In the form y = a sin(bx - c) + d, which parameter controls horizontal shift?

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Multiple Choice

In the form y = a sin(bx - c) + d, which parameter controls horizontal shift?

Explanation:
Horizontal shift comes from the phase part of the inside of the sine function. Look at y = a sin(bx − c) + d. Factor the inside: sin(bx − c) = sin[b(x − c/b)]. This shows the same wave, but shifted to the right by c/b (if c is positive; left if c is negative). So the parameter that controls the horizontal shift is c, with the actual shift amount being c divided by b. The other parameters affect the graph in different ways: a changes the amplitude, d shifts it vertically, and b changes the period (one full cycle is 2π/|b|).

Horizontal shift comes from the phase part of the inside of the sine function. Look at y = a sin(bx − c) + d. Factor the inside: sin(bx − c) = sin[b(x − c/b)]. This shows the same wave, but shifted to the right by c/b (if c is positive; left if c is negative). So the parameter that controls the horizontal shift is c, with the actual shift amount being c divided by b. The other parameters affect the graph in different ways: a changes the amplitude, d shifts it vertically, and b changes the period (one full cycle is 2π/|b|).

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