Determining the second order polynomial from the intersection points

In summary, to determine the ax^2+bx+c polynomial form based on the given points of intersection with the x and y axis, we can use the fact that the point (a, b) lies on the graph of y= f(x) if and only if b= f(a). By setting up and solving three equations using the given points, we can determine the values of a, b, and c. Additionally, we can also use the relationship between factors of a polynomial and its roots to solve the problem.
  • #1
Cinitiator
69
0

Homework Statement


Let's say that we have a second order polynomial function, and we know all of the points where it intersects with the x and y axis. Ex: (-2; 0), (0; 2), (1; 0)

How does on determine the ax^2+bx+c polynomial form based on that?

Homework Equations


-


The Attempt at a Solution



Tried searching for it on Google without any luck.
 
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  • #2
Cinitiator said:

Homework Statement


Let's say that we have a second order polynomial function, and we know all of the points where it intersects with the x and y axis. Ex: (-2; 0), (0; 2), (1; 0)

How does on determine the ax^2+bx+c polynomial form based on that?

Homework Equations


-


The Attempt at a Solution



Tried searching for it on Google without any luck.

What is a second order polynomial? What are both coordinates of the intersections?
 
  • #3
Cinitiator said:

Homework Statement


Let's say that we have a second order polynomial function, and we know all of the points where it intersects with the x and y axis. Ex: (-2; 0), (0; 2), (1; 0)

How does on determine the ax^2+bx+c polynomial form based on that?
I assume that you are saying that this is a general second order polynomial so you have answered sbj-2812's first question. You should also know that the point (a, b) lies on the graph of y= f(x) if and only if b= f(a). If (-2, 0), (0, 2), and (1, 0) are on the graph of [itex]y= ax^2+ bx+ c[/itex] then we must have [itex]0= a(-2)^2+ b(-2)+ c[/itex] or 4a- 2b+ c= 0, [itex]2= a(0)^2+ b(0)+ c[/itex] or [itex]c= 2[/itex], and [itex]0= a(1)^2+ b(1)+ c[/itex] or [itex]a+ b+ c= 0[/itex]
Solve the equations a+ b+ c= 0, c= 0, and 4a- 2b+ c= 0 for a, b, and c.

Homework Equations


-

The Attempt at a Solution



Tried searching for it on Google without any luck.
 
  • #4
Cinitiator said:

Homework Statement


Let's say that we have a second order polynomial function, and we know all of the points where it intersects with the x and y axis. Ex: (-2; 0), (0; 2), (1; 0)

How does on determine the ax^2+bx+c polynomial form based on that?

Homework Equations


-


The Attempt at a Solution



Tried searching for it on Google without any luck.

Do you know the relation between factors of a polynomial and the roots of the polynomial? If you do not, see http://www.sosmath.com/algebra/factor/fac02/fac02.html . Using the relationship makes your problem very easy. That is material well worth knowing.

RGV
 
  • #5
Cinitiator said:

Homework Statement


Let's say that we have a second order polynomial function, and we know all of the points where it intersects with the x and y axis. Ex: (-2; 0), (0; 2), (1; 0)

How does on determine the ax^2+bx+c polynomial form based on that?

Homework Equations



The Attempt at a Solution



Tried searching for it on Google without any luck.
A second order (second degree) polynomial having the form ax2+bx+c and having two real root may be written as a(x-D)(x-F).

From this it should be easy to solve your problem.
 

What is a second order polynomial?

A second order polynomial is a mathematical expression of the form ax^2 + bx + c, where a, b, and c are constants and x is the variable. It is also known as a quadratic function.

How do you determine the second order polynomial from intersection points?

To determine the second order polynomial from intersection points, you will need to have at least two points that the polynomial passes through. Then, you can set up a system of equations using the coordinates of the points and solve for the coefficients a, b, and c.

What does the second order polynomial represent?

The second order polynomial represents a curve on a graph. It is often used to model real-life situations, such as the trajectory of a thrown object or the growth of a population.

What are the possible number of intersection points for a second order polynomial?

A second order polynomial can have a maximum of two intersection points with the x-axis. However, it is possible for the polynomial to have no intersection points or to be tangent to the x-axis, resulting in only one intersection point.

What are some real-life applications of determining the second order polynomial from intersection points?

Determining the second order polynomial from intersection points is commonly used in fields such as physics, engineering, and economics. It can be used to analyze and predict the behavior of various systems, such as projectile motion, structural stability, and market trends.

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