Teacher Guide: Graphing the Tangent Function

Learn to graph y = tan(x), identify asymptotes, period, and key features of the tangent curve.

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10 questions on Trigonometric Graphs. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Graph the parent tangent function $y = \tan(x)$
Identify vertical asymptotes, period, and intercepts of tangent functions
Apply transformations to graph functions of the form $y = a\tan(bx - c) + d$
Calculate the period of transformed tangent functions
Connect the tangent graph to the unit circle and ratio definition

Prerequisites

• Understanding of sine and cosine functions
• Familiarity with the unit circle
• Knowledge of function transformations (shifts, stretches)
• Basic graphing skills on the coordinate plane

Discussion Starters

1. Why do you think the tangent function's graph looks so different from sine and cosine?
2. How can you use the unit circle to predict where the tangent function will be positive or negative?
3. What real-world quantities might produce a tangent-like relationship?
4. Why might engineers prefer expressing slopes as angles rather than ratios?

Common Misconceptions

The tangent function has an amplitude like sine and cosine

Remediation: Show that the tangent curve extends infinitely up and down. The concept of amplitude only applies to bounded periodic functions.

Asymptotes are places where the function equals infinity

Remediation: Clarify that the function is undefined at asymptotes. It approaches infinity but never equals it or crosses the line.

Differentiation Ideas

For Struggling Students:

• Start with a table of values to plot points
• Use technology to explore the graph before sketching by hand
• Focus on one period at a time with clear asymptote boundaries

For On-Level Students:

• Graph transformed tangent functions with varying parameters
• Find asymptotes and periods algebraically
• Connect graph features to the unit circle

For Advanced Students:

• Explore cotangent, secant, and cosecant graphs
• Investigate phase shifts and combined transformations
• Apply tangent functions to real-world modeling problems

Standards Alignment

F-TF.A.4 (CCSS.MATH.CONTENT.HSF.TF.A.4)
Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions
F-TF.B.5 (CCSS.MATH.CONTENT.HSF.TF.B.5)
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline

Lesson Resources

visualInteractive Tangent Graph
Explore how changing parameters affects the tangent curve
activityAsymptote Hunt
Match asymptote locations to transformed tangent functions
worksheetTangent Transformations
Practice graphing various tangent function transformations

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

The tangent function is defined as the ratio of sine to cosine:

\tan(x) = \frac{\sin(x)}{\cos(x)}

Unlike sine and cosine, the tangent function has vertical asymptotes where

\cos(x) = 0

**Key Properties of

y = \tan(x)

:**

Period: $\pi$ (not $2\pi$ like sine and cosine)
Domain: All real numbers except $x = \frac{\pi}{2} + n\pi$ where $n$ is any integer
Range: All real numbers $(-\infty, \infty)$
Asymptotes: Vertical lines at $x = \frac{\pi}{2}, \frac{3\pi}{2}, -\frac{\pi}{2}, ...$
x-intercepts: At $x = 0, \pm\pi, \pm 2\pi, ...$
The function passes through the origin $(0, 0)$

Worked Examples

Sketch one period of $y = \tan(x)$ and identify the asymptotes, intercepts, and behavior.

Find the asymptotes

$\cos(x) = 0$ when $x = -\frac{\pi}{2}$ and $x = \frac{\pi}{2}$ → Vertical asymptotes at $x = \pm\frac{\pi}{2}$

Find the x-intercept

$\tan(x) = 0$ when $\sin(x) = 0$ , so $x = 0$ → x-intercept at $(0, 0)$

Find key points

$\tan\left(-\frac{\pi}{4}\right) = -1$ and $\tan\left(\frac{\pi}{4}\right) = 1$ → Points: $\left(-\frac{\pi}{4}, -1\right)$ and $\left(\frac{\pi}{4}, 1\right)$

Describe the behavior

As $x \to -\frac{\pi}{2}^+$ , $\tan(x) \to -\infty$ ; as $x \to \frac{\pi}{2}^-$ , $\tan(x) \to +\infty$ → Curve rises from $-\infty$ to $+\infty$

Answer: The tangent function has vertical asymptotes at $x = \pm\frac{\pi}{2}$ , passes through $(0,0)$ , and increases continuously from $-\infty$ to $+\infty$ within each period.

Common Mistakes

Thinking the period of tangent is $2\pi$ like sine and cosine

Why it's wrong: The tangent function repeats every $\pi$ radians because $\tan(x + \pi) = \tan(x)$ for all $x$ in the domain.

Correct: The period of $y = \tan(x)$ is $\pi$ . For $y = \tan(bx)$ , the period is $\frac{\pi}{|b|}$ .

Drawing the curve crossing through the asymptotes

Why it's wrong: Asymptotes represent values where the function is undefined. The curve approaches but never touches or crosses them.

Correct: Draw vertical dashed lines for asymptotes. The curve approaches $\pm\infty$ as it nears each asymptote.

Forgetting that tangent has no maximum or minimum value

Why it's wrong: Unlike sine and cosine which are bounded between $-1$ and $1$ , tangent's range is all real numbers.

Correct: The range of $y = \tan(x)$ is $(-\infty, \infty)$ . There is no amplitude for tangent functions.

Why It Matters

The tangent function appears throughout mathematics and real-world applications:

Architecture: Calculating roof pitch and ramp angles
Physics: Analyzing projectile motion and pendulum swings
Navigation: Determining bearings and flight paths
Engineering: Designing roads, bridges, and structures with specific inclines

Understanding the tangent graph helps you visualize how the ratio of vertical to horizontal change behaves as angles vary.

Real World Applications

Road Engineering

Engineers use the tangent function when designing roads with specific gradients and banking angles.

Example:

A road rises 5 meters over a horizontal distance of 100 meters. The grade angle $\theta$ satisfies $\tan(\theta) = \frac{5}{100} = 0.05$ , giving $\theta \approx 2.86°$ .

1Try It Yourself

A wheelchair ramp must have a maximum slope ratio of 1:12 (rise to run).

What is the maximum angle of the ramp?

Step 1: Write the mathematical expression

Calculate $\arctan\left(\frac{1}{12}\right)$ :

Aviation Navigation

Pilots use tangent ratios when calculating descent angles and approach paths.

Example:

A plane descends from 10,000 feet while traveling 30 miles horizontally. The descent angle $\theta$ has $\tan(\theta) = \frac{10000}{30 \times 5280}$ .

2Try It Yourself

A standard 3-degree glide slope is used for landing approaches.

For every mile of horizontal distance, how many feet does the plane descend?

Step 1: Write the mathematical expression

Calculate: $5280 \times \tan(3°)$

Key Takeaways

1The tangent function is defined as $\tan(x) = \frac{\sin(x)}{\cos(x)}$
2The period of $y = \tan(x)$ is $\pi$ (not $2\pi$ )
3Vertical asymptotes occur at $x = \frac{\pi}{2} + n\pi$ where cosine equals zero
4The range is all real numbers $(-\infty, \infty)$ with no maximum or minimum
5For $y = \tan(bx)$ , the period is $\frac{\pi}{|b|}$
6The graph passes through the origin and has x-intercepts at multiples of $\pi$

Frequently Asked Questions

Why does tangent have asymptotes but sine and cosine don't?

Because

\tan(x) = \frac{\sin(x)}{\cos(x)}

, the function is undefined wherever

\cos(x) = 0

. Division by zero creates vertical asymptotes. Sine and cosine are defined for all real numbers with no division involved.

How do I remember where the asymptotes are?

Asymptotes occur where cosine equals zero: at odd multiples of

\frac{\pi}{2}

. That's

\pm\frac{\pi}{2}, \pm\frac{3\pi}{2}, \pm\frac{5\pi}{2}

, etc. Think: halfway between each zero of cosine.

Why is the period $\pi$ instead of $2\pi$ ?

The tangent function repeats after

\pi

radians because both sine and cosine change sign together after

\pi

, so their ratio stays the same:

\frac{-\sin(x)}{-\cos(x)} = \frac{\sin(x)}{\cos(x)}

Glossary

Asymptote: A line that a curve approaches but never touches. For tangent, these are vertical lines where the function is undefined.
Period: The horizontal distance before a function repeats. For $y = \tan(x)$ , the period is $\pi$ .
Tangent function: A trigonometric function defined as $\tan(x) = \frac{\sin(x)}{\cos(x)}$ , representing the ratio of opposite to adjacent sides in a right triangle.