A family of sinusoidal graphs with equations of the form y=a...

A family of sinusoidal graphs with equations of the form $y=a \sin b(x-c)+d$ is created by changing only the vertical displacement of the function. If the range of the original function is $\{y |-3 \leq y \leq 3, y \in \mathrm{R}\}$ determine the range of the function with each given value of $d .$ a) $d=2$ b) $d=-3$ c) $d=-10$ d) $d=8$

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Key Concepts

Vertical Displacement

Vertical displacement is the upward or downward shift of a function’s graph achieved by adding a constant (d) to the function. This transformation moves every point of the graph vertically by the same amount and changes the midline of the sinusoid without altering its shape or amplitude.

Effect on Range Due to Vertical Shift

When a vertical displacement is applied to a function, its range is translated by the same constant. For a function with an original range [m, M], adding a constant d results in a new range of [m + d, M + d]. This concept is crucial in understanding how alterations in the vertical direction affect the output values of the function.

Amplitude and Sinusoidal Function Properties

A sinusoidal function has an amplitude that represents the distance from the midline to its maximum or minimum. In cases where only the vertical displacement is altered, the amplitude and period of the sinusoid stay the same while the entire graph, including all maximum and minimum values, shifts vertically, thereby altering the range.

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