# 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.

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).

## 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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