# How do you write an equation of y=cosx with 3 units to the left and pi units up?

Jun 18, 2017

Add 3 to the argument of the cosine function and add $\pi$ after the cosine function.

#### Explanation:

Here is a graph of the equation $y = \cos \left(x\right)$:

Please observe that I have placed a dot at the point were $x = 0$

We want to change the equation so that point is at x = -3. Lets add a constant, c, to the argument of the cosine function.

$y = \cos \left(x + c\right)$

We want $x + c$ to equal 0 when $x = \frac{3}{2} \pi$

Here is the equation for x plus c equals zero:

$x + c = 0$

Here is the equation forcing x to be -3:

$- 3 + c = 0$

We can solve for d:

$c = 3$

This makes the equation become:

$y = \cos \left(x + 3\right)$

Here is the graph for the equation:

Please observe that the cosine function has shifted 3 units to the left as requested.

Let's add another constant, d, to the equation:

$y = \cos \left(x + 3\right) + d$

We the cosine function is 1, we want y to equal $\pi + 1$.

Here is the equation for that:

$\pi + 1 = 1 + d$

Solve for d:

$d = \pi$

Here is a graph of the equation $y = \cos \left(x + 3\right) + \pi$:

Please observe that I have shifted the curve 3 units to the left and up $\pi$ units.