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Given are the quadratic function f(x)=-x2+6x-1 and the three lineair functions g(x)=2x-1  h(x)=2x+3 and j(x)=2x+5

a. Calculate the coordinates of the points of intersection of the graphs of functions f and g

b. Show that the graphs of function h touches the graph of function f 

c. Reason out without calculation that the graphs of function f and j don't have any points in common

d. Solve f(x) > j(x)

 Jun 8, 2019
 #1
avatar+103973 
+2

Given are the quadratic function f(x)= -x2+6x-1 and the three lineair functions g(x)=2x-1  h(x)=2x+3 and j(x)=2x+5

 

a. Calculate the coordinates of the points of intersection of the graphs of functions f and g

 

 

-x^2 + 6x - 1  = 2x - 1       rearrange as

 

0 =   x^2 -4x

 

x^2 - 4x  = 0        factor

 

x ( x - 4)  = 0

 

Set each factor to 0  and solve for x

 

x = 0         x - 4 = 0

                   x = 4

 

When x = 0,  y = 2(0) - 1  =  - 1       so  (0, -1)  is one intersection point

 

When x = 4, y = 2(4) - 1  = 7          so  (4, 7) is the other

 

 

cool cool cool

 Jun 8, 2019
 #2
avatar+103973 
+2

b. Show that the graphs of function h touches the graph of function f 

 

2x + 3  =  -x^2 + 6x - 1     rearrange as

 

x^2 - 4x + 4  = 0

 

(x + 2)^2  = 0

 

x = 2     and y = 2(2) + 3  = 7

 

So.....the function h touches f  at  (2, 7)

 

 

cool cool cool

 Jun 8, 2019
 #3
avatar+103973 
+1

c. Reason out without calculation that the graphs of function f and j don't have any points in common

 

f = -x^2 + 6x - 1         j  = 2x + 5

 

Set these equal

 

2x + 5 = -x^2 + 6x - 1   rearrange as

 

x^2 -4x + 6  = 0

 

These cannot intersect  because  the discriminant   =  (-4)^2 - 4(6)  =  16 -24  = -8

And when the discriminant is < 0.....we have no real solutions

 

 

cool cool cool

 Jun 8, 2019
 #4
avatar+103973 
+2

d. Solve f(x) > j(x)

 

-x^2 + 6x - 1  < 2x + 5      rearrange as

 

0 <   x^2 -4x + 6

 

x^2 - 4x + 6 > 0

 

This has no real solutions...so.....f(x)  is never > j(x)

 

The graph here confirms this :  https://www.desmos.com/calculator/v7jmn0b707

 

 

cool cool cool

 Jun 8, 2019

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