Lists

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By defining a list of values, you can quickly plot a series of points, lines, or curves, or calculate statistical values such as mean, median, or standard deviation.

Watch the video for an introduction to lists, and read on to learn more.

 

Create a List

Create a list of values in between square brackets and separate elements with commas, such as \([1,3,4]\).

To generate a list of evenly spaced numbers, use an ellipsis. For example, \([1...10]\) represents the list of integers between \(1\) and \(10\) and \([1,3...11]\) represents the list of odd integers between \(1\) and \(11\).

Screenshot of different length of lists.

You can use lists in any expression where you would use a single number, including the definition of a variable. This allows you to graph multiple lines or curves simultaneously.

Screenshot using a list a to graph multiple lines of the form y=ax.

If \(L\) is a list, you can refer to individual elements or subsets of elements by choosing the index of that element:

  • Access the first element of \(L\) with \(L[1]\)
  • Create a new list of the first, third, and fifth elements of \(L\) with \(L[1,3,5]\)
  • \(L[1…5]\) accesses the first \(5\) elements
  • Generate a list of elements of \(L\) from the third to the last with \(L[3…]\)
  • Pull out the positive elements of \(L\) with \(L[L \gt 0]\)
  • If \(M\) is another list, produce a list of elements from \(L\) at the indices specified by \(M\) with \(L[M]\)

See examples of each of these in the Index into a List graph.

Screenshot of a list L with a table showing different subsets of the L.

 

Lists in Geometry

In the Geometry Tool, use lists to define parameters in transformations, create concentric circles, or make “string” art with gliders. Read more about gliders in our Sliders and Movable Points article.

Gif in geometry of concentric circles using a list and string art using gliders and lists.

 

Lists in 3D

In the 3D Graphing Calculator, you can use lists to plot multiple points and graph multiple curves or surfaces.

For instance, if you define a list \(L=[-4…4]\), then the expression \((L,L,L)\) will graph \(9\) points arranged in a diagonal line. Similarly, the expression \((L\sin(t), L\cos(t), L)\) will graph \(9\) circles, each with a radius and position on the \(z\)-axis defined by the values of \(L\).

Graph in 3D with a horizontal line and multiple circles graph with list L.

List Operations

List operations allow you to work with your lists even more specifically. Find list operations in the keypad, or type any function directly into an expression line.
Screenshot of the keypad with function menu open and a highlight around the list opeartions.

 

   Function Inputs Try typing... Description
repeat Two equal sized lists repeat\(\mathbf{([1,5],[3,2])}\)

Repeats each value in the first element according to the values of the corresponding second element.

So, in the example, 1 is repeated three times, and 5 is repeated twice.

Use the repeat function to create frequency tables.
join Two or more lists join\(\mathbf{([1,5],[3,2])}\) Combines two or more lists into one list.
sort One list sort\(\mathbf{([5,1])}\) Sorts the elements from smallest to largest.
shuffle One list shuffle\(\mathbf{([2,3,9])}\)

Randomizes the elements in your list.

When you add the shuffle function to the expression list, the Randomize icon Randomize icon will appear. You can click it to reshuffle your list elements at any time.
unique One list unique\(\mathbf{([1,5,5,10])}\) Outputs a list of each element that is used in the list and removes any repeats.
for Lists or intervals

\(\mathbf{(a,b)}\) for \(\mathbf{a=[1,2], b=[1,2]}\) 

\((\)sin\(\mathbf{(c)}\),cos\(\mathbf{(c)}\)\()\) for \(\mathbf{0 \lt x \lt 2\pi}\) 		Test multiple values for a variable with either a list or an interval.

 

List Comprehensions

You can create new lists based on existing lists using a list comprehension, which is written in the form: \(\left(x,y\right)\operatorname{for}x=\left[1...10\right],y=\left[1...10\right]\). This would create a list of the 100 points with integer coordinates \(1\le x \le 10\), \(1\le y \le 10\) and form a 10x10 grid of points.

A list comprehension consists of a body expression, which in this case \((x,y)\), and one or more input list variables. The resulting list is created by evaluating the body expression for each combination of values from the input lists.

Here are some more examples of list comprehensions and the lists they produce:

Comprehension Result
\(2k\operatorname{for}k=\left[1,2,3\right]\) \( [2, 4, 6]\)
\(1\operatorname{for}k=\left[1...1000\right]\) \([1, 1, 1, ...\)\((1000\) times)\(]\)
\(\operatorname{mean}\left(L.\operatorname{random}\left(10\right)\right)\operatorname{for}k=\left[1...100\right]\) A list of \(100\) mean values, each calculated from \(10\) randomly selected values of list \(L\).

 

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