationships between variables. For example, the trajectory of a projectile can be modeled using a quadratic polynomial, and the growth of a population can be modeled using an exponential function, which can be approximated by a polynomial over a short interval. In economics, polynomial functions can be used to model cost and revenue curves to analyze profitability and optimize production levels. Understanding how to determine the equation of a polynomial that passes through given points is crucial in these applications for accurate modeling and prediction.
To find a polynomial that passes through four points A, B, C, and D, you can assume it's a cubic function: f ( x ) = a x 3 + b x 2 + c x + d . By substituting the coordinates of the points into the polynomial, you can create a system of equations to solve for the coefficients a, b, c, and d. Without specific coordinates, the polynomial's general form remains as stated.
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