Question 1196884: determine the number of intersection points between the two functions below, in terms of the parameter k. Begin with a sketch of the three unique solutions then solve to show how your solutions agrees with your sketch.
y =x (x-8) and y=2x+k
Answer by ikleyn(52771)   (Show Source):
You can put this solution on YOUR website! .
determine the number of intersection points between the two functions below,
in terms of the parameter k. Begin with a sketch of the three unique solutions
then solve to show how your solutions agrees with your sketch.
y =x (x-8) and y=2x+k
~~~~~~~~~~~~~~
First equation represents a parabola on a coordinate plane, opened up.
Second equation represents a straight line with the slope of 2 unit.
The following cases are possible:
(a) straight line intersects the parabola in two points.
(b) straight line touches the parabola (having only one common point with the parabola).
(c) straight line is located below the parabola and does not have intersection points with the parabola.
To analyze these cases, we consider the system of equations
y = x*(x-8) (1)
y = 2x + k (2)
From equations (1) and (2), we have this equation
x*(x-8) = 2x +k.
Simplify it
x^2 - 8x = 2x + k,
x^2 - 10x - k = 0.
The last equation is a standard form quadratic equation.
All the cases are controlled by the discriminant of this quadratic equation.
Referring to the general form quadratic equation, the discriminant is
d = b^2 - 4ac.
In your problem,
d = (-10)^2 - 4*1*(-k) = 100 + 4k.
Case (a) (two solutions and two intersection points) may happen if and only if the discriminant is positive
100+4k > 0, which implies k > -100/4 = -25.
Case (b) (one solution, touching) may happen if and only if the discriminant is zero
100+4k = 0, which implies k = -100/4 = -25.
Case (c) (no solutions, no intersection points) may happen if and only if the discriminant is negative
100+4k < 0, which implies k < -100/4 = -25.
Thus, you have this ANSWER
- there are two intersection point if and only if k > -25;
- there is only one common point (touching) if and only if k = -25;
- there are no intersection points if and only if k < -25.
To get plots, go to web-site www.desmos.com/calculator
and use free of charge plotting tool there.
Print your given equations there, taking appropriate numerical value of k (k= -30; k= -25; k= -20)
and obtain the required plots momentarily.
Happy learning (!)
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