Problem 33 - Algebra 2

Question

State the vertical shift, amplitude, period, and phase shift for each function. Then graph the function. $$ y=\frac{2}{5} \tan \left(6 \theta+135^{\circ}\right)-4 $$

Step-by-Step Solution

Answer

Vertical shift: 4 units down. Amplitude: N/A. Period: \(\frac{\pi}{6}\). Phase shift: 22.5 degrees left.

Step 1: Identify the Vertical Shift

The vertical shift of a trigonometric function is determined by the constant term added or subtracted from the function. In the function \( y = \frac{2}{5} \tan(6\theta + 135^{\circ}) - 4 \), the constant term is \(-4\). Therefore, the vertical shift is 4 units downward.

Step 2: Determine the Amplitude

The amplitude of a tangent function is typically not defined in the same way as for sine and cosine functions, since tangent does not have a maximum or minimum value. Therefore, the concept of amplitude doesn't apply here. However, the multiplier \( \frac{2}{5} \) affects the steepness of the tangent graph, modifying its stretch.

Step 3: Calculate the Period

The period of the tangent function \( \tan(k\theta) \) is given by \( \frac{\pi}{k} \). For this function, \( k = 6 \), so the period is \( \frac{\pi}{6} \).

Step 4: Determine the Phase Shift

The phase shift is determined by the horizontal shift in the function. This is given by solving \( 6\theta + 135^{\circ} = 0 \). Solve for \( \theta \):\[ 6\theta = -135^{\circ} \]\[ \theta = -\frac{135^{\circ}}{6} = -22.5^{\circ} \]The phase shift is -22.5 degrees, which indicates a shift 22.5 degrees to the left.

Step 5: Graph the Function

Graphing the function involves plotting key points and changes per the identified shifts and period. You will mark the origin point shifted vertically by 4 units downwards, then plot the new period as \(\frac{\pi}{6}\), and incorporate the phase shift of 22.5º left. Sketch the tangent curve, ensuring that it is stretched according to the factor \(\frac{2}{5}\), which makes it less steep than \(\tan\theta\).

Key Concepts

Vertical Shift

A vertical shift in a trigonometric function involves moving the entire graph up or down along the y-axis. For the given function \( y = \frac{2}{5} \tan(6\theta + 135^{\circ}) - 4 \), the term \( -4 \) indicates a vertical shift. ● The shift is in the downward direction.
● This affects the position of the whole graph but not its shape.
The vertical shift simply lowers or raises the baseline of the curve.
When analyzing such functions, pay attention to this constant term; it tells us how much the graph moves vertically.

Period of the Trigonometric Functions

The period is the interval over which a periodic function, like a tangent, repeats its values. In tangent functions, the period is influenced by the coefficient of the variable inside the tangent function. The standard period for \( \tan \theta \) is \( \pi \).

To find the period:

• Identify the coefficient \( k \) in the expression \( \tan(k\theta) \).
• Apply the formula \( \frac{\pi}{k} \) to find the actual period.

In our function, \( k = 6 \), resulting in a period of \( \frac{\pi}{6} \).
The graph will repeat itself every \( \frac{\pi}{6} \) units, leading to more frequent oscillations due to the smaller period.

Phase Shift

The phase shift in a trigonometric function deals with horizontal movement along the x-axis. It is critically important for understanding how the function is translated. You can find the phase shift by solving the equation \( k\theta + c = 0 \), where \( c \) is the constant inside the angle of the function.

In our exercise:

• The expression becomes \( 6\theta + 135^{\circ} = 0 \).
• Solving gives \( \theta = -22.5^{\circ} \), a phase shift of 22.5 degrees left.

This leftward phase shift indicates that the entire graph moves in the negative direction by 22.5 degrees, altering the starting points of each cycle.

Tangent Graphing

Graphing the tangent function is often more complex than sine or cosine due to its inherent discontinuities (asymptotes) and periodic nature. Following these steps helps to interpret and construct its graph properly:

• Identify the vertical shift to correctly place the graph on the vertical axis.
• Use the period \( \frac{\pi}{6} \) to determine the frequency of repetition.
• Apply the phase shift to adjust where the function starts repeating.
• Account for any vertical stretch/shrink by the factor \( \frac{2}{5} \); this changes the tangent's steepness.

With these steps, you can accurately sketch the curve, remembering that tangent functions extend to infinity in their asymptotes, seen as the repeating pattern continues. Properly labeling axes and noting these transformations will lead to a successful graph.