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Free Weekly SAT Workshops · Week 12
Desmos 2.
Regressions & Statistics
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Put the data in. Let Desmos find the equation.
Four Desmos 1 moves we will reuse
Tap a gray point
Vertex, intercepts, crossings: tap, then read.
Zoom to fit
Nothing on screen? Zoom out until the story fits.
Overlay the choices
The right choice lands exactly on top.
\(1.41\overline{6}\to\tfrac{17}{12}\)
Fraction toggle
Turn a long decimal into an exact grid-in.
Desmos is a trademark of Desmos Studio PBC. No affiliation.
A
Section A
Table in, equation out.
Type the table. A regression finds the equation.
Table in, exact line out.
SAT-style
The table shows several values of \(x\) and their corresponding values of \(f(x)\) for a linear function \(f\). Which equation defines \(f\)?
| \(x\) | \(f(x)\) |
|---|---|
| 1 | 20 |
| 3 | 12 |
| 5 | 4 |
A\(f(x)=4x+24\)sign
B\(f(x)=24x-4\)swapped
C\(f(x)=-4x+24\)✓
D\(f(x)=-4x+20\)f(1) as intercept
TYPE
expression
table
1
| \(x_1\) | \(y_1\) | |
|---|---|---|
| 1 | 20 | |
| 3 | 12 | |
| 5 | 4 |
2
\(y_{1}\sim mx_{1}+b\)
READ
Desmos is a trademark of Desmos Studio PBC. No affiliation.
Two points? Make a two-row table.
SAT-style · grid-in
In the \(xy\)-plane, line \(\ell\) passes through the points \((-3,\,11)\) and \((5,\,-5)\). The equation of line \(\ell\) can be written as \(y=mx+b\), where \(m\) and \(b\) are constants. What is the value of \(b-m\)?
TYPE
expression
table
1
| \(x_1\) | \(y_1\) | |
|---|---|---|
| -3 | 11 | |
| 5 | -5 |
2
\(y_{1}\sim mx_{1}+b\)
3
\(b-m\)
READ
READ
Two points always fit exactly: \(R^2=1\). \(m=-2\) and \(b=5\), so \(b-m=7\).
Desmos is a trademark of Desmos Studio PBC. No affiliation.
Exponential Fit
SAT-style · first: linear or exponential? Vote
The table gives values of the exponential function \(f\): \(f(2)=90\), \(f(3)=270\), \(f(4)=810\). What is the value of \(f(0)\)?
TYPE
expression
table
1
| \(x_1\) | \(y_1\) | |
|---|---|---|
| 2 | 90 | |
| 3 | 270 | |
| 4 | 810 |
2
\(y_{1}\sim ab^{x_{1}}\)
READ
Desmos: \(a=10\), \(b=3\), so \(f(x)=10\cdot3^x\). Plug in 0: \(f(0)=10\cdot3^0=\) 10.
Desmos is a trademark of Desmos Studio PBC. No affiliation.
Read b as a percent.
SAT-style
The value of an investment account was 2,500 dollars at the start of 2021. The account's value has increased by the same percent each year, and it was 3,025 dollars at the start of 2023. By what percent did the account's value increase each year?
A21two years at once
B10.521 split in half
C1.1b read as the percent
D10✓
TYPE
expression
table
1
| \(x_1\) | \(y_1\) | |
|---|---|---|
| 0 | 2500 | |
| 2 | 3025 |
2
\(y_{1}\sim ab^{x_{1}}\)
READ
b AS A PERCENT\(b=1.1\): each year the value is \(\times1.1\), so it grows 10% a year.
Desmos is a trademark of Desmos Studio PBC. No affiliation.
Fit the line, then plug in 10.
SAT-style · your turn: pick A, B, C, or D; send on GO
A line in the \(xy\)-plane passes through the points \((-2,\,11)\) and \((4,\,-7)\). What is the \(y\)-coordinate of the point on this line whose \(x\)-coordinate is 10?
A47slope sign flipped
B−35
C−9
D−25✓
READ
\(m=-3\), \(b=5\). Typing \(10m+b\) gives \(-25\): D.
TYPE
expression
table
1
| \(x_1\) | \(y_1\) | |
|---|---|---|
| -2 | 11 | |
| 4 | -7 |
2
\(y_{1}\sim mx_{1}+b\)
3
\(10m+b\)
Desmos is a trademark of Desmos Studio PBC. No affiliation.
B
Section B
Facts in, curve out.
Every fact is a point. Pick the model that matches the shape.
Turn every fact into a point.
SAT-style
A quadratic function \(f\) has \(f(-1)=0\), \(f(5)=0\), and \(f(0)=-10\). What is the minimum value of \(f(x)\)?
TRANSLATE
\((-1,\,0)\)\((5,\,0)\)\((0,\,-10)\)TYPE
expression
table
1
| \(x_1\) | \(y_1\) | |
|---|---|---|
| -1 | 0 | |
| 5 | 0 | |
| 0 | -10 |
2
\(y_{1}\sim ax_{1}^{2}+bx_{1}+c\)
3
\(y=2x^{2}-8x-10\)
READ
Type the fitted model so Desmos marks its points; tap the bottom.
READ
Vertex \((2,\,-18)\). Minimum value −18.
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Know the vertex? Type it in.
SAT-style · grid-in
A ball is kicked from a height of 2 feet above the ground. Its path is modeled by a quadratic function \(h\), where \(h(x)\) is the ball's height, in feet, when it is \(x\) feet horizontally from where it was kicked. The ball reaches its greatest height, 20 feet, when \(x=6\). According to the model, what is the ball's height, in feet, when \(x=10\)?
TRANSLATE
start \((0,\,2)\)vertex \((6,\,20)\)
TYPE
expression
table
1
| \(x_1\) | \(y_1\) | |
|---|---|---|
| 0 | 2 | |
| 6 | 20 |
2
\(y_{1}\sim a(x_{1}-6)^{2}+20\)
3
\(a(10-6)^{2}+20\)
READ
READ
\(a=-0.5\), so \(h(10)=\) 12 feet.
WHY TYPE 6 AND 20?
\(h=6\) and \(k=20\) are typed in, so Desmos only finds \(a\).
Desmos is a trademark of Desmos Studio PBC. No affiliation.
C
Section C
Picture in, equation out.
The SAT gives a picture, not the data. Read a few points off it, then let Desmos fit them.
Read two points off the line.
SAT-style
The scatterplot shows the daily high temperature \(x\), in degrees Celsius, and the number of bicycles \(y\) rented from a shop on 10 days. A line of best fit is also shown. Which of the following equations best represents the line of best fit?
A\(y=12x+200\)at x = 15
B\(y=12x+20\)✓
C\(y=20x+12\)swapped
D\(y=-12x+20\)rises
TRANSLATE
TYPE
expression
table
1
| \(x_1\) | \(y_1\) | |
|---|---|---|
| 15 | 200 | |
| 35 | 440 |
2
\(y_{1}\sim mx_{1}+b\)
READ
Use the line, not the dots. \(m=12\), \(b=20\): B.
Desmos is a trademark of Desmos Studio PBC. No affiliation.
Predict: type the model, plug in.
SAT-style
The scatterplot shows the height, in centimeters, of a bamboo shoot on each of the first 8 days after planting, along with a line of best fit. Based on the line of best fit, what is the predicted height of the shoot, in centimeters, 10 days after planting?
A10that is x
B20
C27a data dot
D32✓
TRANSLATE
READ
Between gridlines at day 10; Desmos: 32. D.
TYPE
expression
table
1
| \(x_1\) | \(y_1\) | |
|---|---|---|
| 4 | 20 | |
| 9 | 30 |
2
\(y_{1}\sim mx_{1}+b\)
3
\(10m+b\)
Desmos is a trademark of Desmos Studio PBC. No affiliation.
Read a few dots. Fit, then match.
SAT-style
The scatterplot shows the number of customers \(y\) in a store \(x\) hours after it opened on one day. Which of the following equations is the most appropriate quadratic model for the data?
A\(y=0.5x^2+6x+10\)opens up
B\(y=-0.5x^2-6x+10\)peak at x < 0
C\(y=-0.5x^2+6x+10\)✓
D\(y=-0.5x^2+6x+28\)starts at 28
TRANSLATE
READ
The fit uses only our 3 dots. C has nearby numbers and follows the whole picture.
TYPE
expression
table
1
| \(x_1\) | \(y_1\) | |
|---|---|---|
| 0 | 11 | |
| 6 | 29 | |
| 12 | 10 |
2
\(y_{1}\sim ax_{1}^{2}+bx_{1}+c\)
Desmos is a trademark of Desmos Studio PBC. No affiliation.
D
Section D
Lists and statistics.
Type a list. Desmos finds the median, the mean, and the spread.
Median of the list: let Desmos sort.
SAT-style · your turn: pick A, B, C, or D; send on GO
The mean hourly wage of all 140 employees of a restaurant chain is 19.40 dollars. The hourly wages, in dollars, of the 10 employees at one location are: 18, 22, 16, 31, 20, 17, 24, 19, 13, 30. What is the median hourly wage of these 10 employees?
A18.5not sorted
B19one middle
C19.5✓
D21the mean
TYPE
1
\(L=[18,22,16,31,20,17,24,19,13,30]\)
2
\(\operatorname{median}(L)\)
3
\(\operatorname{mean}(L)\)
READ
Median \(=19.5\) (C). The mean of 21 is a trap answer, and the chain's 19.40 is extra information you never use.
Desmos is a trademark of Desmos Studio PBC. No affiliation.
Same range, different spread.
SAT-style · first: which list looks more spread out? Vote
List \(P\): \(5, 9, 10, 10, 11, 15\). List \(Q\): \(5, 5, 5, 15, 15, 15\). Which statement about the standard deviations of \(P\) and \(Q\) is true?
TYPE
1
\(P=[5,9,10,10,11,15]\)
2
\(Q=[5,5,5,15,15,15]\)
3
\(\operatorname{stdev}(P)\)
4
\(\operatorname{stdev}(Q)\)
AP's is greater than Q's.
BP's is less than Q's.✓
CThey are equal.same range
DThey cannot be compared.
WHAT SD MEASURES
How far values sit from their own mean. Both ranges are 10, so range can't decide.
Desmos is a trademark of Desmos Studio PBC. No affiliation.
One big value drags the mean.
SAT-style · first: mean or median, which moves more? Vote
The data set \(18, 20, 21, 23, 26, 28\) shows the number of customers a food truck served on each of six days. On a seventh day, the food truck served 63 customers, and that value is added to the data set. Which of the following statements about the resulting data set is true?
TRANSLATE
add one value: \(\operatorname{join}(L,[63])\)
TYPE
1
\(L=[18,20,21,23,26,28]\)
2
\(M=\operatorname{join}(L,[63])\)
3
\(\operatorname{mean}(M)-\operatorname{mean}(L)\)
4
\(\operatorname{median}(M)-\operatorname{median}(L)\)
AThe mean and the median both increase, and the mean increases by more than the median.✓
BBoth increase, and the median increases by more than the mean.
CThe mean increases, and the median does not change.median moves 1
DThe mean and the median increase by the same amount.
READ
The mean rises 5.76; the median rises 1. The mean feels extreme values: A.
Desmos is a trademark of Desmos Studio PBC. No affiliation.
Frequency table? Weight each value.
SAT-style
The table shows the number of goals scored last season by each of the 20 players on a soccer team. What is the mean number of goals scored per player?
| Number of goals | Number of players |
|---|---|
| 0 | 3 |
| 1 | 6 |
| 2 | 5 |
| 3 | 4 |
| 4 | 2 |
A1.8✓
B2mean of 0 to 4
C4
D7.236 ÷ 5
TRANSLATE
values \(\to V\)counts \(\to F\)
TYPE
1
\(V=[0,1,2,3,4]\)
2
\(F=[3,6,5,4,2]\)
3
\(\dfrac{\operatorname{total}(VF)}{\operatorname{total}(F)}\)
READ
\(\operatorname{total}(VF)=36\) goals over \(\operatorname{total}(F)=20\) players: 1.8. A.
Desmos is a trademark of Desmos Studio PBC. No affiliation.
Unpack the table with repeat.
SAT-style
The table shows the number of pets owned by each of the 20 students in a class. What is the median number of pets owned by the students?
| Number of pets | Number of students |
|---|---|
| 0 | 4 |
| 1 | 7 |
| 2 | 3 |
| 3 | 2 |
| 4 | 4 |
TRANSLATE
values \(\to V\)counts \(\to F\)\(\operatorname{repeat}(V,F)\to\) full list
A4biggest count
B2middle row
C1.75the mean
D1✓
TYPE
1
\(V=[0,1,2,3,4]\)
2
\(F=[4,7,3,2,4]\)
3
\(L=\operatorname{repeat}(V,F)\)
4
\(\operatorname{median}(L)\)
READ
20 values: the median is the 10th and 11th. The first \(4+7=11\) are 0s and 1s, so both are 1: D.
Desmos is a trademark of Desmos Studio PBC. No affiliation.
Drop one 6, then find the median.
SAT-style · your turn: pick A, B, C, or D; send on GO
The table shows the frequency of each score on a 10-point quiz for 11 students. One of the scores of 6 is removed from the data. What is the median of the remaining 10 scores?
| Score | Frequency |
|---|---|
| 6 | 2 |
| 7 | 3 |
| 8 | 1 |
| 9 | 3 |
| 10 | 2 |
A8old median
B8.2the mean
C8.5✓
D9
TRANSLATE
one 6 removed: count 2 becomes 1
TYPE
1
\(V=[6,7,8,9,10]\)
2
\(F=[1,3,1,3,2]\)
3
\(\operatorname{median}(\operatorname{repeat}(V,F))\)
Desmos is a trademark of Desmos Studio PBC. No affiliation.
Skills we learned today
Table in, line out
Type the table, then fit it with ~.
Two points
A two-row table, then the fit.
Exponential fit
Equal ratios? Fit \(y_1\sim ab^{x_1}\); \(a\) is the start.
\(b=1.1\to+10\%\)
Read b as a percent
1.1 means \(\times1.1\): up 10% a period.
Facts become points
Each fact is a table row.
Vertex + a point
Type h and k in; fit only a.
Line from a picture
Two gridline corners on the line.
Fit, then match
Read a few dots, fit, pick the closest choice.
\(\operatorname{median}(L)\)
Let Desmos sort
Name the list; ask for the median.
\(\operatorname{stdev}(P)\)
Compare spread
Same range? Compare stdev.
\(\operatorname{join}(L,[63])\)
Outliers
One big value drags the mean, not the median.
\(\operatorname{repeat}(V,F)\)
Frequency tables
Weight with total; unpack with repeat.
Desmos is a trademark of Desmos Studio PBC. No affiliation.
Practice this week.
Q
Practice questions from tonight, with worked answers.
Replay tonight's moves in the same Desmos the test uses: desmos.com/testing/cb-digital-sat
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NEXT WEEK
Study Habits & Test Anxiety
Thursday, Nov 12 · 5:00 PM PT
A plan for the weeks before the test, and what to do when nerves hit.
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