Write an equation of the tangent function with period 3pi/8, phase shift -pi/5, and vertical shift -2.
Answer: y=tan(8x/3 - 8pi/15)-2
Thanks
correct, except for the missing PI in the frequency term.
You're welcome! I'm glad you found my answer helpful. And remember, if the tangent function starts telling you jokes instead of giving you answers, don't be too surprised. It's just a funny side effect of dealing with trigonometry!
To write an equation of the tangent function with a given period, phase shift, and vertical shift, you can start with the standard equation of the tangent function and apply the appropriate transformations.
The standard equation of the tangent function is y = tan(x).
To determine the period of the tangent function, you can use the formula period = 2π / b, where b is the coefficient of x in the equation. In this case, the period is given as 3π/8, so we need to find b such that 2π / b = 3π/8. Simplifying this equation, we have b = 16/3.
To determine the phase shift, you can use the formula phase shift = -c/b, where c is the constant term in the equation. In this case, the phase shift is given as -π/5, so we have -c/(16/3) = -π/5. Solving for c, we get c = 16π/15.
Therefore, the equation of the tangent function with the given period, phase shift, and vertical shift is y = tan((8/3)x - 16π/15) - 2.
However, we can simplify the equation further by expressing -16π/15 as -8π/30, which reduces to -4π/15. So, the final simplified equation is:
y = tan((8/3)x - 4π/15) - 2.
Please note that the equation assumes x is measured in radians.
To write the equation of a tangent function with a given period, phase shift, and vertical shift, follow these steps:
Step 1: Period
The period of a tangent function is given by 2π divided by the coefficient of x. In this case, the period is 3π/8. Therefore, the coefficient of x is 8/3.
Step 2: Phase Shift
The phase shift determines how the graph is shifted horizontally. A positive phase shift moves the graph to the right, and a negative phase shift moves it to the left. In this case, the phase shift is -π/5.
To account for the phase shift, you need to subtract (or add) the phase shift from the argument of the tangent function. Since the coefficient of x is 8/3, the adjusted expression for the phase shift is (8/3)x - (8/3)(π/5).
Step 3: Vertical Shift
The vertical shift determines how the graph is shifted vertically. A positive vertical shift moves the graph upward, and a negative vertical shift moves it downward. In this case, the vertical shift is -2.
Step 4: Putting it all together
Now, we can write the equation of the tangent function using the information we gathered. The equation is:
y = tan((8/3)x - (8/3)(π/5)) - 2
To simplify, you can also rewrite the equation as:
y = tan(8x/3 - 8π/15) - 2
And that's the final answer: y = tan(8x/3 - 8π/15) - 2.
Remember, it's important to understand the concepts behind the equations so that you can solve similar problems on your own.