What a Gubby Is and How to Build One in Desmos

A gubby is a playful stick figure drawn entirely from mathematical equations in Desmos. It consists of a circular head, a body made from line segments, and limbs positioned at angles — all created by entering functions and inequalities into the calculator. The result looks like a simple cartoon character standing on the coordinate plane.

Building a gubby teaches you how Desmos handles different equation types: circles for the head, linear equations for the body and limbs, and domain restrictions to control where each piece appears. You do not need advanced math knowledge — just the ability to enter a few basic formulas and adjust numbers to position each part.

Key Takeaways

  • A gubby is made from a circle for the head, line segments for the body and arms, and line segments for the legs, each entered as a separate equation.
  • The circle equation (x – h)² + (y – k)² = r² creates the head, where you choose the center point and radius size.
  • Limbs are drawn using linear equations with domain restrictions that limit where each line appears on the graph.
  • Adjusting the numbers in each equation moves parts around, resizes them, or changes their angle without rewriting the whole thing.
  • Desmos shows your gubby in real time as you type, so you can see mistakes immediately and fix them.

Drawing the Head with a Circle Equation

Start by creating the head. Open Desmos and enter the equation for a circle. The standard form is (x – h)² + (y – k)² = r², where h and k are the center coordinates and r is the radius. For a head centered at the origin, type x² + y² = 9. This draws a circle with radius 3 at the point (0, 0).

If you want the head higher or to the side, change the center. For example, (x – 0)² + (y – 5)² = 9 moves the circle up 5 units. Make the head bigger by increasing the radius — x² + (y – 5)² = 16 creates a circle with radius 4. Desmos draws the circle instantly as you type, so you can see the size and position right away.

Adding a Body with Line Segments

The body is typically a vertical line segment below the head. In Desmos, you draw a line by entering a linear equation with a domain restriction. Type y = 5 – x {0 ≤ x ≤ 0} to create a vertical line. The part in curly braces limits where the line appears — in this case, only at x = 0, which gives you a vertical segment.

For a simple vertical body, use y = 5 – t {0 ≤ t ≤ 2}, where t is a parameter. In Desmos, you can also type x = 0 {2 ≤ y ≤ 5} to draw a vertical line from y = 2 to y = 5 along the x = 0 axis. This creates a body segment that connects the head to where the legs will start. Adjust the numbers to make the body longer or shorter, or move it left or right.

Creating Arms and Legs with Restricted Lines

Arms and legs are line segments drawn at angles using linear equations with domain restrictions. For a left arm, enter something like y = 3 + x {-3 ≤ x ≤ 0}. This draws a line with slope 1 that only appears between x = -3 and x = 0, creating a diagonal segment from the body outward.

For a right arm, use y = 3 – x {0 ≤ x ≤ 3} to mirror the left arm. Legs work the same way but start lower on the body. Try y = 2 – x {-2 ≤ x ≤ 0} for a left leg and y = 2 + x {0 ≤ x ≤ 2} for a right leg. Change the slope (the number multiplying x) to make limbs steeper or shallower, and adjust the domain range to make them longer or shorter.

Positioning Each Part to Look Like a Figure

Once you have all the pieces entered, step back and look at how they fit together. The head should sit above the body, the body should connect to the arms and legs, and all four limbs should point outward. If something is in the wrong place, edit that single equation — you do not have to start over.

Common adjustments: if the arms are too high or low, change the y-value in the equation (the constant term). If a leg is too steep, change the slope. If a limb is too short, expand the domain range. Desmos updates the graph as you type, so you can see each change instantly and decide whether it looks right.

Adding Details Like Eyes and a Mouth

Once the basic figure is complete, you can add a face. Eyes are small circles — enter (x + 1)² + (y – 7)² = 0.25 for a left eye and (x – 1)² + (y – 7)² = 0.25 for a right eye, adjusting the center coordinates and radius to fit your head size. A mouth can be a small arc or line segment using the same techniques as the limbs.

You can also color each part differently by clicking on the equation in the left panel and choosing a color from the menu. This makes it easier to see which equation draws which part and makes your gubby more visually interesting.

Frequently Asked Questions

Can I save my gubby after I draw it?

Yes. Desmos saves your work automatically if you create a free account and log in. You can also share a link to your graph by clicking the Share button, which lets others view your gubby without editing it. If you do not have an account, take a screenshot to keep a copy.

What if my lines do not show up where I expect?

Check the domain restriction in curly braces — it controls where the line appears. If you typed y = x {0 ≤ x ≤ 2}, the line only shows between x = 0 and x = 2. Also check that your equation is mathematically correct: a typo like y = x {0 ≤ x ≤ 2 (missing closing brace) will cause an error.

How do I make the gubby bigger or smaller?

Multiply all the numbers in your equations by the same factor. To double the size, change the circle radius from 3 to 6, change body coordinates from 5 to 10, and so on. This scales the whole figure proportionally without changing its shape.

Can I make the gubby move or animate?

Desmos has a slider feature that lets you add a variable and animate it. Instead of typing fixed numbers, use a slider variable like a, then reference it in your equations — for example, x² + (y – a)² = 9 lets you slide the head up and down. Click the play button next to the slider to animate.

What if I want to draw a more complex figure?

You can add more line segments, curves, and shapes using the same methods. Parabolas, exponential curves, and piecewise functions all work in Desmos. The more equations you add, the more detailed your figure becomes — but each one still follows the same basic pattern of an equation plus a domain restriction.