Given are the quadratic functions *f(x) = -2x ^{2} + 4x - 2* and

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.

Guest Jun 8, 2019

#1**+2 **

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

CPhill Jun 8, 2019

#2**+2 **

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

CPhill Jun 8, 2019

#3**+2 **

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

CPhill Jun 8, 2019