91Ó°ÊÓ

The period of a function tells us how often the function repeats itself. For tangent functions, this is a straightforward concept. The standard period for the tangent function \( y = \tan(x) \) is \( \pi \). However, our problem involves an altered tangent function: \( y = -2 \tan(2x - \frac{\pi}{3}) \). To find the period of such a function, we use the formula \( \frac{\pi}{B} \), where \( B \) is the coefficient of \( x \) inside the function.

In this function, \( B = 2 \). Therefore, substituting the value in the formula gives us:

• Period = \( \frac{\pi}{2} \)

This means the function repeats itself every \( \frac{\pi}{2} \) units along the x-axis. The reduced period compared to the standard tangent function is due to the horizontal compression caused by the \( 2 \) factor.

Phase shift refers to the horizontal movement of a function along the x-axis. It's influenced by the \( C \) value in the transformation form \( y = A \tan(Bx - C) + D \). For the given function \( y = -2 \tan(2x - \frac{\pi}{3})\), the phase shift is calculated by \( \frac{C}{B} \).

Here, we have:

• \( C = \frac{\pi}{3} \)
• \( B = 2 \)

Thus, substituting the values, the phase shift is \( \frac{\frac{\pi}{3}}{2} = \frac{\pi}{6} \).

The positive result indicates that the graph of our tangent function shifts \( \frac{\pi}{6} \) units to the right. It's important for locating the starting point of the function's cycle on the graph.

The vertical stretch in a tangent function affects how much the graph stretches or compresses vertically by a factor determined by \( A \). If \( |A| > 1 \), it means the function's graph is stretched, making it taller, while \( |A| < 1 \) would compress it, making it shorter.

For our function \( y = -2 \tan(2x - \frac{\pi}{3}) \), \( A = -2 \), giving a stretch factor of \( |A| = 2 \).

• This means every point on the tangent curve is twice as far from the x-axis as it would be in the parent graph of \( y = \tan(x) \).

The effect of vertical stretching can make the curve appear steeper as peaks and valleys are exaggerated.

Reflection is one way a function's graph can be transformed. In particular, a reflection across the x-axis inverts all y-values of the function. This occurs in our function due to the negative sign in the \( A \) value.

Given the function \( y = -2 \tan(2x - \frac{\pi}{3}) \), the negative sign before the 2 (i.e., \( A = -2 \)) indicates:

• All positive y-values of a corresponding point in the parent graph become negative.
• All negative y-values of a corresponding point in the parent graph become positive.

This can easily be visualized as flipping the graph of the tangent function upside down, creating an inverted image about the x-axis. It's a key step in accurately sketching transformed trigonometric functions like this one.