Question

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

Answer 1

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(x)`:

![Desmon.com]()

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(x+ c)`

We want `x+c` to equal 0 when `x=3/2pi`

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(x+3)`

Here is the graph for the equation:

![Desmos.com]()

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(x+3)+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(x+3)+pi`:
![Desmos.com]()

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