The graphs of y=3-x^2+x^3 and y=1+x^2+x^3 intersect in multiple points. Find the maximum difference between the y-coordinates of these intersection points.
let's find their intersection points
1+x^2+x^3 = 3-x^2+x^3
2x^2 = 2
x^2 = 1
x = ± 1
if x = 1, y =1+2+1 = 3
if x = -1, y = 1 + 1 - 1 = 1
They intersect at (1,3) and (1,1)
so what is the difference between their y values ?
To find the maximum difference between the y-coordinates of the intersection points of the two graphs, we need to first find the x-coordinates of these points and then calculate the corresponding y-coordinates.
To find the x-coordinates, we set the two equations equal to each other and solve for x:
3 - x^2 + x^3 = 1 + x^2 + x^3
By rearranging terms and canceling out common terms:
2 - 2x^2 = 0
Now we solve for x by isolating the variable:
2x^2 = 2
x^2 = 1
x = ±1
So the x-coordinates of the intersection points are x = -1 and x = 1.
Next, we substitute these x-values back into either equation to find the corresponding y-coordinates.
For y = 3 - x^2 + x^3:
When x = -1:
y = 3 - (-1)^2 + (-1)^3 = 3 - 1 - 1 = 1
When x = 1:
y = 3 - (1)^2 + (1)^3 = 3 - 1 + 1 = 3
For y = 1 + x^2 + x^3:
When x = -1:
y = 1 + (-1)^2 + (-1)^3 = 1 + 1 - 1 = 1
When x = 1:
y = 1 + (1)^2 + (1)^3 = 1 + 1 + 1 = 3
So, the y-coordinates of the intersection points are y = 1 and y = 3.
Now we can find the maximum difference between the y-coordinates:
Max difference = |3 - 1| = 2
Therefore, the maximum difference between the y-coordinates of the intersection points is 2.