Calculate the intercepting points between f(x)=x^2+x+1 and f(x) = 2x + 1
f(x) = x^2 + x + 1
g(x) = 2x + 1
The intercepting point is where f(x) = g(x)
x^2 + x + 1 = 2x + 1
Now grouo similar terms:
==> x^2 - x = 0
Now factor x:
==> x( x-1) = 0
==> x1= 0 ==> f(x1) = f(0) = 1
==> x2= 1 ==> f(x2) = f(1) = 3
Then we have two intercepting points at:
A(0, 1) and B(1, 3)
f(x) = x^2+x+1 and f(x) = 2x+1.
To find the inerceting points.
Let us denote f(x) by y on y axis. Then at the intercepting point the y coordinate is same on both graph.
So we can write x^2+x+1 = 2x+1.
x^2+x+1-2x-1 = 0
x^2-x = 0
x(x-1) = 0
x = 0 or x=1 are x coordinatesof the point of intersection.
When x = 0 , the 2nd equatio y = 2x+1 gives y = 2*0+1 =1.
When x=1, y =2x+1 gives: y = 2*1+1 = 3.
So the interception coordinates at 2 points are: (0 ,1) and (1, 3).
To determine the intercepting points between the graphs of the given fucntions, we'll have to solve the system formed by the equations of the functions:
y = 2x + 1 (1)
y = x^2+x+1 (2)
The coordinates of the intercepting points are the solutions of the system, so they have to verify both equations.
We'll put (1)=(2):
2x + 1 = x^2+x+1
We'll eliminate like terms:
We'll put each factor as 0:
x=0 and x-1=0, x=1
Now, we'll substitute the x values in one of the 2 equation, and we'll do it in the linear equation,because it's more easy to calculate:
So the intercepting points are: A(0,1) and B(1,3).