Desmos regression and statistics for the SAT

On test day

Two exact points and a question about a line: put them in a table, type y1~mx1+b on the row below, and read $m$ and $b$ in the box under it.
A table with one empty cell: fit the line to the complete rows only. Then type f(x)= your , and f(8) if the empty cell is at $x=8$ .
Three points on a and a question about its : fit the parabola and click the gray dot where the curve turns. The given point with the biggest doesn't have to be the vertex.
A short list of numbers: type mean([3,5,10]), and put the same list inside median or stdev. In a frequency table, type each value as many times as its frequency.
A , a box plot, or "What does 0.8 represent?" gives you nothing to type. Read the display or the story.

When to reach for a table

A means typing points into a Desmos table and letting Desmos find the equation through them. You tell it the shape, such as a line or a parabola, and it works out the numbers.

The points don't always come as a table. A value like $f(2)=11$ is the point $(2,11)$ , and "after 3 minutes the tank holds 210 liters" is the point $(3,210)$ .

A line needs two points and a parabola three. With exactly that many, the equation Desmos finds goes through every point, so its numbers are exact. An through two points needs no table: divide the outputs and take a root, as the nonlinear functions lesson shows.

Leave Desmos closed when there's nothing exact to type. On a scatterplot you can only guess where each dot sits, and the question is about the already drawn, so read the line. The two-variable data lesson shows how. Read a box plot or a the same way. And when a question asks what a number means in the story, no fit can tell you.

Fitting a line through two points

A water tank drains at a steady rate. Three minutes after a valve opens, the tank holds 210 liters. Eight minutes after it opens, the tank holds 110 liters. Which equation gives the number of liters $y$ in the tank $x$ minutes after the valve opens?

A. $y=-20x+270$
B. $y=-0.05x+210.15$
C. $y=-20x+210$
D. $y=20x+150$

The two readings are the points $(3,210)$ and $(8,110)$ .

Fit a line

Open the + menu at the top of the rows and choose table.
Type 3 under $x_1$ and 210 under $y_1$ . Put 8 and 110 in the next row.
On the next empty row below the table, type y1~mx1+b. Desmos lowers each 1 by itself, and the ~ needs Shift.
A Regression Parameters box appears under that row, showing $m=-20$ and $b=270$ . You can ignore the other numbers in it.

The line is $y=-20x+270$ , so the answer is A. To check it, put in the second reading: $-20(8)+270=110$ .

You could work out the by hand and then solve for $b$ , but the table gets both at once. Don't type an underscore for the small 1. Desmos basics shows what it does to the row.

Trap: The first reading as the starting value

Choice C, $y=-20x+210$ , has the right slope but starts the tank at 210 liters. That's the amount at minute 3.

What to do: The starting value is the amount at minute 0. Take it from b in the box.

Choice B divides the wrong way: 5 minutes over a drop of 100 liters gives $-0.05$ . Choice D has a positive slope, but the tank is losing water.

Filling in a missing value

The table gives values of the linear function $f$ . What is the value of $k$ ?

$x$	$f(x)$
1	7
3	17
8	$k$

Type only the two complete rows into the table and fit y1~mx1+b. The box shows $m=5$ and $b=2$ , so $f(x)=5x+2$ .

Find the missing value

On a new row, type f(x)=5x+2.
On the row under it, type f(8). It prints = 42.

So $k=42$ . For more than one gap, give each its own row, like f(2) and f(5).

Leave the row with the empty cell out of the table. If you type a 0 there to fill it, Desmos treats the 0 as a real value, and the line it fits comes out wrong.

Finding a parabola's vertex

A parabola passes through $(1,0)$ , $(2,6)$ and $(6,-10)$ . What is the $x$ -coordinate of its vertex?

Type the three points into a table. On the row below it, type y1~ax1^2+bx1+c straight through. Desmos lowers each 1, and the + after the 2 drops you back out of the .

The box shows $a=-2$ , $b=12$ and $c=-10$ , and Desmos draws the curve through the three points. Click the curve, then click the gray dot at its highest point. It reads $(3,8)$ , so the answer is 3.

Trap: The biggest given output

Of the three points, $(2,6)$ has the biggest output, so 2 looks like the answer. The curve keeps climbing past it and turns at $x=3$ .

What to do: Read the vertex off the fitted curve. None of the given points has to be the vertex.

If the question wants , keep the same $a$ and use the vertex: $y=-2(x-3)^2+8$ . The nonlinear functions lesson shows how to find a vertex by hand.

Statistics from a short list

For a short list of numbers, skip the table and put the numbers in square brackets. mean([3,5,10]) prints 6, and median([3,5,10]) prints 5. stdev works the same way for the . If the numbers are already in a table column, mean(y1) uses them as they are.

A frequency table says how many times each value appears:

Value	Frequency
2	3
5	2
10	2
12	1

Type each value that many times. mean([2,2,2,5,5,10,10,12]) prints 6, and median of the same list prints 5. If you type each value once, mean([2,5,10,12]) prints 7.25, because the 2, which appears three times, now counts only once.

To compare spread, put each set's stdev on its own row. The sets $1,5,9$ and $1,1,9$ have the same smallest and largest values, but stdev([1,1,9]) prints the bigger number, so that set is more spread out.

Skip the list when the question only describes the numbers, as in "40 whole numbers, all between 10 and 50." There's nothing to type. Skip it too when a frequency table adds up to more than about 20 values. The one-variable data lesson covers what these numbers mean.

Practice

Work through the three examples, then try five more questions. Before you open Desmos on each one, decide whether there's anything exact to type.

Worked example

At Xiomara’s woodworking shop, the linear function $f$ gives the number of board feet used for $x$ cabinet frames.

$x$ | $f(x)$ 7 | 87.5 12 | 125 16 | 155 21 | 192.5

Which equation defines $f$ ?

A
$f(x) = 7.5x + 35$
B
$f(x) = 37.5x + 35$
C
$f(x) = 7.5x - 35$
D
$f(x) = 35x + 7.5$

Show the answer

A
$f(x) = 7.5x + 35$
Correct answer
Using the points $(7,87.5)$ and $(12,125)$ , the is $\frac{125-87.5}{12-7}=7.5$ . Substituting $(7,87.5)$ into $f(x)=7.5x+b$ gives $b=35$ , so $f(x)=7.5x+35$ .
B
$f(x) = 37.5x + 35$
This uses the increase, $125-87.5=37.5$ , as the without dividing by the increase, $12-7=5$ . The slope is $37.5/5=7.5$ .
C
$f(x) = 7.5x - 35$
This subtracts the $35$ instead of adding it. Substitution gives $87.5=7.5(7)+35$ , so the intercept is positive.
D
$f(x) = 35x + 7.5$
This swaps the and , placing $35$ as the of $x$ and $7.5$ as the . The calculated slope is $7.5$ , and the intercept is $35$ .

Solution

Use two points from the table to find the :

$m=\frac{125-87.5}{12-7}=\frac{37.5}{5}=7.5$ .

Write the as $f(x)=7.5x+b$ . Substitute $(7,87.5)$ :

$87.5=7.5(7)+b$

$87.5=52.5+b$

$b=35$ .

Therefore, the is $f(x)=7.5x+35$ . It also gives $f(21)=7.5(21)+35=192.5$ , matching the table.

Report a problem

Worked example

In the $xy$ -plane, a line passes through the points $(84, 95)$ and $(98, 89)$ .

Which equation defines this line?

A
$y = -\frac{7}{3}x + 291$
B
$y = -\frac{3}{7}x + 95$
C
$y = -\frac{3}{7}x + 131$
D
$y = \frac{3}{7}x + 59$

Show the answer

A
$y = -\frac{7}{3}x + 291$
This divides the horizontal change, $14$ , by the vertical change, $-6$ , giving $-\frac{7}{3}$ . is vertical change divided by horizontal change.
B
$y = -\frac{3}{7}x + 95$
This uses the first point's $y$ -coordinate, $95$ , as the $y$ -intercept. The point $(84,95)$ is not on the $y$ -axis because its $x$ -coordinate is not $0$ .
C
$y = -\frac{3}{7}x + 131$
Correct answer
The is $\frac{89-95}{98-84}=-\frac{6}{14}=-\frac{3}{7}$ . Substituting $(84,95)$ into $y=-\frac{3}{7}x+b$ gives $95=-36+b$ , so $b=131$ .
D
$y = \frac{3}{7}x + 59$
This treats the decrease from $95$ to $89$ as a positive vertical change. Since $y$ decreases as $x$ increases, the must be negative.

Solution

Calculate the using vertical change divided by horizontal change:

$m=\frac{89-95}{98-84}=\frac{-6}{14}=-\frac{3}{7}$ . Use $y=mx+b$ and substitute $(84,95)$ :

$95=-\frac{3}{7}(84)+b$

$95=-36+b$

$b=131$ . Therefore, the line is $y=-\frac{3}{7}x+131$ .

Report a problem

Worked example

The mass of silt in each load from a dredger, $L(d)$ , measured in metric tons, is a quadratic function of the dredging depth $d$ , in meters, for $2\leq d\leq 6$ . Measurements give $L(2)=136$ , $L(4)=156$ , and $L(6)=160$ .

Which equation defines $L$ in a form that displays the maximum silt mass per load as a constant?

A
$L(d) = -2(d + 5.5)^2 + 160.5$
B
$L(d) = -2(d + 22)^2 + 100$
C
$L(d) = -2(d - 6)^2 + 160$
D
$L(d) = -2(d - 5.5)^2 + 160.5$

Show the answer

A
$L(d) = -2(d + 5.5)^2 + 160.5$
Using $b/(2a)$ instead of $-b/(2a)$ gives a depth of $-5.5$ meters. This sign error produces $(d+5.5)^2$ .
B
$L(d) = -2(d + 22)^2 + 100$
This transfers the from $-2d^2+22d+100$ directly into $-2(d+22)^2+100$ . Completing the square requires factoring out $-2$ from the terms and adjusting the to preserve the .
C
$L(d) = -2(d - 6)^2 + 160$
This treats $(6,160)$ , the measurement with the largest recorded mass, as the . The largest of the three measured values does not establish the 's maximum.
D
$L(d) = -2(d - 5.5)^2 + 160.5$
Correct answer
The measurements give $L(d)=-2d^2+22d+100=-2(d-5.5)^2+160.5$ . Since $-2(d-5.5)^2\leq 0$ , the maximum silt mass is the displayed $160.5$ metric tons, attained at $d=5.5$ meters.

Solution

Write $L(d)=ad^2+bd+c$ . Substituting the measurements gives $4a+2b+c=136$ , $16a+4b+c=156$ , and $36a+6b+c=160$ .

Subtract consecutive to get $12a+2b=20$ and $20a+2b=4$ . Subtracting these gives $8a=-16$ , so $a=-2$ . Then $-24+2b=20$ gives $b=22$ , and $-8+44+c=136$ gives $c=100$ .

Complete the square: $L(d)=-2(d^2-11d)+100=-2[(d-5.5)^2-30.25]+100=-2(d-5.5)^2+160.5$ .

The squared term is smallest at $d=5.5$ , where it equals zero. Multiplying it by $-2$ makes every other smaller, so the displayed $160.5$ is the maximum silt mass per load in metric tons.

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5 questions on this skill, with every choice explained.

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Related lessons

Desmos for data and regression

When to reach for a table

Fitting a line through two points

Filling in a missing value

Finding a parabola's vertex

Statistics from a short list

Practice