How to Sketch Trigonometric Graphs?
Tutor-style math help
Sketch Trigonometric Graphs: what to notice and how to work it
Trigonometry skill
A sine graph is a repeating wave. To sketch it well, mark the midline, amplitude, period, and five key points in one cycle.
What to notice first
For \(y=A\sin(Bx)+D\), the amplitude is \(|A|\), the period is \(2\pi/|B|\), and the midline is \(y=D\).
Common student mistake
Do not space the five key points randomly. Divide one period into four equal parts so the wave starts, rises, returns, falls, and returns again.
Key formulas and cues
\(y=A\sin(Bx)+D\)
\(\text{amplitude}=|A|\)
\(\text{period}=\frac{2\pi}{|B|}\)
\(\text{midline}=y=D\)
A reliable path
- Choose the modelUse a right triangle, the unit circle, or a transformed graph.
- Track unitsConvert degrees and radians when needed.
- Use identitiesReplace complicated trig expressions with equivalent simpler ones.
Worked examples
Read a sine graph rule
Example: \(y=3\sin(2x)-1\)
- Amplitude is |3|.
- Period is 2pi/2 = pi.
- Midline is y = -1.
Answer: Amplitude \(3\), period \(\pi\), midline \(y=-1\).
Place five key points
Example: Graph one cycle of \(y=\sin x\).
- Start at (0, 0).
- Use quarter-period steps: pi/2, pi, 3pi/2, 2pi.
- The y-values are 0, 1, 0, -1, 0.
Answer: \((0,0),(\pi/2,1),(\pi,0),(3\pi/2,-1),(2\pi,0)\).
Try one before moving on
Try: Find the amplitude and period of \(y=2\sin(4x)\).
Answer: Amplitude \(2\), period \(\frac{\pi}{2}\).
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
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Sketch Trigonometric Graphs: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. For \(y=4\sin x\), the amplitude is:
2. For \(y=\sin(3x)\), the period is:
3. A sine graph's midline is:
- If the trigonometric function is of the form\(y=a\:sin\:b\:\theta\), \(y=a\:cos\:b\:\theta\) or \(y=a\:tan\:b\:\theta\) identify the values of a and b and calculate the values of the amplitude and period.
- Create a table of ordered pairs for the points you are plotting. The first value in the ordered pair corresponds to the value of the angle \(θ\), and the values of y correspond to the value of the trigonometric function for the angle \(θ\), for example, \(sin\: θ\), so the ordered pair will be \((θ, sin\: θ)\). The values of \(θ\) can be either in degrees or radians.
Sketching Trigonometric Graphs – Example 1:
Exercises for Sketching Trigonometric Graphs
Graph each trigonometric function.
- \(\color{blue}{y=\frac{1}{2}\:cos\:2\:\theta}\)
- \(\color{blue}{y=\:3\:cot\:\theta}\)
- \(\color{blue}{y=\frac{1}{2}\:cos\:2\:\theta}\)
- \(\color{blue}{y=\:3\:cot\:\theta}\)
Original price was: $109.99.$54.99Current price is: $54.99.
Original price was: $109.99.$54.99Current price is: $54.99.
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