How to Write the Equation of a Sine Graph
The general equation of a sine graph is \(y = a sin(b(x – h)) + k\), where \(a\) is the amplitude, \(b\) is the period, \(h\) is the horizontal shift, and \(k\) is the vertical shift.
Related Topics
- How to Graph the Sine Function
- How to Graph the Cosine Function
- How to Graph Inverse of the Sine Function
- How to Graph Inverse of the Cosine Function
Step-by-step to write the equation of a sine graph
To find out how to write the equation of a sine graph, follow the step-by-step guide below:
- Amplitude: The amplitude is the distance from the maximum or minimum value of the function to the midline (average of the maximum and minimum values). It is represented by the variable \(“a”\) in the equation.
- Period: The period is the distance between consecutive maximum or minimum values of the function. The period of the sine function is \(2π\). It is represented by the variable \(“b”\) in the equation.
- Horizontal Shift: The horizontal shift is the movement of the graph horizontally. It is represented by the variable \(“h”\) in the equation.
- Vertical Shift: The vertical shift is the movement of the graph vertically. It is represented by the variable \(“k”\) in the equation.
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For example, if you want to write the equation of a sine graph with an amplitude of \(2\), a period of \(4π\), a horizontal shift of \(3\) units to the right and a vertical shift of \(1\) unit up, the equation would be \(y = 2 sin(\frac{(x – 3)}{(4π)} ) + 1\)
You can also use a graphing calculator or a software like Desmos to help you graph the function and find the equation.
It’s important to note that sine and cosine functions have an amplitude between \(-1\) and \(1\) and a period of \(2×pi\), and that the domain of these functions is all the real numbers.
Tutor-style math help
Write the Equation of a Sine Graph: 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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Write the Equation of a Sine Graph: 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:
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