Tables, lists, and calculated columns

Your feedback matters:

Lesson progressPractice problems 0/4

Use Desmos tables and lists to organize paired data, evaluate functions, create calculated columns, and compare row-by-row results.

Difficulty: Intermediate
Estimated time: 50 minutes
Domains: Algebra Advanced Math

What you’ll learn

Keep each input and its output together in the same table row.
Read table columns like x_1 and y_1 as lists.
Type a formula in a column header so Desmos fills in every row.
Apply a function to a whole list in square brackets.
Compare given values with calculated ones without redoing the arithmetic.
Answer with exactly what the question asks for: an input, an output, a choice, or a unit.
Tell when typing one value, or working by hand, beats setting up a table.

Prerequisites

Why this matters on the SAT

Type the rule once, check every value

The SAT can hand you a function and ask which table matches it, which of several inputs gives a certain output, or what new value each row of a data table produces. All three come down to one move: type the rule once and let Desmos run it on every value. A table does that in labeled columns. A list does it in one line, when you don't need the labels.

SAT example

The function $f$ is defined by

f(x)=3x^2-4.

Which table shows the corresponding values of $f(x)$ for $x=-2$ , $x=0$ , and $x=3$ ?

$x$	$f(x)$
$-2$	$8$
$0$	$-4$
$3$	$23$

$x$	$f(x)$
$-2$	$16$
$0$	$4$
$3$	$31$

$x$	$f(x)$
$-2$	$-16$
$0$	$-4$
$3$	$-31$

$x$	$f(x)$
$-2$	$8$
$0$	$4$
$3$	$23$

Fast Desmos solution

Type f(x)=3x^2-4 and add a table. Put $-2$ , $0$ , and $3$ in the x_1 column, then change the y_1 header to f(x_1). Desmos fills in $8$ , $-4$ , and $23$ , which is table A.

The table paid off because you needed the same function three times. If the question asked only for $f(3)$ , typing f(3) would be quicker than building a table.

A table doesn't mean you need regression, either. Regression finds a model, or its unknown numbers, from data, and it's the next lesson. When the question already gives you the rule and wants its outputs, skip regression and evaluate.

Calculator loads as you approach

One calculated column checks all three pairs at once.

When should you use a table or list?

Both run one rule on many values. The difference is what you need to keep in view: a table keeps labels and rows, and a list fits on one line.

Use a table

Each input has to stay with its own output, like the pairs $(2,11)$ and $(5,20)$ .
The question asks which table matches a function, like the $f(x)=3x^2-4$ question above.
Several inputs need the same calculation, like four answer choices to plug into one rule.
You want to compare two expressions at the same inputs, so each one gets its own column.
The question gives you data and a new quantity to work out for every row, like $2g(x)-3x$ from values of $g$ .

Use a list

You want to test several inputs and don’t need labeled rows.
The values fit easily in one line, like L=[-2,0,3,5].
The outputs come back as a short list you can read at a glance, so it’s easy to see which position hits the target.

Skip both and use direct evaluation or hand calculation when you need only one simple value, like the $f(3)$ above, when a pattern you can see already gives the answer, or when one short equation is faster than typing every choice. If $f(x)=2x+1$ and the question asks when $f(x)=9$ , solving $2x+1=9$ gets you $x=4$ before you could fill in a table.

Check your understanding:

Say $f(x)=4x-1$ and you need only $f(3)$ . What would you type? Then a different question gives $q(x)=x^3-5x+9$ and asks which of $-3$ , $-1$ , $2$ , or $4$ gives $q(x)=7$ . How would you set that one up?

Keep matching values in the same row

Desmos usually names the columns of your first table x_1 and y_1. Each full row is one ordered pair:

(x_1,y_1).

Say the question gives you the pairs $(2,11)$ , $(5,20)$ , and $(8,29)$ . Each $x$ goes in the same row as its $y$ :

`x_1`	`y_1`
$2$	$11$
$5$	$20$
$8$	$29$

Each column is also a list. Here x_1 is the list $[2,5,8]$ , and y_1 is $[11,20,29]$ . Desmos matches them by position: first with first, second with second, third with third. That's why a row has to stay together.

Add a second table, and its columns may be called x_2 and y_2. So x_1 doesn't mean "the first column of any table." Use the names you see in the table you're working in.

Check your understanding:

The pairs are $(2,11)$ , $(5,20)$ , and $(8,29)$ . A student types x_1=[2,5,8] and y_1=[11,29,20]. What went wrong?

Common mistake:

It’s tempting to sort or move one column so it looks tidy. If the values paired with it don’t move too, the table still looks neat, but it holds the wrong ordered pairs.

Calculator loads as you approach

Each row keeps one given pair together.

Calculate a whole column

A calculated column is a column that one formula fills in for you. Define

g(x)=3x^2-2,

type $-2$ , $0$ , $1$ , and $4$ in x_1, then change the next column's header to

g(x_1).

Desmos fills in the outputs:

`x_1`	`g(x_1)`
$-2$	$10$
$0$	$-2$
$1$	$1$
$4$	$46$

The output cells turn gray, because the header formula makes them. You can't fix one by typing over it. To change an output, change its input or the header formula.

Try it yourself:

Change the last input from $4$ to $3$ . Which output should change, and to what? Make your guess, then check that the last output becomes $g(3)=25$ .

Check your understanding:

Your inputs are in x_1. Why type g(x_1) in the header and not g(x)?

Common mistake:

Leaving the header as y_1 and typing each output yourself. That’s the same arithmetic done by hand, and if you change an input later, it’s easy to miss that its output is now wrong.

Calculator loads as you approach

The formula in the second header fills in every output.

Use a list when you don't need labels

Square brackets make a list:

L=[-2,0,3,5].

Define

f(x)=x^2-3x+2,

and type f(L). Desmos applies $f$ to each item, in order:

f(L)=[12,2,2,12].

Here's a shortcut, if you want it: three dots fill in a range of integers, and both ends are included. For example,

[1...6]=[1,2,3,4,5,6].

Start with two numbers, and Desmos counts by the step between them:

[2,4...10]=[2,4,6,8,10].

Keep a list short enough to compare at a glance. When labels, units, or more than one calculated value matter, a table is clearer.

Try it yourself:

Change the $5$ in $L$ to $4$ . Predict the new last output before you look. You should get $f(4)=6$ .

Check your understanding:

If $h(x)=2x+5$ , what does h([1,2,3,4]) give you?

Calculator loads as you approach

The outputs come out in the same order as the inputs.

Combine given values row by row

Sometimes a question gives you a few values of one function and builds a new function from them. Say you're given

$x$	$g(x)$
$-1$	$7$
$2$	$13$
$5$	$25$

and

h(x)=2g(x)-3x.

Put the $x$ -values in x_1 and their $g(x)$ values in y_1, so in each row, y_1 stands for $g(x)$ . In a third header, type

2y_1-3x_1.

Desmos gives you the three values of $h$ you can find:

`x_1`	`y_1`	`2y_1-3x_1`
$-1$	$7$	$17$
$2$	$13$	$20$
$5$	$25$	$35$

Careful here. The table gives you $g$ only at these three inputs. It says nothing about $g(x)$ anywhere else, so the new column works only for rows where the question gives you $g(x)$ .

Check your understanding:

Using the table above, why can you find $h(2)$ but not $h(4)$ ?

Common mistake:

Reading $2g(x)-3x$ as $g(2x-3)$ . They do different jobs: $2g(x)-3x$ doubles the output of $g$ and then subtracts $3x$ , while $g(2x-3)$ changes the input before $g$ acts. At $x=2$ , the first gives $2(13)-6=20$ , but the second asks for $g(1)$ , which the table doesn’t even give you.

Calculator loads as you approach

The third column combines each input with its own given output.

Example: Test every choice in one table

Worked example

The function $p$ is defined by

p(x)=(x+3)(x-2)(x-5).

Which candidate input, $-2$ , $1$ , $3$ , or $6$ , gives a value of $34$ for

p(x-1)+4?

A
$-2$
B
$1$
C
$3$
D
$6$

Step 1

Keep the choices together

Add a table and type $-2$ , $1$ , $3$ , and $6$ in x_1. Those four are the only values the question asks you to test.

Step 2

Type the function once

Define p(x)=(x+3)(x-2)(x-5). In the table's next header, type p(x_1-1)+4. That runs the whole rule, the $-1$ inside the parentheses and the $+4$ outside, on every choice.

Calculator loads as you approach

The whole rule, worked out for every choice.

Step 3

Find the target in its row

The outputs are $4$ , $34$ , $4$ , and $4$ . The $34$ sits in the row where $x=1$ , so the answer is B.

Why do three rows show $4$ ? For those choices, $x-1$ is $-3$ , $2$ , or $5$ . Those are the zeros of $p$ , so $p$ gives $0$ and only the $+4$ is left.

Step 4

Check the answer

For $x=1$ , the input to $p$ is $x-1=0$ . So

\begin{aligned} p(1-1)+4 &=p(0)+4\\[1.4em] &=(0+3)(0-2)(0-5)+4\\[1.4em] &=3(-2)(-5)+4\\[1.4em] &=34. \end{aligned}

That's the target, so $\boxed{x=1}$ .

Common mistake: Working out $p(x)-1+4$ instead of $p(x-1)+4$ . The $-1$ sits inside the parentheses, so it changes the input to $p$ , not the output. At $x=1$ , the wrong version gives $p(1)+3=19$ , not $34$ .

A table wins here because you have to check a transformed cubic at four inputs. For one simple substitution, direct evaluation would be shorter.

Finish the solution

The four candidate inputs are already in the table. Define $r$ , add one calculated column, and submit the input from the row that hits the target.

Finish the solution

The function $r$ is defined by

r(x)=\frac{x^2+11}{x+1}.

Which of the candidate inputs $1$ , $2$ , $4$ , or $9$ gives

r(x)=\frac{27}{5}?

First steps

Define the function $r$ .
Type r(x_1) in the empty table header.
Find the row whose output is $5.4$ , which is $\frac{27}{5}$ written as a decimal.
Submit the input from that row.

Finish it

Your answer

Calculator loads as you approach

Type r(x₁) in the empty header.

Practice problems

Now it's your turn. Try each problem before you open its hint, and reread the last line of the question before you submit.

Practice problem

The function $q$ is defined by

q(x)=12\left(\frac{1}{2}\right)^{x-1}.

Which table shows the corresponding values of $q(x)$ for $x=0$ , $x=1$ , and $x=3$ ?

$x$	$q(x)$
$0$	$6$
$1$	$12$
$3$	$48$

$x$	$q(x)$
$0$	$24$
$1$	$12$
$3$	$6$

$x$	$q(x)$
$0$	$24$
$1$	$12$
$3$	$3$

$x$	$q(x)$
$0$	$3$
$1$	$12$
$3$	$24$

Calculator loads as you approach

Evaluate q at all three inputs, then find the matching table.

Practice problem

The table gives three values of the function $m$ .

$x$	$m(x)$
$-2$	$3$
$1$	$-1$
$4$	$5$

The function $n$ is defined by

n(x)=[m(x)]^2-x.

What is the value of

n(-2)+n(1)+n(4)?

Your answer

Calculator loads as you approach

Build a column from the given values, then add up the outputs.

Practice problem

The function $w$ is defined by

w(x)=\frac{x^2+7x-5}{x-1}.

Which choice is a value of $x$ for which $w(x)=13$ ?

Calculator loads as you approach

Evaluate w at all four choices and look for 13.

Finish the lesson

4 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Each table row keeps one input with its output, and each named column is a list.
A formula like f(x_1) in a header fills in every row and updates when you change an input.
A list in square brackets runs one rule on several inputs, without the labels of a table.
A new column can combine given values row by row, but it can't fill in function values the question doesn't give.
For one simple output, type it directly. Use a table or list when the same calculation repeats or rows need to stay paired.
Submit what the question asks for, which isn't always the number Desmos shows most clearly.

Next lesson

Desmos regression from a supplied table

Fit linear, quadratic, and exponential models after entering paired data correctly.

Start next lesson

Practice

Practice this lesson

35 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

Start practice