Investigating the Cosine Function
Name _______________________
Cosine function f(x) = a cos (bx + h) + v
Goal: We will investigate how the cosine function changes by changing the values of ‘a’, ‘b’, ‘h’ and ‘v’
by adjust each slider one at a time.
1. Open file GV_cos2.ggb
2. Note the parent function f(x) = cos(x) will remain visible as a dash line so you can compare it to
the function changes by changing the parameters of ‘a’, ‘b’, ‘h’ and ‘v’.
3. Set ‘a’ = 1, ‘b’ = 1, ‘h’ = 0, and ‘v’ = 0 (Note: Do not adjust slider ‘c’)
4. Adjust the ‘a’ slider to the left and right.
a) What observation can you note as you change the values of ‘a’?
b) How does the range of the function changes as the ‘a’ amplitude changes?
c) Why does the function become the x-axis when ‘a’ equals zero? Prove algebraically why this
happens?
d) When ‘a’ and ‘b’ are non-zeros do the x-intercepts change?
e) What happens when ‘a’ is zero and ‘v’ is 2? Prove algebraically why this happens?
** Note the amplitude of the function is the absolute value of ‘a’. **
5. Set ‘a’ = 1, ‘b’ = 1, ‘h’ = 0, and ‘v’ = 0 and adjust the ‘b’ slider to the left and right.
a) The cosine function completes one full cycle from 0 to 2𝜋. How many cycles are completed
when ‘b’ is 2? When ‘b’ is 3?
b) What observation can you conclude as the values of ‘b’ changes?
c) Why do you think this occurs? (Hint the unit circle)
6. Set ‘a’ = 1, ‘b’ = 1, ‘h’ = 0, and ‘v’ = 0 and adjust the ‘h’ slider to the left and right.
a) What happens to the function as ‘h’ increases in value?
b) What happens to the function as ‘h’ decreases in value?
c) What can you concluded from your observation?
7. Set ‘a’ = 1, ‘b’ = 1, ‘h’ = 0, and ‘v’ = 0 and adjust the ‘v’ slider to the left and right.
a) What happens as ‘v’ increases in value?
b) What happens as ‘v’ decreases in value?
c) What can you concluded from your observations?
8. Sketch the following graphs, then adjust the sliders to check your answers.
a) f(x) = 2 cos(x)
b) f(x) = cos(3x)
c) f(x) = cos(x + 2)
d) f(x) = cos(x) – 1
e) f(x) = 2 cos(x) + 1
f)
9.
f(x) = ½ cos(x – 𝜋) – 1 note 𝜋 is approximately equal to 3.14 and you can’t check this one.
Move slider c to the number 1.