Step 1: Find the zeros and vertical asymptotes
The zeros of the function occur when the numerator, (x - 1)(x + 1), equals 0. This happens when x = 1 and x = -1.
The vertical asymptotes occur when the denominator, (x - 2)(x + 2), equals 0. This gives us vertical asymptotes at x = 2 and x = -2.
Step 2: Test points on both sides of each zero and vertical asymptote. Identify the sign of the function at each of those values.
Let's test points on the intervals created by the zeros and vertical asymptotes.
For x < -2, we can choose x = -3. Plugging this into the function gives:
F(-3) = ((-3 - 1)(-3 + 1))/((-3 - 2)(-3 + 2)) = ((-4)(-2))/((-5)(-1)) = 8/5
This means that F(-3) is positive.
For -2 < x < -1, we can choose x = -1.5. Plugging this into the function gives:
F(-1.5) = ((-1.5 - 1)(-1.5 + 1))/((-1.5 - 2)(-1.5 + 2)) = ((-2.5)(-0.5))/((-3.5)(-0.5)) = 1/7
This means that F(-1.5) is positive.
For -1 < x < 1, we can choose x = 0. Plugging this into the function gives:
F(0) = ((0 - 1)(0 + 1))/((0 - 2)(0 + 2)) = (-1)(1)/(-2)(2) = -1/4
This means that F(0) is negative.
For 1 < x < 2, we can choose x = 1.5. Plugging this into the function gives:
F(1.5) = ((1.5 - 1)(1.5 + 1))/((1.5 - 2)(1.5 + 2)) = (0.5)(2.5)/(-0.5)(3.5) = -5/7
This means that F(1.5) is negative.
For x > 2, we can choose x = 3. Plugging this into the function gives:
F(3) = ((3 - 1)(3 + 1))/((3 - 2)(3 + 2)) = (2)(4)/(1)(5) = 8/5
This means that F(3) is positive.
Step 3: Make the sign chart on a number line. Label the zeros and vertical asymptotes and the sign of the function in between.
We can now create the sign chart on the number line:
-2 -1 1 2
-------------------------------------------
(-) (+) (-) (+) (+)
Zeros: x = -1, x = 1
Vertical asymptotes: x = -2, x = 2
This sign chart represents the intervals and the sign of the function between each value.
