Teacher Guide: Graphing the Tangent Function
Learn to graph y = tan(x), identify asymptotes, period, and key features of the tangent curve.
Use this lesson with your class
Free, no student accounts needed.
Share with students
Students open the lesson and practise with instant feedback.
Class quiz
10 questions on Trigonometric Graphs. Students join with a name, you see everyone's score.
For Teachers
- Graph the parent tangent function
- Identify vertical asymptotes, period, and intercepts of tangent functions
- Apply transformations to graph functions of the form
- Calculate the period of transformed tangent functions
- Connect the tangent graph to the unit circle and ratio definition
- • Understanding of sine and cosine functions
- • Familiarity with the unit circle
- • Knowledge of function transformations (shifts, stretches)
- • Basic graphing skills on the coordinate plane
- 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?
The tangent function has an amplitude like sine and cosine
Asymptotes are places where the function equals infinity
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
- 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
- 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.
Lesson Content
Everything students see: definition, examples, common mistakes, applications. Tap to open.
Definition
- Period: (not like sine and cosine)
- Domain: All real numbers except where is any integer
- Range: All real numbers
- Asymptotes: Vertical lines at
- x-intercepts: At
- The function passes through the origin
Worked Examples
Sketch one period of and identify the asymptotes, intercepts, and behavior.
Find the asymptotes
when and → Vertical asymptotes at
Find the x-intercept
when , so → x-intercept at
Find key points
and → Points: and
Describe the behavior
As , ; as , → Curve rises from to
Answer: The tangent function has vertical asymptotes at , passes through , and increases continuously from to within each period.
Common Mistakes
Thinking the period of tangent is like sine and cosine
Why it's wrong: The tangent function repeats every radians because for all in the domain.
Correct: The period of is . For , the period is .
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 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 and , tangent's range is all real numbers.
Correct: The range of is . There is no amplitude for tangent functions.
Why It Matters
- 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
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 satisfies , giving .
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 :
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 has .
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:
Key Takeaways
- 1The tangent function is defined as
- 2The period of is (not )
- 3Vertical asymptotes occur at where cosine equals zero
- 4The range is all real numbers with no maximum or minimum
- 5For , the period is
- 6The graph passes through the origin and has x-intercepts at multiples of
Frequently Asked Questions
Why does tangent have asymptotes but sine and cosine don't?
How do I remember where the asymptotes are?
Why is the period instead of ?
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 , the period is .
- Tangent function
- A trigonometric function defined as , representing the ratio of opposite to adjacent sides in a right triangle.