Problem 229 - Precalculus

Question

For the following exercises, find the period and horizontal shift of each function. $$ g(x)=3 \tan (6 x+42) $$

Step-by-Step Solution

Answer

Period is \( \frac{\pi}{6} \), horizontal shift is \(-7\).

Step 1: Identify the Standard Form

The standard form for the tangent function is \( f(x) = a \tan(bx + c) + d \). Comparing \( g(x) = 3 \tan (6x + 42) \) with the standard form, we can identify parameters: \( a = 3 \), \( b = 6 \), \( c = 42 \), and \( d = 0 \).

Step 2: Calculate the Period

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

Step 3: Determine the Horizontal Shift

The horizontal shift of the tangent function is calculated using the formula \( -\frac{c}{b} \). For this function, \( c = 42 \) and \( b = 6 \), so the horizontal shift is \( -\frac{42}{6} = -7 \).

Key Concepts

Horizontal Shift

A horizontal shift is a transformation that moves a function left or right along the x-axis. This is a crucial concept in understanding how functions behave across different points on the graph.
When we examine a trigonometric function, such as the tangent function, the horizontal shift can be determined using the formula:

• Horizontal Shift = \(-\frac{c}{b}\)

Let's break this down using the example function provided, \( g(x) = 3 \tan(6x + 42) \):

• Identify the values of \(c\) and \(b\) from the function. Here, \(c = 42\) and \(b = 6\).
• Insert these values into the formula: \(-\frac{42}{6} = -7\).
• The negative sign indicates the function shifts 7 units to the left.

This shift means that all values on the function's graph are moved 7 units to the left from their original positions.

Standard Form of Tangent Function

Understanding the standard form of the tangent function is key to analyzing its graph features and transformations. The standard form is expressed as:

• \(f(x) = a \tan(bx + c) + d\)

Each parameter in this equation represents a different transformation:

• \(a\): The vertical stretch or compression factor.
• \(b\): Affects the period of the function.
• \(c\): Determines the horizontal shift.
• \(d\): A vertical shift, moving the graph up or down.

For the function \(g(x) = 3 \tan(6x + 42)\), comparing it with the standard form helps us identify:

• \(a = 3\): This means the function is vertically stretched by a factor of 3.
• \(b = 6\): This will influence the function’s period.
• \(c = 42\): Indicates the horizontal shift as calculated previously.
• \(d = 0\): No vertical shift occurs.

Recognizing these components makes it much simpler to understand the function's graph.

Parameters in Trigonometric Functions

Trigonometric functions are characterized by various parameters, which significantly affect their appearance and behavior. In the broader spectrum of sinusoidal functions, these parameters allow us to customize the function for different real-world applications and mathematical analysis.

• Amplitude: Primarily relevant for sine and cosine, not tangent, but in a broader context, it's represented by \(a\).
• Period: Determined by \(b\) and influences how often the function repeats. For tangent, the formula \(\frac{\pi}{b}\) is used.
• Phase Shift (or Horizontal Shift): Given by \(-\frac{c}{b}\), affecting where the period starts.
• Vertical Shift: Defined by \(d\), which lifts or lowers the graph.

In our example, \( g(x) = 3 \tan(6x + 42) \), the parameters tell us:

• The amplitude-like effect for tangent is due to coefficient \(a = 3\).
• The function’s period, \(\frac{\pi}{6}\), indicates how frequently it repeats within \([0, 2\pi]\).
• A substantial horizontal shift of -7, shown by \(-\frac{42}{6}\).
• No vertical shift, as \(d = 0\), confirming no movement up or down.

Understanding these parameters is critical for manipulating and correctly interpreting trigonometric functions.