< Previous Section Home Next Section >

Section 3.9
Coordinate Systems and Graphs

A two-dimensional coordinate system is a method for locating points in the plane. It is also a method for representing the values of two quantities simultaneously. It was by Descartes’ ingenious method of representing the location of points in a plane or in space that we can visually represent relationships between quantities’ values.

Cartesian Coordinate System

The Cartesian, or rectangular, coordinate system locates points by representing quantities’ values separately on two number lines that are perpendicular to each other, and then combining them to locate a point in the plane whose location represents the quantities’ values simultaneously.


Every person has a height and has a weight (mass). Suppose we want to represent individuals’ heights and weights simultaneously. Figure 3.8a shows Bill’s weight (84.1 kg) on the horizontal axis and his height (1.83 m) on the vertical axis. Figure 3.9.1a represents Bill’s height and weight, but separately.

Figure 3.9.1b combines the two separate representations into one representation—the point whose location is 84.1 weight-units from the vertical axis and 1.83 height-units from the horizontal axis.

The point with coordinates (84.1, 1.83) is called the correspondence point for these particular values of Bill’s weight and height. It is a point that represents Bill’s weight and height simultaneously.

More generally, a correspondence point is determined by taking the values that co-occur in the quantities you are relating as the coordinates of a point in your coordinate system. A correspondence point’s location in any coordinate system represents values of two (or more) quantities simultaneously.

Bill's height on two graphs

Figure 3.9.1 Bill’s height, in meters is represented on the horizontal axis. Bill’s weight, in kg, is represented by the segment on the vertical axis. The correspondence between Bill’s height and weight is represented by the location of the point having coordinates (84.1, 1.83).

If we could record Bill’s height and weight continuously, we would have a record of of every correspondence point between his height and weight. Figure 3.9.2 is a hypothetical graph of Bill height (in meters) in relation to his weight (in kg) recorded continuously over his first 20 years. Each point on this graph represents Bill’s height when he was a particular weight.

Graph of Bill's height
Figure 3.9.2. Bill’s height at each number of kilograms that he weighed.

Reflection 3.9.1. The point with coordinates (10.55 0.80) is on the graph in Figure 3.9.2. What does the location of this point represent about Bill’s weight and height?

Reflection 3.9.2. What is your best estimate of Bill’s birth height and birth weight? What assumption must you make?

Reflection 3.9.3. Why does the displayed graph in Figure 3.9.2 start at (3, 0.5) instead of starting at (0,0)?

Reflection 3.9.4. Bill claimed that his weight and height stayed at 25.68 kg and 1.28 m for two years. Is his claim inconsistent with the graph in in Figure 3.9.2? Explain.

Reflection 3.9.5. The displayed graph of Bill’s height in relation to his weight is not realistic in several ways. Describe one way in which it is not realistic.

Independent and Dependent Variables

Sometimes you will identify a variable in a mathematical statement as the variable that is under your control, in the sense that you may choose values for it. Variables whose values you consider under your control are called independent variables.

A dependent variable is a variable whose value becomes determined once you assign a value to an independent variable. It is customary in a rectangular coordinate system to put values of an independent variable on the horizontal axis and values of a dependent variable on the vertical axis. But this is just a custom. It is not necessary. However, if you break custom, you should state which variable you are taking as independent and which you are taking as dependent.

In some situations, variables will be neither dependent nor independent. The situation in Figure 3.9.2 is an example. Bill’s weight does not depend on his height, nor does Bill’s height depend on his weight. The graph in Figure 3.9.2 simply keeps track of them together. You could put either quantity on either axis.

A rectangular coordinate systems is like a street map of Phoenix (an exceptionally flat city). Locations are determined by intersections of lines that are perpendicular to axes.

Polar Coordinate System

A polar system works like radar. A point’s location is recorded according to its distance from a reference point and its angle from a reference direction.

The reference point is called the system’s pole and the ray from the pole in the reference direction is called the polar axis.

Figure 3.9.3 illustrates the polar coordinate system. Angles are measured in radians (see the Sections 2.2, 2.3, and 2.4). Distances from the pole are measured in units of the radius of the unit circle. Coordinates in the polar plane are customarily in the form $(r,\theta)$, where r is the point’s distance from the pole and θ is the measure in radians of the angle formed by a ray from the pole to the point and the reference direction.

A point with coordinates (3, 1.2) in the polar system is 3 unit radius lengths from the pole in a direction of 1.2 radians from the polar axis.

Polar coordinate system

Figure 3.9.3 The polar coordinate system. Rays point in directions, measured in radians from the polar axis. Circles designate distances from the pole, measured in units of the radius of the unit circle. Coordinates of a point are given as (distance from pole, direction from polar axis).

The animation in Figure 3.9.4 shows a way to think about determining a point’s location in the polar plane. First determine its distance from the pole. Once the distance is settled, determine the direction from the polar axis.

Figure 3.9.4. Locate a point in the polar coordinate system by first determining its distance from the pole and then determining its direction from the polar axis.

Many people find the convention for ordered pairs in polar coordinates unnatural. The first coordinate is r, but it is easier to think first of a point’s direction.

Imagine a ray pointing in the direction of θ, then imagine going r units in that direction (backing up if r is negative).

Thinking of a point’s direction first also fits with the fact that θ often is taken as the independent variable when displaying a graph in polar coordinates.

You are free to think of locating a point by any method you find most natural. Just remember that the order of the coordinates is (r, θ) when you represent a point in the polar plane.

Points in the polar plane do not have unique coordinates. Figure 3.9.5 shows a point in the polar plane located in two ways, both in the form (r, θ). The point in Figure 3.9.5 can be located by the pair (3.7 miles, 0.69 radians) and it can be located by the pair (‑3.7 miles, 0.69‑π radians).

Figure 3.9.5. Points in the polar plane do not have unique coordinates.

Graphs in Polar Coordinates

To imagine a graph in the polar plane, imagine varying the independent variable a little at a time while simultaneously keeping track of the dependent variable’s value. Record the covariation using the conventions of your coordinate system. In most situations in which we use polar coordinates, we will treat θ as the independent variable and r as the dependent variable.

Imagine two quantities whose values are represented by r and θ, and that they have the relationship that r = 2θ – 5. To represent this relationship in the polar plane, vary the value of θ, keeping track of its value and the value of r.

Recall that in the polar plane, θ is a number of radians measured from the reference direction and r is a number of unit lengths from the pole in the direction of θ. Figure 3.9.6 shows points with coordinates (r, θ) where r = 2θ ‑ 5. Be sure that you understand what the coordinates of each ordered pair represent and why these points are located where they are.

Points for r=-*theta-5

Figure 3.9.6. Points for r = 2θ - 5 plotted in the polar plane. They appear jumbled, making it hard to see a pattern in their generation.

It is quite hard to envision the complete graph of r = 2θ - 5 in polar coordinates just from plotted points. The plotted points are too jumbled to convey what happens as θ varies. A complete graph of f in polar coordinates for 0 ≤ θ < 2π is given in Figure 3.9.7.

Imagine a ray, originating from the pole, being swept counter-clockwise as θ varies from 0 to 2π radians. At the same time that you sweep a ray from 0 to 2π radians, imagine the location at a distance of 2θ – 5 units in the direction that the ray points—going forward when 2θ – 5 is positive, and backward when 2θ – 5 is negative.

Figure 3.9.7. Graph of $r=2\theta-5$ as $\theta$ varies from 0 to $2\pi$.

Exercise Set 3.9

  1. The animation below shows values of two quantities, u and v. The value of u is on the vertical axis. The value of v is on the horizontal axis.

    Sketch a graph that displays a record of the value of u in relation to the value of v.

    Imagine a correspondence point, then keep track of it by imagining it sprinkled with Pixie dust. The Pixie dust will leave a trace of everywhere the correspondence point has been.

  2. (click for solution)

  3. Each animation in 2.a - 2.d portrays a bottle being filled with liquid.

    Sketch a graph for each bottle that displays the variation in the water’s volume with respect to the water’s height in the bottle.

    Click the solution link for each bottle after you sketch your graph to see the actual graph constructed as the liquid's height varies.

    Be sure to understand why each graph behaves as it does. Reflect on your thinking if your graph differs from what is shown.

  4. a) ( click for solution)

    b) ( click for solution)

    c) ( click for solution)

    d) ( click for solution)

  5. The left coordinate system is standard rectangular. The right system has a skewed and flipped vertical axis.

    The arrows in each system indicate the positive direction for each axis.

    The highlighted point in the left system is plotted in the right system.

    Sketch the displayed graph on the left within the coordinate system on the right. Do the two displayed graphs represent the same relationship?

  6. Two graphs same or not  ( solution)

  7. For each of a-e, the coordinates on both sides of the equal sign are in the polar system. Fill in the blank so that the polar coordinates on the right locate the point represented by the polar coordinates on the left.

    Remember that polar coordinates are in the form $(r,\theta)$.

    1. $(1, 2) = (-1,\_ \_)$
    2. $(-3, \sqrt 2) = (3, \_ \_)$
    3. $(3, -\pi/4) = ( \_ \_, 3\pi/4)$
    4. $(4.3, 27) = (-4.3,  \_ \_)$
    5. $(-4.3, -27) = (4.3,  \_ \_)$
  8. Download and print this set of polar coordinate papers or open it in a drawing program.
    1. Sketch a graph of each of I-IV on both types of coordinate paper. Try not to plot points, but rather draw the graph smoothly while mentally estimating values of variables.

      Check your sketched graph with GC (type “theta” to get “θ”). Change the domains for r and θ in GC as needed.
      1. What to type in GC
      2. $r = 0.1\cdotθ,\text{ } 0 ≤ θ < 10π$
      3. $r = 5-θ, \text{ } 0 ≤ θ < 2π$
      4. $r = \mathrm{sin}(θ), \text{ } 0 ≤ θ < 2π$
      5. $θ = \mathrm{sin}(r), \text{ } -8 ≤ r ≤ 8$
    2. In Part I you graphed $r = 0.1\cdotθ,\text{ } 0 ≤ θ < 10π$

      How far apart are two successive points that lie on the same ray from the pole? (Graphs of r = , m a non-zero real number, are called Archimedean Spirals)

    3. In Part II you graphed $r = 5-θ, \text{ } 0 ≤ θ < 2π$

      What are the coordinates of the point on the graph that is at the pole? Why does the graph never pass through (0, 0) even though it passes through the pole?

    4. In Part III you graphed $r = \sin(\theta),\, 0 ≤ θ < 2π$

      Why does this graph make a circle? What is its center and radius?

    5. In Part IV you graphed $θ = \sin(r),\, -8 ≤ r ≤ 8$

      This graph is called the Batman Curve. Explain how the variation in θ relative to r over the following intervals produces a piece of the displayed graph (It might help to print a graph of y = sin(x) in Cartesian coordinates.):

      -8 ≤ r ≤ -7

      -7 ≤ r ≤ -6

      -6 ≤ r ≤ -5

      5 ≤ r ≤ 6

      6 ≤ r ≤ 7

      7 ≤ r ≤ 8

  9. The animation below shows a machine gun rotating rapidly as it fires bullets. A bullet, once fired, travels at a constant speed in the direction it leaves the gun’s barrel.

    1. Why do the bullets make a spiral?

    2. Pause the movie. Do the bullets’ positions lie on an Archimedean spiral? Explain.
  10. Type the statement cos(0.8) into GC. It will display the value 0.69670671 (if you have GC set to display 8 significant digits).

    What does this tell you about a point on the graph of y = cos(x)?

    What does it tell you about a point on the graph of r = cos(θ)?

  11. Use GC to display the graph of $y=x^2$ in rectangular coordinates. In a new window, display the graph of $x=y^2$.

    The displayed graphs are different. Are the graphs themselves different? Explain.

    (Recall the meaning of a statement’s graph from Section 3.5. Also recall that coordinate pairs in rectangular coordinates are in the order value of independent variable, value of dependent variable.)

      1. Homer Simpson's Distances from Springfield and Shelbyville

  12. The animation below depicts Homer Simpson driving his car along a straight road from its beginning to its end.

    Your task is to

    Imagine a correspondence point in your coordinate system for the two distances, then keep track of it by imagining it sprinkled with Pixie dust. The Pixie dust will leave a trace of everywhere the correspondence point has been.

    Make a concerted effort to do this before clicking on the solution link.

      1. Solution

  13. The animation below shows Homer driving along the same road as in Exercise 3.9.9 except that he occasionally backs up. How will this affect the graph of Homer's distance from Springfield in relation to his distance from Shelbyville? Explain.


  14. The animation below shows Homer driving a curved road that passes between Springfield and Shelbyville.

    Your task is to