Consider the functions f and g defined by f(x)=(x−1)2+2 and g(x)=−x2+7x−4.

Question

The graphs of f and g intersect at two points. 

Let (x1,y1), and (x2,y2) be these two points.

find the distance between the points (x1,y1) and (x2,y2)

Answer 1

Consider the functions f and g defined by f(x)=(x−1)2+2 and g(x)=−x2+7x−4.

The graphs of f and g intersect at two points. 

Let (x1,y1), and (x2,y2) be these two points.

find the distance between the points (x1,y1) and (x2,y2)

d = √([X₂-X₁)²+(Y₂-Y₁)²] * that was alot of work!

points of intersection, set equal and solve for x:

(x-1)²+2 = -x²+7x-4

x² - 2x+1 + 2 = -x²+7x-4

2x²-9x+7 = 0 *factor

(2x - 7)(x - 1) = 0

x = 7/2, x = 1

Now substitute the 2 values into either expression to get their respective y-values at the point of intersection.

(7/2-1)²+2 =

(5/2)²+2 =

25/4 + 2 =

25/4 + 8/4 =

33/4

intersection at (7/2 , 33/4)

(1-1)²+2 =

0²+2 =

2

intersection at (1,2)

Now use distance fromula:

d = √[(7/2- 1))²+(33/4-2)²]

d = √[(25/4 + 25/4)]

d = √[(50/4)]

d =(5√2)/2