How to Write the Equation of a Cosine Graph?
The equation of a cosine graph is based on the mathematical function \(cos(x)\), where \(x\) is the angle in radians.
Write the Equation of a Cosine Graph: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
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 cosine rule
- Amplitude is |2|.
- Period is 2pi/4 = pi/2.
- Midline is y = 3.
Place parent cosine points
- Start at (0, 1).
- Use quarter-period steps.
- The y-values are 1, 0, -1, 0, 1.
Try one before moving on
Write the Equation of a Cosine Graph: pop-up practice
The equation of a cosine graph with amplitude \(A\), period \(T\), phase shift (horizontal shift) of \(d\), and vertical shift of \(k\) is given by:
\(y = A × cos(2 × pi × \frac{(x – d)}{T}) + k\)
Where \(x\) is the independent variable (usually time or angle), \(y\) is the dependent variable (the value of the function), \(A\) is the amplitude (the maximum value of the function), \(T\) is the period (the distance between consecutive maximum or minimum values), \(d\) is the phase shift (the horizontal shift of the graph), and \(k\) is the vertical shift (the amount the graph is shifted up or down).
Related Topics
- How to Graph the Sine Function
- How to Graph the Cosine Function
- How to Write the Equation of a Sine Graph
- How to Graph Inverse of the Sine Function
- How to Graph Inverse of the Cosine Function
A step-by-step to write the equation of a cosine graph
To find out how to write the equation of a cosine graph, follow the step-by-step guide below:
The equation of a cosine graph is based on the mathematical function \(cos(x)\), where \(x\) is the angle in radians. However, in the equation of a cosine graph, we often use \(x\) as the independent variable, which could represent time, distance, or any other variable.
The amplitude \((A)\) of the cosine graph is the maximum value of the function, it tells us how high or low the graph oscillates.
The period \((T)\) is the distance between consecutive maximum or minimum values. In other words, it tells us how many units of the independent variable \((x)\) it takes for the graph to repeat its pattern.
The phase shift \((d)\) is a horizontal shift of the graph. It tells us how much the graph has been shifted to the right or left along the \(x\)-axis.
The vertical shift \((k)\) is the amount the graph is shifted up or down along the \(y\)-axis.
For example, if we have a cosine graph with amplitude of \(2\), period of \(4\), phase shift of \(1\), and vertical shift of \(3\), the equation would be:
\(y = 2 × cos(2 × pi × \frac{(x – 1)}{4}) + 3\)
This equation would represent a cosine graph that oscillates between \(1\) and \(5 (2+3)\), completes one full oscillation every \(4\) units of \(x\) and has been shifted \(1\) unit to the right along the \(x\)-axis.
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