Help with curves please!

Question

Find the area of the region bounded by the graphs of y = x, y = -x + 4, and y = 0.

Options:

1)	1
2)	2
3)	4
4)	None of these

Answer 1

y=0 is the horizontal axis

y=x is a 45 degree line through the origin

y=-x+4 is a diagonal line going down hill, from (0,4) to (4,0)

y=x and y=-x+4 intersect at x=2, or (2,2) just set x=x=4 and solve for x which also equals y

The area is that of a triangle going from (0,0) to (2,2) to (4,0)

It helps to plot the points and sketch the two lines

those two lines are perpendicular, creating a right triangle with the x axis.

They're perpendicular because their slopes are negative inverses, 1 and -1.

y=x has slope = 1, y=-x+4 has slope = -1

the coefficient of the x term is the slope.

inverse of 1 is 1/1 = 1 and the negative inverse is -1

It has a hypotenuse of 4, the line segment of the x-axis from the origin to (4,0) and two sides of equal length 2 times square root of 2

The triangle has area 1/2 times the product of those other two sides

or (1/2) times 2sqr2 squared = 1/2 times 8 = 4

Or, when you find the 3 points of the triangle, whether it's a right triangle or not. You have (0,0), (2,2) and (0,4) You automatically have it's base = 4 That's the difference in x coordinates of (0,0) and (0,4) and you have the height =2, That's the difference between the x coordinate of (2,2) and the x-axis point directly below it at (2,0) With the height and base, multiply them together and divide by 2. 4x2/2 = 8/2 = 4