Calculate area of rectangle

Question

Let

f(x)= lx-1l

g(x)= -lx+1l+4

Calculate the area of the rectangle marked by the 2 vertices and the 2 intersections as shown in the plot below.
(Shown in the photo in the link below.)

https://hishsaka.smugmug.com/Math/i-7kBBzVJ/

Answer 1

Looking at the graph we can see that the rectangle's vertices are at four points: A)The discontinuity point of g(x) where the line flips, B)The x-intercept of f(x), C)the intersection of the two lines right of the y-axis and D)the intersection of the two lines left of the y-axis.

1)Find these four points.

A)the absolute value of |x - 1| shifts values at -1. We can confirm this by using test points: when x < -1, |x - 1| = -(x - 1, when x > -1 |x - 1| = x - 1.

g(-1) = -|-1 + 1| + 4 = 4.

Thus point A is (-1, 4).

B)Set f(x) = |x - 1| = 0 -> x = 1. -> (1,0)

C)Set f(x) = g(x); Solving this equation will give the graph intercepts.

|x - 1| = -|x + 1| + 4.

|x - 1| + |x + 1| - 4 = 0

To solve this equation, there are three intervals of interest:

x = {(-∞,-1),(-1,1),(1,+∞)}, using test points we find that signs for |x - 1| and |x + 1|

are (-,-),(-,+),(+,+) respectively.

Looking at the graph, the intersection point to the right of the Y-axis will be to the right of point B and thus in the third interval: (1,+∞).

x - 1 + x + 1 - 4 = 0, solve for x. 2x - 4 = 0, x = 2,

plug x = 2 into both functions should yield the same result.


|2 - 1| = 1
-|2 + 1| + 4 = 1

The intersection point is (2,1).

D)The point to the left of the y-axis will be in the interval: ((-∞,-1)}. We can deduce this by looking at the graph. At point D both functions are in range where the absolute value is the negative of the phrase inside the absolute value, i.e. |x| = -x.

f(x) = |x - 1| = 1 - x
g(x) = -|x + 1| + 4 -> x + 5

1 - x = x + 5 -> 2x + 4 = 0 -> x = -2.

Plugging -2 back into the functions:

|-2 - 1| = 3

-|-2 + 1| + 4 = 3.

The intersection point D is (-2,3)

So, the four points are: A(-1,4),B(1,0),C(2,1),D(-2,3).

 

2)Find the distances between two points to find the lengths of the sides of the rectangle.

 

AD = distance from (-1,4) to (-2,3) = √(3-4)²+(-2-(-1))²) = √(10)

 

AC = distance from (-1,4) to (2,1) = √(1-4)²+(2-(-1))²) = √(18)

 

Area of Rectangle = Base * Width = √(180) = 3√(20).