Graphable Expressions

## This is a list of all the graphable expressions in Desmos, along with instructions on how to graph them, examples, and other comments.

Without:

## Regular Expressions

You can write an expression with any two variables, so just as

# y
2
x
5

{\displaystyle y=2x+5}

works - so does

# a
2
b
5

{\displaystyle a=2b+5}

. You can also create functions such as

f
(
x
# )
2

x
2

4
x

{\displaystyle f(x)=2x^2+4x}

and

t
(
x
# )
4

x
3

19
x
3

{\displaystyle t(x)=4x^3+19x+3}

.

# x

1
−

y
2

{\displaystyle x=\sqrt{1-y^2}}

.

## Points, Point Lists and Movable Points

To plot points, just use the format (x,y). To plot point lists write several points, separating each with commas. To create a movable point follow one of these formats: (a,b), (x,b), (a,y), where a and b are variables and x and y are consonants. To read more about movable points, visit the Variables and Sliders page.

# y
2
x
[
1
,
2
,
3
,
4
,
5
]

{\displaystyle y=2x+[1,2,3,4,5]}

, you're creating five different graphs:

# y
2
x
1

{\displaystyle y=2x+1}

,

# y
2
x
2

{\displaystyle y=2x+2}

, etc.

## Inequalities

When making strict inequalities, using

<

{\displaystyle < }

or

>

{\displaystyle > }

, the lines will be dashed. When making non-strict inequalities, using

≤

{\displaystyle \le}

or

≥

{\displaystyle \geq}

, the lines will not be dashed. You can make all sorts of inequalities from simple linear inequalities like

2
x
4
<
y

{\displaystyle 2x+4<y}

, to quadratic inequalities like

2

x
3

4
x
1

≤

y

{\displaystyle 2x^3+4x+1 {\leq} y}

, to implicit inequalities such as

y
2

x
2

<
25

{\displaystyle y^2+x^2<25}

.

## Polar

To graph a polar expression, make the subject

r

{\displaystyle r}

and express it in terms of

θ

{\displaystyle \theta}

, which can be inserted in your equation by typing 'theta'. A classic example is classic polar flower, formed by the equation

# r
s
i
n
(
5
θ
)

{\displaystyle r=sin(5\theta)}

. You might also want to switch the grid layout of the Graphing Calculator to suit the polar expressions. To switch between the cartesian and polar grid, go to settings of the graph - represented by the wrench button on the top right - and click the polar grid icon as resented in Fig. 1.

## Piecewise

To graph a piecewise function, you need to input the format of {condition: value, default}, here's what this means. The condition is the area of the graph you want to be different, the value is the equation/value you want that area to change to, the default is just the rest of the graph. Confused? Let's go through an example. You want a graph of

# y
2
x

{\displaystyle y=2x}

, except from -2 to 2 x-coordinates you want

x
2

{\displaystyle x^2}

, here's how you write it:

{
−
2
<
x
<
2
:

x
2

,
2
x
}

{\displaystyle \{-2<x<2: x^2, 2x\}}

.

You can even make multiple pieces in one piece wise function using this format: {condition1: value1, condition2, value2, ..., default}, for example:

{
−
1
<
x
<
1
:
3
x
,
3
<
x
<
4
:

x
2

,
x
}

{\displaystyle \{-1<x<1: 3x, 3<x<4: x^2, x \}}

, which is shown in Fig. 2.

## Domain and Range Restrictions

Add restrictions at the end of your expression with {}. For example:

# y
2
x
3
{
−
3
<
x
<
6
}

{\displaystyle y=2x+3\{-3<x<6\}}

, or,

# y
s
i
n
(
x
)
{
−
0.5
<
y
<
0.5
}

{\displaystyle y=sin(x)\{-0.5<y<0.5\}}

, shown in Fig. 3.

## Parametric and Implicit

You can write implicit equations such as

y
2

x
2

=
25

{\displaystyle y^2+x^2=25}

. And you can write parametric similar to the way you can write points, for example

(
s
i
n
(
2
t
)
,
c
o
s
(
3
t
)
)

{\displaystyle (sin(2t), cos(3t))}

. You have to use the letter t.

## Regressions

To create a regression you need two sets, in the form of two lists or in the form of a table. You then use ~ to make a regression. For example, if you have two sets

y
1

{\displaystyle y\_1}

and

x
1

{\displaystyle x\_1}

, you can create a regression:

y
1

∼
m

x
1

c

{\displaystyle y\_1\sim mx\_1+c}

for a linear regression, or

y
1

∼
a

x
1
2

b

x
1

c

{\displaystyle y\_1 \sim ax\_1^2+bx\_1+c}

for a quadratic regression, you can make any type you want!