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Understanding the concept of absolute maxima is critical in various fields of mathematics, especially when graphing functions. An absolute maximum refers to the highest point over the entire domain of a function. In the context of trigonometric functions, such as the sine function, the absolute maxima are the peak points that the function reaches.
For the sine function, which oscillates between -1 and 1, the absolute maxima occur at the points where the function reaches 1. When we adjust the sine function to only take non-negative values - by taking the absolute value of the sine - this means that every peak becomes an absolute maximum. The exercise requires sketching a graph with infinitely many absolute maxima, thus the sine function is a suitable choice due to its periodic nature.
In summary, by modifying the sine function to take its absolute value, we guarantee that every wave peak becomes an absolute maximum, and these points will occur infinitely often due to the periodicity of the sine wave.

The sine function is a periodic function defined for all real numbers. Periodicity means that the function repeats its values in regular intervals, which is essential for creating a graph with infinitely many absolute maxima.
The standard sine function is given by the equation \(y = \text{sin}(x)\) and it has a range of [-1, 1]. Its graph is a smooth, continuous wave that crosses the x-axis at multiples of \(\pi\) and reaches its maximum and minimum at \(\frac{\pi}{2}\) and \(-\frac{\pi}{2}\) + n\pi, where n is any integer respectively.
When working with sine functions, it's important to note the key properties such as amplitude, period, phase shift, and vertical shift. Each of these properties can be adjusted to alter the graph of the sine function, which is particularly useful when an exercise asks you to create a function with specific characteristics.

Amplitude adjustment is a fundamental concept when graphing trigonometric functions. The amplitude of a trigonometric function is the distance from the horizontal axis to the peak of the wave. For the standard sine function, the amplitude is 1. Altering this amplitude changes how tall or short the peaks and troughs of the graph are.
By taking the absolute value of the sine function, the amplitude effectively becomes the distance from the horizontal axis to the absolute maxima of the function. This is because the absolute value reflects all negative values of the sine function to positive values, resulting in a graph that only consists of peaks.
In the exercise, the modification \(y = |\text{sin}(x)|\) maintains the amplitude consistently at 1 for all peaks, thus fulfilling the requirement of having infinitely many absolute maxima with the same height. This type of manipulation is a powerful tool to meet specific criteria in graphing exercises.

Sketching graphs of trigonometric functions involves understanding their behavior and characteristics. To sketch the graph of the adjusted sine function, you start by drawing the x and y axes. Mark the x-axis at intervals of \(\pi\), reflecting the sine function's period.
Since every wave of the sine function has been made positive by the absolute value operation, we place absolute maxima at every multiple of \(\pi\). Each of these maxima will be at a y-value of 1 due to the amplitude adjustment. By doing this, the graph will have peaks at regular intervals, illustrating the property of infinite absolute maxima.
Finally, to complete the graph, draw smooth curves connecting these maxima, ensuring that the curves remain above the x-axis throughout. This results in a visual representation of a sine wave that oscillates above the axis, with each peak representing an absolute maximum, in accordance with the given exercise.