SOLUTION: Let c be a real number. What is the maximum value of c such that the graph of the parabola y = 2x^2 has at most one point of intersection with the line y = x+c?

Algebra ->  Coordinate-system -> SOLUTION: Let c be a real number. What is the maximum value of c such that the graph of the parabola y = 2x^2 has at most one point of intersection with the line y = x+c?      Log On


   

Question 1209308: Let c be a real number. What is the maximum value of c such that the graph of the parabola y = 2x^2 has at most one point of intersection with the line y = x+c?
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

We can set those two right hand sides equal to each other
x+c = 2x^2
-2x^2+x+c = 0

We have a quadratic that fits the template ax^2+bx+c = 0
a = -2
b = 1
c = some real number constant

We'll generate exactly one solution to -2x^2+x+c = 0 when the discriminant is zero.
d = b^2-4ac = discriminant
b^2-4ac = 0
(1)^2-4(-2)c = 0
1+8c = 0
8c = -1
c = -1/8 = -0.125
This is the largest possible value of c such that x+c and 2x^2 have at most one point of intersection.
If c > -0.125 then the two curves intersect at two different locations.
If c < -0.125 then the two curves don't intersect at all.

Here's an interactive Desmos graph to explore.
https://www.desmos.com/calculator/kq2vvhcccx
Move the slider around for the parameter c to see how the red line moves.



Answer: -1/8 = -0.125