Unit 6 Worksheet 22 Graphing Tangent Functions

Graphing tangent functions might seem daunting at first, but understanding their unique characteristics and how they differ from sine and cosine functions makes the process manageable and even insightful. The tangent function, represented as tan(x), plays a crucial role in trigonometry, calculus, and various fields of engineering and physics. This comprehensive guide will walk you through the intricacies of graphing tangent functions, ensuring you master the skills needed to tackle Unit 6 Worksheet 22 and beyond.

Understanding the Tangent Function

Before we dive into graphing, let's solidify our understanding of the tangent function itself. Remember that the tangent function is defined as the ratio of the sine to the cosine of an angle:

tan(x) = sin(x) / cos(x)

This definition has profound implications for the graph of the tangent function:

• Periodicity: Unlike sine and cosine, which have a period of 2π, the tangent function has a period of π. This means the graph repeats itself every π units along the x-axis.

• Vertical Asymptotes: The tangent function is undefined where cos(x) = 0. This occurs at x = π/2 + nπ, where n is an integer. At these points, the graph of the tangent function has vertical asymptotes.

• Range: The range of the tangent function is all real numbers, meaning it extends from negative infinity to positive infinity.

• Key Points: Knowing the tangent values at key angles is crucial. For example:

  • tan(0) = 0
  • tan(π/4) = 1
  • tan(-π/4) = -1

The Basic Tangent Function: y = tan(x)

The simplest form of the tangent function is y = tan(x). Let's outline its key features:

• Period: π

• Vertical Asymptotes: x = π/2 + nπ (e.g., x = -π/2, x = π/2, x = 3π/2)

• x-intercepts: x = nπ (e.g., x = -π, x = 0, x = π)

• Key Points:

  • Between x = -π/2 and x = π/2:
    • At x = -π/4, y = -1
    • At x = 0, y = 0
    • At x = π/4, y = 1

Graphing y = tan(x):

1. Draw Vertical Asymptotes: Start by drawing vertical dashed lines at x = -π/2, x = π/2, x = 3π/2, and so on. These lines indicate where the function is undefined.

2. Plot Key Points: Plot the key points mentioned above, such as (-π/4, -1), (0, 0), and (π/4, 1).

3. Sketch the Curve: Remember that the tangent function increases from negative infinity to positive infinity between each pair of asymptotes. Sketch a smooth curve that passes through the key points and approaches the asymptotes without touching them.

4. Repeat the Pattern: Repeat this pattern for each interval of length π to complete the graph.

Transformations of the Tangent Function

Now, let's explore how to graph more complex tangent functions involving transformations. The general form of a transformed tangent function is:

y = A tan(B(x - C)) + D

Where:

• A (Amplitude): Although tangent functions don't have a true amplitude like sine and cosine (because their range is infinite), A vertically stretches or compresses the graph. A negative A also reflects the graph across the x-axis.

• B (Period Change): B affects the period of the function. The new period is given by π/|B|.

• C (Phase Shift): C shifts the graph horizontally. A positive C shifts the graph to the right, and a negative C shifts the graph to the left.

• D (Vertical Shift): D shifts the graph vertically. A positive D shifts the graph upward, and a negative D shifts the graph downward.

Let's break down how each transformation affects the graph.

1. Vertical Stretch/Compression and Reflection (A)

• |A| > 1: Vertical stretch. The graph becomes steeper.

• 0 < |A| < 1: Vertical compression. The graph becomes less steep.

• A < 0: Reflection across the x-axis. The graph is flipped upside down.

Example: y = 2 tan(x)

• A = 2, so the graph is vertically stretched. It's steeper than y = tan(x).

Example: y = -tan(x)

• A = -1, so the graph is reflected across the x-axis.

2. Period Change (B)

• |B| > 1: Period decreases. The graph is compressed horizontally.

• 0 < |B| < 1: Period increases. The graph is stretched horizontally.

The new period is π/|B|. The vertical asymptotes change accordingly.

Example: y = tan(2x)

• B = 2, so the period is π/2. The asymptotes are now at x = π/4 + n(π/2).

Example: y = tan(x/2)

• B = 1/2, so the period is 2π. The asymptotes are now at x = π + 2nπ.

3. Phase Shift (C)

The phase shift C shifts the entire graph horizontally.

• C > 0: Shift to the right by C units.

• C < 0: Shift to the left by |C| units.

Example: y = tan(x - π/4)

• C = π/4, so the graph is shifted π/4 units to the right.

Example: y = tan(x + π/2)

• C = -π/2, so the graph is shifted π/2 units to the left.

4. Vertical Shift (D)

The vertical shift D moves the entire graph up or down.

• D > 0: Shift upward by D units.

• D < 0: Shift downward by |D| units.

Example: y = tan(x) + 1

• D = 1, so the graph is shifted 1 unit upward.

Example: y = tan(x) - 2

• D = -2, so the graph is shifted 2 units downward.

Steps to Graphing Transformed Tangent Functions

Here’s a step-by-step guide to graphing transformed tangent functions:

1. Identify A, B, C, and D: From the equation y = A tan(B(x - C)) + D, identify the values of A, B, C, and D.

2. Determine the Period: Calculate the new period using the formula Period = π/|B|.

3. Find the Vertical Asymptotes:

   • The basic asymptotes of y = tan(Bx) are at Bx = -π/2 and Bx = π/2.
   • Solve for x to find the asymptotes: x = -π/(2B) and x = π/(2B).
   • Account for the phase shift C: The asymptotes become x = -π/(2B) + C and x = π/(2B) + C. Add multiples of the period to find additional asymptotes as needed.

4. Determine the Phase Shift and Vertical Shift: Note the values of C and D. These will shift your entire graph horizontally and vertically, respectively.

5. Find Key Points:

   • Find the midpoint between the asymptotes. This is where the transformed tangent function will have a value of D (the vertical shift).
   • Find the points halfway between the midpoint and each asymptote. At these points, the y-value will be D + A and D - A.
   • In other words, consider these three x-values within one period:
     • x₁ = C (Midpoint: y = D)
     • x₂ = C - π/(4B) (y = D - A)
     • x₃ = C + π/(4B) (y = D + A)

6. Sketch the Graph:

   • Draw the vertical asymptotes as dashed lines.
   • Plot the key points calculated in step 5.
   • Sketch the tangent curve between the asymptotes, passing through the key points. Remember that the tangent function increases (or decreases if A is negative) from negative infinity to positive infinity.
   • Repeat the pattern for additional periods as needed.

Example Problems and Solutions

Let’s work through some example problems to solidify your understanding.

**Example 1: Graph y = 3 tan(x) **

1. Identify A, B, C, and D:

   • A = 3
   • B = 1
   • C = 0
   • D = 0

2. Determine the Period:

   • Period = π/|1| = π

3. Find the Vertical Asymptotes:

   • x = -π/2 and x = π/2

4. Determine the Phase Shift and Vertical Shift:

   • No phase shift (C = 0)
   • No vertical shift (D = 0)

5. Find Key Points:

   • Midpoint: x = 0, y = 0
   • x = -π/4, y = -3
   • x = π/4, y = 3

6. Sketch the Graph: Draw the asymptotes at x = -π/2 and x = π/2. Plot the points (-π/4, -3), (0, 0), and (π/4, 3). Sketch the curve, remembering that it approaches the asymptotes.

**Example 2: Graph y = tan(2x - π/2) **

First, rewrite the equation in the standard form: y = tan(2(x - π/4))

1. Identify A, B, C, and D:

   • A = 1
   • B = 2
   • C = π/4
   • D = 0

2. Determine the Period:

   • Period = π/|2| = π/2

3. Find the Vertical Asymptotes:

   • x = -π/(22) + π/4 = 0*
   • x = π/(22) + π/4 = π/2*

4. Determine the Phase Shift and Vertical Shift:

   • Phase shift: π/4 to the right
   • No vertical shift (D = 0)

5. Find Key Points:

   • Midpoint: x = π/4, y = 0
   • x = 0, y = -1
   • x = π/2, y = 1

6. Sketch the Graph: Draw the asymptotes at x = 0 and x = π/2. Plot the points (0, -1), (π/4, 0), and (π/2, 1). Sketch the curve.

**Example 3: Graph y = -2 tan(x + π/4) + 1 **

1. Identify A, B, C, and D:

   • A = -2
   • B = 1
   • C = -π/4
   • D = 1

2. Determine the Period:

   • Period = π/|1| = π

3. Find the Vertical Asymptotes:

   • x = -π/(21) - π/4 = -3π/4*
   • x = π/(21) - π/4 = π/4*

4. Determine the Phase Shift and Vertical Shift:

   • Phase shift: π/4 to the left
   • Vertical shift: 1 unit upward

5. Find Key Points:

   • Midpoint: x = -π/4, y = 1
   • x = -3π/4, y = 3
   • x = π/4, y = -1

6. Sketch the Graph: Draw the asymptotes at x = -3π/4 and x = π/4. Plot the points (-3π/4, 3), (-π/4, 1), and (π/4, -1). Sketch the curve, remembering that the negative A reflects the graph.

Common Mistakes to Avoid

• Incorrectly Identifying A, B, C, and D: This is the most common source of errors. Double-check your values before proceeding. Remember to factor out B if necessary.

• Incorrectly Calculating the Period: Always use the formula Period = π/|B|.

• Incorrectly Finding the Asymptotes: Make sure to solve for x after setting Bx - C equal to ±π/2.

• Forgetting the Phase Shift: The phase shift can significantly alter the position of the graph.

• Not Accounting for Vertical Shifts: Vertical shifts can be easily overlooked. Remember to shift the entire graph up or down by D units.

• Sketching the Wrong Direction for Negative A: If A is negative, the graph is reflected across the x-axis and decreases from left to right between the asymptotes.

Practical Applications of Tangent Functions

Tangent functions aren't just abstract mathematical concepts; they have practical applications in various fields:

• Navigation: Tangent functions are used in triangulation to determine distances and positions.

• Physics: They appear in calculations involving angles of elevation and depression, projectile motion, and optics.

• Engineering: Civil engineers use tangent functions to calculate slopes and angles in road and bridge construction. Electrical engineers use them in circuit analysis.

• Computer Graphics: Tangent functions are used in creating realistic perspectives and lighting effects.

Conclusion

Graphing tangent functions involves understanding the basic properties of the tan(x) function and how transformations affect its graph. By correctly identifying A, B, C, and D, calculating the period and asymptotes, and plotting key points, you can accurately sketch transformed tangent functions. Remember to practice regularly and pay attention to common mistakes to solidify your skills. Mastering these concepts will not only help you excel in your math courses but also provide you with valuable tools for various real-world applications. Tackle Unit 6 Worksheet 22 with confidence, knowing you have a solid grasp of graphing tangent functions.