Given are the quadratic functions f(x) = -2x2 + 4x - 2 and g(x) = x2 - 2x + 3. Also is given the family of lines y = -2x + p
a. At p = 0,5 there is one of the lines. This line cuts the graph of f in two points. Calculate the x coordinates of those two points of intersections.
b. There is a line that touches the graph of f. Calculate for which value of p the line touches the graph of f.
c. Calculate for which values of p the line has two points of intersection with the graph of g.
+128474 Given are the quadratic functions f(x) = -2x2 + 4x - 2 and g(x) = x2 - 2x + 3. Also is given the family of lines y = -2x + p
a. At p = 0,5 there is one of the lines. This line cuts the graph of f in two points. Calculate the x coordinates of those two points of intersections.
So we have
y = -2x + .5 and y = -2x^2 + 4x - 2
Put the first function into the second for y and we have that
-2x + .5 = -2x^2 + 4x - 2 rearrange as
2x^2 -6x + 2.5 = 0
2x^2 -6x + 5/2 = 0 multiply through by 2
4x^2 - 12x + 5 = 0 factor
(2x - 1) ( 2x - 5) = 0
Set each factor to 0 and solve for x
2x -1 =0 2x - 5 = 0
2x = 1 2x = 5
x = 1/2 x = 5/2 = 2.5
+128474 b. There is a line that touches the graph of f. Calculate for which value of p the line touches the graph of f.
y = -2x + p and y = -2x^2 + 4x - 2 set these equal
-2x + p = -2x^2 + 4x - 2 rearrange as
2x^2 -6x + (2 + p) = 0
If the line just touches the quadratic.....then the discriminant must = 0
So we have that
(-6)^2 - 4(2)(2 + p) = 0
36 - 8(2 + p) = 0
36 - 16 - 8p = 0
20 -8p =0
20 = 8p divide both sides by 8
20/8 = p = 5/2
See the graph here : https://www.desmos.com/calculator/cxek0jmo32
+128474 c. Calculate for which values of p the line has two points of intersection with the graph of g.
y = 2x + p y = x^2 - 2x + 3
x^2 - 2x + 3 = 2x + p rearrange as
x^2 - 4x + (3 - p) = 0
If the discriminant is > 0....we will have two intersection points so
(-4)^2 - 4(3 -p) > 0
16 - 12 + 4p > 0
4 + 4p > 0
1 + p > 0
p > -1
