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The cosine function is one of the fundamental trigonometric functions, often written as \(f(x) = \cos x\). It originates from the unit circle, where it represents the x-coordinate of a point on a circle of radius 1 as the angle moves around the circle. It's a periodic function, meaning it repeats its values in regular intervals or periods. Its standard period is \(2\pi\), meaning after an interval of \(2\pi\), the function starts again from the same point.

• The amplitude, or height of the wave, is always between -1 and 1 for the standard cosine function.
• Its graph starts at (0, 1), drops to (\(\pi\), -1), and returns to (\(2\pi\), 1), indicating one full cycle.
• The maximum points are at the beginning and end of the cycle, and the minimum point is right in the middle.

Understanding the standard behavior of cosine functions helps us when modifications are applied, like period changes and vertical shifts. This foundation is crucial for effectively maneuvering through trigonometric graph transformations.

The period of a trigonometric function is the length required for the function to repeat its pattern. For the cosine function, the standard period is \(2\pi\). However, adjustments to the function's formula can alter this period.A period change occurs when the variable inside the cosine function is multiplied by a coefficient. Such a transformation is seen in our exercise where the function \(f(x) = \cos \pi x\) results in a period change:

• The coefficient \(\pi\) compresses the period by the same factor.
• To calculate the adjusted period, use the formula \(\frac{2\pi}{b}\), where \(b\) is the coefficient inside the trigonometric function.
• In our case, \(b = \pi\), so the period is \(\frac{2\pi}{\pi} = 2\).

This means the function's cycle completes in an interval of length 2. Thus, visually it appears more tightly packed compared to the standard cosine, showing how coefficients affect periodicity.

A vertical shift moves the entire graph of a function up or down without affecting its shape or horizontal position. This transformation is applied to a function \(y = a + f(x)\) where \(a\) is the amount of shift.In our exercise, the function \(g(x) = 1 + \cos \pi x\) represents a vertical shift:

• The graph of \(f(x) = \cos \pi x\) is shifted vertically upwards by 1 unit.
• This means every y-value on the graph increases by 1.
• If the original graph, \(f(x)\), passed through (0, 1), the new graph, \(g(x)\), will pass through (0, 2).

Vertical shifts are handy for adjusting the position of the function to better fit certain conditions or represent specific data. They are straightforward but impactful changes to any cosine or trigonometric graph.