Graphing Tangent
A free Trigonometry lesson from the “Graphs of Trig Functions” unit, with a worked example and practice problems including step-by-step solutions.
The graph of y = tan(x) repeats every pi units and has vertical asymptotes wherever cosine is zero, at x = pi/2 + k*pi. Between consecutive asymptotes the curve rises continuously from negative infinity to positive infinity, crossing zero at every multiple of pi; tangent has no amplitude because its range is all real numbers. For y = tan(bx), the period compresses or stretches to pi/|b|.
What you'll learn
- Graph y = tan(x) using its period, asymptotes, and intercepts
- Locate vertical asymptotes and x-intercepts of tangent graphs
- Find how a coefficient b changes the period to pi/|b|
Worked example
Problem. Find the period and the location of the vertical asymptotes for y = tan(2x).
- Period of tan(bx) is pi/|b|, so with b = 2 the period is pi/2
- Asymptotes occur where the inside equals pi/2 + k*pi: 2x = pi/2 + k*pi
- Solve for x: x = pi/4 + k*pi/2
Answer: Period = pi/2; asymptotes at x = pi/4 + k*pi/2
Practice problems
1. What is the period of y = tan(x)?
Show solution
- Warm-up: First identify exactly what the question is asking: What is the period of y = tan(x)?
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- The basic tangent function repeats every pi units
- So the period is pi
- Check the result by substituting or estimating: the response should match pi and make sense in the original problem.
Answer: pi
2. Which best describes the amplitude of y = tan(x)?
Choices: Amplitude is 1 · Amplitude is pi · Tangent has no amplitude (its range is all real numbers) · Amplitude is 1/2
Show solution
- Warm-up: First identify exactly what the question is asking: Which best describes the amplitude of y = tan(x)?
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Amplitude measures half the distance between max and min for bounded waves
- Tangent increases without bound toward both infinities, so no amplitude is defined
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Tangent has no amplitude (its range is all real numbers)
3. At what x-value between -pi/2 and pi/2 does the graph of y = tan(x) cross the x-axis?
Show solution
- Warm-up: First identify exactly what the question is asking: At what x-value between -pi/2 and pi/2 does the graph of y = tan(x) cross the x-axis?
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- x-intercepts of tan(x) occur at multiples of pi
- The only multiple of pi in (-pi/2, pi/2) is 0
- Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.
Answer: 0
4. Give the equation of the vertical asymptote of y = tan(x) that lies between x = 0 and x = pi.
Show solution
- Core Practice: First identify exactly what the question is asking: Give the equation of the vertical asymptote of y = tan(x) that lies between x = 0 and x = pi.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Asymptotes of tan(x) are at x = pi/2 + k*pi
- The one between 0 and pi is x = pi/2
- Check the result by substituting or estimating: the response should match x = pi/2 and make sense in the original problem.
Answer: x = pi/2
5. What is the period of y = tan(x/2)?
Show solution
- Core Practice: First identify exactly what the question is asking: What is the period of y = tan(x/2)?
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Period of tan(bx) is pi/|b| with b = 1/2
- pi divided by 1/2 equals 2pi
- Check the result by substituting or estimating: the response should match 2pi and make sense in the original problem.
Answer: 2pi
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