write an equation of the tangent function with a period 3pi and a vertical translation of 5 units down
The period of tan(kx) is π/k, so
y = tan(1/3 x) - 5
To write the equation of the tangent function with a period of 3π and a vertical translation of 5 units down, we need to consider the general equation of the tangent function and then modify it accordingly.
The general equation of the tangent function is:
y = A * tan(Bx + C) + D
Where:
- A is the amplitude
- B determines the period
- C shifts the function horizontally
- D shifts the function vertically
In this case, the given period is 3π. The period of the tangent function is π, so we need to modify B accordingly. The period is given by the formula:
Period = 2π / |B|
From the given information, Period = 3π, which implies:
3π = 2π / |B|
To find the value of B, we can rearrange the equation:
|B| = 2π / (3π)
|B| = 2/3
Since B determines the period, we want the positive value:
B = 2/3
So, our equation becomes:
y = A * tan((2/3)x + C) + D
Now, we also need to account for the vertical translation of 5 units down. D represents the vertical shift, so in this case, D = -5. The equation becomes:
y = A * tan((2/3)x + C) - 5
Note that we don't have enough information to determine the amplitude or the horizontal shift, so A and C remain variables.