Tables, lists, and calculated columns

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Difficulty: Intermediate
Estimated time: 50 minutes
Domains: Algebra Advanced Math
Techniques: TablesListsCalculated-columnsFunction-evaluationCandidate-testing

What you’ll learn

Keep corresponding values paired in the same table row.
Recognize table columns as ordered lists.
Enter one formula in a column header to calculate every row.
Apply a function to a bracketed list.
Compare supplied and calculated values without repeating the same arithmetic.
Answer with the exact input, output, choice, or unit requested.
Choose direct evaluation or hand calculation when a table adds unnecessary setup.

Prerequisites

Why this matters on the SAT

Check several values with one accurate rule

The SAT can give you a function and ask which table matches it, which listed input produces a target output, or how a new quantity is calculated from each row of supplied data. A Desmos table lets you enter the rule once and calculate every row. A list does the same job in a more compact form when you do not need labeled columns.

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 $x=3$ ?

A
$x$
$f(x)$
$-2$
$8$
$0$
$-4$
$3$
$23$
B
$x$
$f(x)$
$-2$
$16$
$0$
$4$
$3$
$31$
C
$x$
$f(x)$
$-2$
$-16$
$0$
$-4$
$3$
$-31$
D
$x$
$f(x)$
$-2$
$8$
$0$
$4$
$3$
$23$

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

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

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

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

Fast Desmos solution

Define f(x)=3x^2-4, add a table, and enter $-2$ , $0$ , and $3$ in the x_1 column. Replace the y_1 header with f(x_1). Desmos fills the second column with $8$ , $-4$ , and $23$ , so the answer is A.

The method is valuable because the same function is evaluated several times. If the question asked only for $f(3)$ , entering f(3) would be faster than building a table.

This lesson uses tables and lists to organize and calculate known values. The next lesson introduces regression, which fits a model to supplied data. Creating a table does not automatically mean that regression is needed.

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One calculated column checks all three input-output pairs.

When should you use a table or list?

Match the setup to the information you need to preserve. Tables keep labels and rows visible. Lists are compact. Direct evaluation is strongest for one simple output.

Use a table when…

corresponding values must stay paired by row;
the question asks which table represents a function;
several inputs need the same calculation;
two expressions must be compared for the same inputs; or
supplied data need a new quantity calculated in every row.

Use a list when…

you need to test several inputs without labeled rows;
the candidate values fit clearly in one expression; or
a compact output list makes the position match easy to read.

Use direct evaluation or hand calculation when only one simple function value is needed, a visible pattern gives the answer immediately, or one short equation is faster than entering several candidates. Do not begin with regression when the rule is already given and you only need its outputs. Regression finds a model or unknown parameters from data.

Check your understanding:

For $f(x)=4x-1$ , what is the best Desmos entry for finding only $f(3)$ ? A different question gives $q(x)=x^3-5x+9$ and asks which candidate, $-3$ , $-1$ , $2$ , or $4$ , gives $q(x)=7$ . Which setup is stronger for that question?

1. Keep matching values in the same row

In the first Desmos table, the columns are usually named x_1 and y_1. Each complete row represents one ordered pair:

(x_1,y_1).

If the supplied pairs are $(2,11)$ , $(5,20)$ , and $(8,29)$ , enter them like this:

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

The columns are also lists. Here, x_1 means the list $[2,5,8]$ , and y_1 means the list $[11,20,29]$ . Desmos pairs entries by position: first with first, second with second, and third with third.

If you add another table, its columns may be named x_2 and y_2. Use the names shown in the table you are working with. x_1 never refers to the first column of every table.

Check your understanding:

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

Common mistake:

Sorting or moving one column without moving the values paired with it. A table may still look tidy while representing the wrong ordered pairs.

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Each row preserves one supplied pair.

2. Calculate an entire column

A calculated column uses one formula to generate every output. Define

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

enter $-2$ , $0$ , $1$ , and $4$ in x_1, then replace the next column header with

g(x_1).

Desmos calculates:

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

The calculated cells are gray because their values come from the column formula. You cannot correct one computed cell by typing over it. Edit the source input or the header formula instead.

Try it yourself:

Change the final input from $4$ to $3$ . Predict which output will change, then confirm that the last computed value becomes $g(3)=25$ .

Check your understanding:

A table contains inputs in x_1. Why should the calculated header use g(x_1) instead of g(x)?

Common mistake:

Leaving the header as y_1 and manually typing outputs. That repeats the arithmetic and makes later input changes easier to miss.

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The formula in the second header generates every output.

3. Use a list when labels are unnecessary

Square brackets create a list:

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

After defining

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

enter f(L). Desmos applies the function to every list item in order:

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

As an optional shortcut, use three dots to build an inclusive integer range. For example,

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

You can also begin with a different step:

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

Keep the list short enough to compare accurately. If labels, units, or several calculated quantities matter, a table is clearer.

Try it yourself:

Replace $5$ in $L$ with $4$ . Before checking the result, predict the new final output. It should be $f(4)=6$ .

Check your understanding:

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

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The outputs stay in the same order as the inputs.

4. Transform supplied values row by row

Sometimes a question supplies values of one function and defines a new quantity from them. Suppose:

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

and

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

Enter the $x$ -values in x_1 and the corresponding $g(x)$ values in y_1. In a third header, enter

2y_1-3x_1.

Desmos calculates the three known values of $h$ :

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

The table tells you only the listed values of $g$ . It does not define $g(x)$ at every other input. Use the calculated column only for rows where the problem supplies $g(x)$ .

Check your understanding:

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

Common mistake:

Replacing $g(x)$ with $g(2x-3)$ instead of calculating $2g(x)-3x$ . The coefficients outside a function output and the operations inside its input do different jobs.

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The third column combines each input with its matching supplied output.

5. Example: Test every listed input

Worked example

The function $p$ is defined by

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

Which candidate input gives a value of $34$ for

p(x-1)+4?

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

Step 1

Keep the candidates together

Add a table and enter $-2$ , $1$ , $3$ , and $6$ in x_1. These are the only candidates the question asks you to test.

Step 2

Enter the function once

Define p(x)=(x+3)(x-2)(x-5). In the table's next header, enter p(x_1-1)+4. This applies the complete transformed rule to every candidate.

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The transformed rule is evaluated for every listed input.

Step 3

Match the target by row

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

Step 4

Check the winning input

For $x=1$ , the inside input is $x-1=0$ . Therefore,

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

Thus, $\boxed{x=1}$ .

Common mistake:

Calculating $p(x)-1+4$ instead of $p(x-1)+4$ . The subtraction inside the parentheses changes the input to $p$ .

The table is faster here because the transformed cubic must be checked at four listed inputs. For one simple substitution, direct evaluation would be shorter.

Finish the solution

The candidate inputs are already entered. Define the function, add one calculated column, and submit the input paired with the target output.

Match a rational output

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 $r$ .
Enter r(x_1) in the empty table header.
Find the row whose output is $5.4$ , the decimal equivalent of $\frac{27}{5}$ .
Submit the input from that row.

Finish it

Your answer

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Replace the empty header with the function evaluated at x₁.

Practice problems

SAT practice problems

Use a table when rows must stay paired or several inputs need the same rule. Use a compact list when labels do not matter. Return to the prompt before submitting so that you answer with the requested input, output, choice, and unit.

Match an exponential function to a table

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 $x=3$ ?

Table A

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

Table B

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

Table C

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

Table D

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

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Use the calculator to identify the table that matches all three inputs.

Transform supplied function values

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

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Use the calculator to determine the requested sum from the supplied values.

Test a candidate list

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 $w(x)=13$ ?

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Use the calculator to identify which candidate produces the target output.

Finish the lesson

4 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Each table row preserves one input-output pair, and each named column is a list.
A formula such as f(x_1) in a column header calculates every row and updates when an input changes.
A bracketed list applies one rule to several inputs without the labels of a table.
New columns can combine supplied values row by row, but they do not define missing function values.
Use direct evaluation for one simple output. Use a table or list when repeated calculation or row pairing makes it faster and clearer.
Submit the quantity the question asks for, not automatically the most prominent calculator output.

Next lesson

Regression from a supplied table

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

Start next lesson

Free Practice

What you’ll learn

Prerequisites

Check several values with one accurate rule

Fast Desmos solution

When should you use a table or list?

Use a table when…

Use a list when…

1. Keep matching values in the same row

2. Calculate an entire column

3. Use a list when labels are unnecessary

4. Transform supplied values row by row

5. Example: Test every listed input

Keep the candidates together

Enter the function once

Match the target by row

Check the winning input

Finish the solution

Match a rational output

First steps

Finish it

Practice problems

SAT practice problems

Match an exponential function to a table

Transform supplied function values

Test a candidate list

4 practice examples left

Quick recap

Regression from a supplied table

Free question bank practice problems