Mathwords: Cartesian Form

Key Formula

y = f(x)

Where:

$x$ = The horizontal coordinate (independent variable) along the x-axis
$y$ = The vertical coordinate (dependent variable) along the y-axis
$f$ = The function rule that maps each x-value to a y-value

Worked Example

Problem: Convert the polar equation r = 6cos(θ) into Cartesian form.

Step 1: Recall the conversion relationships between polar and Cartesian coordinates.

x = r\cos\theta, \quad y = r\sin\theta, \quad r^2 = x^2 + y^2

Step 2: Multiply both sides of the polar equation by r to create terms we can substitute.

r^2 = 6r\cos\theta

Step 3: Replace r² with x² + y² on the left side, and replace r cos θ with x on the right side.

x^2 + y^2 = 6x

Step 4: Rearrange into standard circle form by completing the square on x.

x^2 - 6x + 9 + y^2 = 9 \implies (x - 3)^2 + y^2 = 9

Answer: The Cartesian form is

(x - 3)^2 + y^2 = 9

, which is a circle centered at

(3, 0)

with radius 3.

Another Example

This example converts from parametric form instead of polar form, showing that Cartesian form means eliminating the parameter to get one equation in x and y.

Problem: Convert the parametric equations x = 2t + 1 and y = 4t² into a single Cartesian equation.

Step 1: Solve the x-equation for the parameter t.

x = 2t + 1 \implies t = \frac{x - 1}{2}

Step 2: Substitute this expression for t into the y-equation.

y = 4t^2 = 4\left(\frac{x - 1}{2}\right)^2

Step 3: Simplify by squaring the fraction and multiplying.

y = 4 \cdot \frac{(x-1)^2}{4} = (x - 1)^2

Answer: The Cartesian form is

y = (x - 1)^2

, a parabola with vertex at

(1, 0)

.

Frequently Asked Questions

What is the difference between Cartesian form and polar form?

Cartesian form locates points using horizontal and vertical distances

(x, y)

measured from the origin along perpendicular axes. Polar form locates points using a distance

r

from the origin and an angle

\theta

measured from the positive x-axis. The two forms describe the same points but use different coordinate systems, and you can convert between them using

x = r\cos\theta

and

y = r\sin\theta

.

Why is Cartesian form also called rectangular form?

The name "rectangular" comes from the fact that the x- and y-axes are perpendicular (at right angles), creating a rectangular grid. Any point

(x, y)

sits at the corner of a rectangle whose sides are parallel to the axes. The term is especially common when working with complex numbers, where

a + bi

is the rectangular form.

When should you use Cartesian form instead of other forms?

Cartesian form is best when a curve has a straightforward relationship between x and y, such as lines (

y = mx + b

) or parabolas (

y = ax^2 + bx + c

). It is also the standard form for graphing on a typical coordinate plane. Polar form tends to be more convenient for curves with rotational symmetry, like circles centered at the origin or spirals.

Cartesian Form vs. Polar Form

	Cartesian Form	Polar Form
Coordinates used	$(x, y)$ — horizontal and vertical distances	$(r, \theta)$ — distance and angle from origin
Example: unit circle	$x^2 + y^2 = 1$	$r = 1$
Example: vertical line	$x = 3$	$r\cos\theta = 3$ (less natural)
Best suited for	Lines, parabolas, polynomials	Circles, spirals, rose curves
Conversion	$x = r\cos\theta,\; y = r\sin\theta$	$r = \sqrt{x^2+y^2},\; \theta = \arctan(y/x)$

Why It Matters

Cartesian form is the default way equations are written throughout algebra, geometry, and calculus. When you graph

y = 2x + 3

or find the intersection of two curves, you are working in Cartesian form. Converting other representations — parametric equations, polar equations, or complex numbers — into Cartesian form is a core skill in precalculus and is essential for standardized tests and college-level mathematics.

Common Mistakes

Mistake: Confusing the conversion formulas: using

x = r\sin\theta

instead of

x = r\cos\theta

.

Correction: Remember that

x

(horizontal) pairs with

\cos\theta

and

y

(vertical) pairs with

\sin\theta

. Think of the unit circle: at

\theta = 0

, the point is

(1, 0)

, so

x = \cos 0 = 1

.

Mistake: Forgetting to eliminate the parameter completely when converting from parametric to Cartesian form, leaving t in the final equation.

Correction: The whole purpose of Cartesian form is to express the relationship using only x and y. Solve one parametric equation for t, substitute into the other, and simplify until no t remains.

Related Terms

Cartesian Coordinates — The coordinate system Cartesian form is built on
Polar Coordinates — Alternative coordinate system using r and θ
Parametric Equations — Equations using a parameter, often converted to Cartesian form
Coordinates — General concept of locating points in space
Function — A relation often expressed in Cartesian form as y = f(x)
Relation — Any set of ordered pairs, expressible in Cartesian form