91Ó°ÊÓ

The tangent function, represented by the equation \( y = \tan(x) \), is a fundamental trigonometric function renowned for its periodic oscillating behavior. Unlike the sine and cosine functions which are known for their smooth and continuous wave-like patterns, the tangent function exhibits a different pattern, characterized by intersecting the x-axis at regular intervals and having vertical asymptotes where the function is undefined.

Some unique characteristics of the tangent function include:

• Intersects the x-axis at integer multiples of \( \pi \).
• The function has no maximum or minimum points, as its values extend to positive and negative infinity near its vertical asymptotes.
• Unlike sine and cosine, it is an odd function, meaning it is symmetrical about the origin.

In the context of transformations, when you have a function like \( y = \tan(bx + c ) + d \), each parameter (b, c, and d) can modify its graph: b affects the period, c causes horizontal shifts, and d results in vertical shifts.

The period of a trigonometric function is the interval over which the function's pattern repeats. For the basic tangent function \( y = \tan(x) \), this period is \( \pi \), reflecting how it repeats its vertical cycloid path every \( \pi \) units along the x-axis.

To find the period of a transformed tangent function of the form \( y = \tan(bx) \), use the formula:

• Period = \( \frac{\pi}{|b|} \), where b is the coefficient of x.

For \( y = \frac{1}{3}\tan(-2x - \pi) + 1 \), the coefficient \( b = -2 \) and thus, the period becomes \( \frac{\pi}{|-2|} = \frac{\pi}{2} \).

This reduced period signifies that the rapid oscillating cycle of the tangent occurs twice as often compared to the standard function, completing a full cycle every \( \frac{\pi}{2} \) units.

Graph transformations involve modifying the basic graph of a function to produce shifts, stretches, shrinks, or reflections. When examining \( y = \frac{1}{3} \tan(-2x - \pi) + 1 \), we see multiple transformations applied to the standard tangent function.

• Amplitude Modification: The coefficient \( \frac{1}{3} \) scales the tangent values, decreasing the steepness of rise and fall.
• Horizontal Phase Shift: Determined by solving the equation \( -2x - \pi =0 \), results in a shift to the left by \( \frac{\pi}{2} \) units.
• Vertical Shift: Occurs by adding \(+1\), lifting the entire graph up by 1 unit, thereby adjusting the centerline of the oscillation.

These transformations interplay to create a modified graph where patterns repeat more frequently, the centerline is shifted upwards, and the graph is less steep due to the coefficient \( \frac{1}{3} \). Visualizing these changes requires considering both the original function shape and the effect of each transformation layer applied.