Question: The sine function has an amplitude of 2, a phase shift of π/6 to the right, a period of 8π, and a vertical translation of 2 units down. Write an equation for the function
The function I've made is: y = 2sin (8π(x+π/6))-2
Is it accurately based on the guidelines in the question?
Yes, the equation you provided is accurate based on the given guidelines. Here's a breakdown of each component of the equation:
Amplitude: The amplitude of 2 is correctly included as a coefficient in front of the sine function.
Phase shift: The phase shift of π/6 to the right is represented correctly by adding π/6 inside the parentheses.
Period: The period of 8π is correctly represented by the coefficient of x inside the parentheses.
Vertical translation: The vertical translation of 2 units down is correctly represented by the constant term of -2 at the end of the equation.
Overall, your equation accurately incorporates all the specified characteristics of the sine function.
Yes, the equation you have written, y = 2sin(8π(x+π/6)) - 2, accurately represents the given guidelines for the sine function. Here's an explanation of how to arrive at this equation:
1. Amplitude: The amplitude of a sine function determines the maximum distance it reaches from its midline. In this case, the amplitude is given as 2, which means the function oscillates between 2 units above and 2 units below the midline.
2. Phase Shift: The phase shift of a sine function determines the horizontal shift of the entire graph. In this case, the phase shift is given as π/6 to the right. This means that the graph is shifted to the right by π/6 units.
3. Period: The period of a sine function is the distance it takes for the pattern to repeat. In this case, the period is given as 8π, which means the graph completes one full oscillation every 8π units.
4. Vertical Translation: The vertical translation of a sine function determines the up or down shift of the entire graph. In this case, the translation is given as 2 units down, which means the graph is shifted downward by 2 units.
Putting all of this information together, the equation y = 2sin(8π(x+π/6)) - 2 accurately represents the given guidelines.
close, but no cigar
period of sin(kx) is 2π/k
so if the period is 8π,
2π/k = 8π
k = 1/4
f(x-h) shifts h to the right
y = 2 sin(1/4 (x-π/6)) - 2