Question

How do you graph `r = 12cos(theta)`?

Answer 1

Set your compass to a radius of 6, put the center point at `(6, 0)`, and draw a circle.

Explanation:

Multiply both sides of the equation by r:

`r^2 = 12rcos(theta)`

Substitute `x^2 + y^2` for `r^2` and `x` for `rcos(theta)`

`x^2 + y^2 = 12x`

The standard form of this type of equation (a circle) is:

`(x - h)^2 + (y - k)^2 = r^2`

To put the equation in this form, we need to complete the square for both the x and y terms. The y term is easy, we merely subtract `0` in the square:

`x^2 + (y - 0)^2 = 12x`

To complete the square for the x terms, we add `-12x + h^2` both sides of the equation:

`x^2 - 12x + h^2 + (y - 0)^2 = h^2`

We can use the pattern, `(x - h)^2 = x^2 - 2hx + h^2` to find the value of h:

`x^2 - 2hx + h^2 = x^2 - 12x + h^2`

`-2hx = -12x`

`h = 6`

Substitute 6 for h into the equation of the circle:

`x^2 - 12x + 6^2 + (y - 0)^2 = 6^2`

We know that the first three terms are a perfect square with h = 6:

`(x - 6)^2 + (y - 0)^2 = 6^2`

This is a circle with a radius of 6 and a center point `(6, 0)`