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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.

 Jun 8, 2019
 #1
avatar+105184 
+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

 

 

cool cool cool

 Jun 8, 2019
 #2
avatar+105184 
+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

 

 

cool cool cool

 Jun 8, 2019
 #3
avatar+105184 
+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

 

 

cool cool cool

 Jun 8, 2019

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