Graph the given function over the interval [-2 p, 2 p], where p is the period of the function. y

step 1 All right, we're graphing y equals negative 5, sine of one -half x. So our amplitude is going to

Graph the given function over the interval [-2 p, 2 p],...

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function that models periodic, oscillatory behavior. Its waveform, characterized by repetitive cycles, is foundational in mathematics and applied sciences for describing waves and harmonic motion.

Amplitude

Amplitude refers to the peak value from the midline of a periodic function and indicates the extent of its vertical displacement. In transformations, a negative amplitude, such as -5, not only scales the function’s magnitude by a factor of 5 but also reflects it across the horizontal axis.

Period of a Sine Function

The period of a sine function is the length of the interval over which the function completes one entire cycle. For the standard sine function, this period is 2?. When the function’s argument is scaled, the period is altered accordingly, which is essential when graphing over specified intervals.

Horizontal Scaling and Transformation

Horizontal scaling alters the input of a function, effectively stretching or compressing its graph along the x-axis. In the function sin(½x), the factor of ½ causes the period to increase (calculated by dividing the standard period by ½), which plays a critical role in determining the function’s cycle length and the interval over which it is plotted.

Graphing Over a Specific Interval

Graphing a function over a specified interval involves plotting the function’s values across that range to capture its behavior over complete cycles. By setting the interval in terms of the function’s period (as in [-2p, 2p]), one ensures that the graph shows multiple full oscillations, aiding in the visualization of its periodic nature and transformations.

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