Recall these three tricks in Desmos

Vertex

Tap the turning point.

x-intercepts

Tap where the graph crosses $y=0$.

Intersection

Tap where two graphs cross.

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A

Section A

Set up the screen.

Four habits so the screen tells you the truth.

Radians or degrees?

TYPE

The expression list (type here)The graph (the wrench is top right)

READ

1

$\sin (30)$

A

Reverse contrast

Braille Mode

Grid

Axis Numbers

Minor Gridlines

Arrows

X-Axis

−10 ≤ x ≤ 10

Y-Axis

−7 ≤ y ≤ 7

Complex Mode

Radians

Degrees

Radians

Degrees

Zoomed in: bottom of the settings panel

Radians

Degrees

THE RULE

Desmos starts in radians. If the question uses degrees ($^\circ$), tap the wrench and pick Degrees before any trig.

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Zooming in and out for clarity

TYPE

1

$y=0.5x+3$

2

$y=30-x$

How many solutions do you see?

READ

TAP THE GRAY POINT

The crossing was just off the screen: $(18,\,12)$. Desmos may round what it shows, so when an answer must be exact, plug it back in or match it to the choices.

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Rename the letters.

SAT-style

A ball's height $h$, in feet, $t$ seconds after it is thrown is $h=-16t^2+64t+5$. What is the ball's maximum height, in feet?

TRANSLATE

$t \to x$$h \to y$

WHY

Desmos graphs $x$ against $y$. Most other letters become sliders instead. Rename the input and output only; leave real constants alone.

TYPE

1

$y=-16x^{2}+64x+5$

READ

Tap the top: $(2,\,69)$. The maximum height is 69 feet, at 2 seconds.

READ

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Long decimal? Toggle to a fraction.

Grid-in

A grid-in (no answer choices) works out to $\tfrac{2}{3}+\tfrac{3}{4}$. Desmos shows a long decimal. What exactly do you type into Bluebook?

TYPE

1

$\dfrac{2}{3}+\dfrac{3}{4}$

Tap the fraction button at the left of the row.

In Bluebook's answer box:

17/12

Answer preview: $\dfrac{17}{12}$

GRID-IN ENTRY

Enter 17/12. Or, for this repeating decimal, use all five characters: 1.416 or 1.417.

NEVER

A mixed number like 1 5/12. Bluebook reads it wrong.

FITS?

Up to 5 characters for a positive answer, 6 for a negative one (the minus counts).

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B

Section B

Solve by seeing.

One method: graph each side. The solutions are the x-values where the graphs meet.

Solve any equation

Desmos presents the solutions as vertical lines

TYPE

1

$3(x-2)=2x+5$

READ

A vertical line at $x=11$ means that the solution is $x=11$.

READ

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Another method for solving equations

SAT-style

$\sqrt{3x+10}=x+2$ Which of the following is the set of all solutions to the given equation?

A$\{2\}$✓

B$\{-3\}$fake root

C$\{-3,\ 2\}$extraneous

D$\{-2,\ 3\}$sign slip

THE ALGEBRA

$\sqrt{3x+10}=x+2$

→$3x+10=x^2+4x+4$square both sides

→$x^2+x-6=0$

→$(x+3)(x-2)=0$

→$x=-3$ or $x=2$two candidates

TYPE

1

$y=\sqrt{3x+10}$

2

$y=x+2$

READ

The graphs cross once, at $(2,\,4)$. Only $x=2$ is a solution.

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Where does a line meet a parabola?

SAT-style · first: 0, 1, or 2 crossings? Vote

Consider the system $y=x^2-4x+1$ and $y=x-3$. If $(x_1,y_1)$ and $(x_2,y_2)$ are the solutions to the system, what is $y_1+y_2$?

TRANSLATE

Solutions of a system = the crossing points.

TYPE

1

$y=x^{2}-4x+1$

2

$y=x-3$

READ

Two crossings: $(1,\,-2)$ and $(4,\,1)$.
So $y_1+y_2=-2+1=-1$.

READ

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A circle in disguise.

SAT-style

$x^2+18x+y^2-8y+48=0$ The given equation defines a circle in the $xy$-plane. What is the length of the circle's radius?

A49that is r²

B$\sqrt{145}$kept the 48

C7✓

D$\sqrt{17}$signs slipped

TYPE

1

$x^{2}+18x+y^{2}-8y+48=0$

READ THE EDGES

The left edge is at $x=-16$ and the right edge is at $x=-2$. So the diameter is 14, which means the radius is 7.

READ

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Which point is in the shaded region?

SAT-style

Consider the system of inequalities $y>x+3$ and $y\le -2x+6$. Which point lies in the solution region of the system?

A(2, 6)above red

B(0, −1)below blue

C(1, 4)on dashed line

D(−3, 4)✓

TYPE

1

$y>x+3$

2

$y\le -2x+6$

3

$(2,6),(0,-1),(1,4),(-3,4)$

READ

The answer lives where both colors overlap. D is inside. C sits on the dashed line, and dashed means not included.

READ

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C

Section C

Systems.

Put the constant on a slider. Watch solutions appear, merge, and vanish.

Slide k until the lines never meet.

SAT-style · first: what does “no solution” look like? Vote

In the equations $2x+3y=11$ and $6x+ky=27$, $k$ is a constant. If the system has no solution, what is the value of $k$?

TYPE

1

$2x+3y=11$

2

$6x+ky=27$

add slider: k

3

$k=1$

-10

10

-10≤ k ≤Step:

A3copied the 3

B6scaled only x

C9✓

D−9sign slip

READ

As $k$ grows, the crossing slides away. At $k=9$ the lines are parallel: same slope, different line, so no solution.

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Slide k until the lines match.

SAT-style · your turn: 45 sec, then GO

In the equations $y=4x-7$ and $12x-ky=21$, $k$ is a constant. If the system has infinitely many solutions, what is the value of $k$?

TYPE

1

$y=4x-7$

2

$12x-ky=21$

add slider: k

3

$k=1$

-10

10

-10≤ k ≤Step:

A−3sign slip

B3✓

C$\tfrac{1}{3}$flipped the scale

D12wrong unknown

READ

At $k=3$ the red line lands exactly on the blue: the same line, so every point is a solution. (Algebra: $12x-3y=21$ is $y=4x-7$ times 3.)

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Slide k until the parabola just touches.

SAT-style · first: at k = 1, how many crossings? Vote

In the equations $y=-2$ and $y=x^2-6x+k$, $k$ is a constant. If the system has exactly one real solution, what is the value of $k$?

TYPE

1

$y=-2$

2

$y=x^{2}-6x+k$

add slider: k

3

$k=1$

-10

10

-10≤ k ≤Step:

A2setup slip

B5arithmetic slip

C7✓

D11that is 0 crossings

READ

As $k$ grows the parabola rises: 2 crossings, then they merge into 1 at $k=7$, then 0. Confirm exactly: the vertex is at $x=3$, where $y=k-9$; touching $y=-2$ means $k-9=-2$, so $k=7$.

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The answer is past 10.

SAT-style · first: will a slider from -10 to 10 find it? Vote

In the equations $y=-4x+7$ and $ax+6y=-6y+4$, $a$ is a constant. If the system has no solution, what is the value of $a$?

TYPE

1

$y=-4x+7$

2

$ax+6y=-6y+4$

add slider: a

3

$a=1$

-10

10

-10≤ a ≤Step:

A48✓

B−48sign flip

C24missed the other 6y

D−4copied the slope

READ

At $a=48$ the red line runs parallel to the blue: no solution. (Combine the $y$ terms: $48x+12y=4$ has slope $-4$.)

READ

OUTSIDE THE SLIDER?

Answer outside $-10$ to $10$? Type it, or tap the end number to reset the range. A fraction? Type it.

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Fraction choices? Type each one.

SAT-style · translate: graph each side; exactly one = the line just touches

$x(2x+9)=4x-k$ In the given equation, $k$ is a constant. The equation has exactly one distinct real solution. What is the value of $k$?

TYPE

1

$y=x(2x+9)$

2

$y=4x-k$

add slider: k

3

$k=1$

-10

10

-10≤ k ≤Step:

A$\tfrac{25}{4}$misses

B$\tfrac{81}{8}$misses

C$-\tfrac{25}{8}$two crossings

D$\tfrac{25}{8}$✓ touches

READ

Only $k=\tfrac{25}{8}$ makes the line just touch: D. Exact check: $(4x+5)^2=0$, one solution.

READ

OUTSIDE THE SLIDER?

Answer outside $-10$ to $10$? Type it, or tap the end number to reset the range. A fraction? Type it.

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Count the points inside the region.

SAT-style · grid-in · your turn: 45 sec, then type your answer in chat

Consider the system of inequalities $y\ge 2x-9$ and $y<x+4$. In the $xy$-plane, the point $(b,\,5)$ is a solution to the given system of inequalities, where $b$ is an integer. How many values of $b$ are possible?

TRANSLATE

$(b,\,5)$ means $x=b$ and $y=5$: it lives on the line $y=5$.

TYPE

1

$y\ge 2x-9$

2

$y<x+4$

3

$y=5$

On $y=5$: $1<x\le 7$ → b = 2, 3, 4, 5, 6, 7

ANSWER

6 values

READ

On $y=5$, the overlap runs from $x=1$ (dashed red: not included) to $x=7$ (solid blue: included). Integers 2 through 7: 6.

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D

Section D

Match the curve.

Equivalent expressions make the same graph.

Which choice lands exactly on top?

SAT-style · first: vote A-D

Which of the following is equivalent to $\dfrac{x^2+5x-3}{x+2}$?

A$x+3+\dfrac{9}{x+2}$misses

B$x+3-\dfrac{9}{x+2}$✓ on top

C$x+3-\dfrac{3}{x+2}$misses

D$x+7+\dfrac{11}{x+2}$misses

TYPE

1

$y=\dfrac{x^{2}+5x-3}{x+2}$

2

$y=x+3+\dfrac{9}{x+2}$

3

$y=x+3-\dfrac{9}{x+2}$

READ

Green covers the blue completely, and both are undefined at $x=-2$: B. (The algebra is the proof.)

READ

Compare each choice with the original (blue).

Desmos is a trademark of Desmos Studio PBC. No affiliation.

Slide p until the curves match.

SAT-style

In the expression $2(x^2+px+5)-3x(p+2)$, $p$ is a constant. The expression is equivalent to $2x^2-16x+10$. What is the value of $p$?

TYPE

1

$y=2x^{2}-16x+10$

2

$y=2(x^{2}+px+5)-3x(p+2)$

add slider: p

3

$p=1$

-10

10

-10≤ p ≤Step:

DO

Type both → tap add slider → drag until red covers blue: $p=10$.

CHECK

$-(p+6)=-16$, so $p=10$.

READ

Drag p: the red parabola moves until it covers the blue

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E

Section E

Transformations.

Type f once. Then every shift and flip is one short row.

Define f once. Reuse it.

SAT-style

The functions $f$ and $g$ are defined by $f(x)=3x-5$ and $g(x)=x^2+2$. What is the value of $f(g(3))$?

A4only f(3)

B22dropped the +2

C28✓

D33dropped the −5

TRANSLATE

Copy the definitions word for word. Desmos keeps them for the whole module, so on a new question edit the old $f$ row instead of adding a second $f$.

TYPE

1

$f(x)=3x-5$

2

$g(x)=x^{2}+2$

3

$f(g(3))$

4

$f(f(2))$

INSIDE FIRST

$g(3)=11$, then $f(11)=28$.

REUSE

$f(f(2))$: $f(2)=1$, then $f(1)=-2$. No retyping.

Desmos is a trademark of Desmos Studio PBC. No affiliation.

Inside: sideways. Outside: vertical.

Predict · commit in chat, then GO

Predict before we type: compared with $y=f(x)$, where does $y=f(x+3)$ go, left or right?

TYPE

1

$f(x)=x^{2}$

2

$y=f(x+3)$

3

$y=f(x)-2$

INSIDE: OPPOSITE

$f(x+3)$ moves left 3. The vertex goes from $(0,0)$ to $(-3,0)$.

OUTSIDE: SAME WAY

$f(x)-2$ moves down 2, to $(0,-2)$.

READ

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Where does the minus sign go?

TYPE

1

$f(x)=\sqrt{x}$

2

$y=-f(x)$

3

$y=f(-x)$

4

$y=2f(x)$

−f(x)

Minus outside: flips over the $x$-axis (red, upside down).

f(−x)

Minus inside: flips over the $y$-axis (green, points left).

2f(x)

Times 2 outside: twice as tall (purple).

READ

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Shift right 6: which equation?

SAT-style · your turn: 45 sec, then GO

The function $f$ is defined by $f(x)=2x^2$. The function $g$ is the result of shifting $f$ to the right 6 units. Which equation defines $g$?

A$g(x)=2(x+6)^2$wrong direction

B$g(x)=2(x-6)^2$✓ covers it

C$g(x)=2x^2-6$moved down

D$g(x)=2x^2+6$moved up

TYPE

1

$f(x)=2x^{2}$

2

$y=f(x-6)$

3

$y=2(x-6)^{2}$

READ

$y=f(x-6)$ is the shift itself (red). Choice B (green) lands exactly on it. Right 6 means minus 6 inside.

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Shift, then read the new x-intercepts.

SAT-style · grid-in · your turn: 45 sec, then type your answer in chat

The function $f$ is defined by $f(x)=(x-5)(x+5)$. The function $g$ is defined by $g(x)=f(x-2)$. The equation $y=g(x)$ has $x$-intercepts at $x=p$ and $x=q$. What is the value of $p+q$?

TRANSLATE

$g(x)=f(x-2)$: define $f$, then type the shift.

TYPE

1

$f(x)=(x-5)(x+5)$

2

$y=f(x-2)$

ANSWER

$p+q=-3+7=4$

READ

The red parabola crosses at $x=-3$ and $x=7$. So $p+q=4$. (Each intercept moved right 2: $-5\to-3$, $5\to7$.)

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F

Section F

Absolute Values.

Solve an absolute value equation

SAT-style

What are the solutions to the equation $|x-3|=5$?

TYPE

1

$y=|x-3|$

2

$y=5$

V-SHAPE

$y=|x-3|$ is a V with its point at $(3,\,0)$.

READ

The line $y=5$ crosses the V twice: $x=-2$ and $x=8$. Two solutions, no case-splitting.

READ

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Read both crossings, then add.

SAT-style · your turn: 45 sec, then GO

$|2x+1|=9$ What is the sum of the solutions to the given equation?

A4only one case

B9the right side

C−1✓

D1sign flip

TYPE

1

$y=|2x+1|$

2

$y=9$

READ

Crossings at $x=-5$ and $x=4$. Sum: $-5+4=-1$.

READ

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All solutions: read the band.

SAT-style · your turn: 45 sec, then GO

$|x-7|<4$ Which of the following describes all solutions to the given inequality?

A$3<x<11$✓

B$x<3$ or $x>11$flipped

C$x<11$one case only

D$-4<x<4$dropped the 7

TYPE

1

$|x-7|<4$

READ

The band runs from 3 to 11 with dashed edges: $3<x<11$.

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G

Section G

Graphs in context.

Every gray point means something. Translate, zoom to fit, then say what it means.

Three gray points, three answers.

SAT-style

A ball is thrown upward from a roof. Its height $h$, in feet, $t$ seconds after it is thrown is $h=-16t^2+48t+64$.

TRANSLATE

$t\to x$$h\to y$

TYPE

1

$y=-16x^{2}+48x+64$

Y-INTERCEPT = START

$(0,\,64)$: the height when $t=0$.

VERTEX = MAXIMUM

$(1.5,\,100)$: highest point, and when.

POSITIVE X-INTERCEPT = LANDS

$(4,\,0)$: height 0. The flight is only $0\le t\le 4$: ignore $(-1,\,0)$ and the curve below ground.

READ

Zoom so the whole flight fits

Desmos is a trademark of Desmos Studio PBC. No affiliation.

Break-even, best, and zero sales.

SAT-style

A company's profit $P$, in dollars, from selling $x$ units is $P=-2x^2+120x-1000$.

TYPE

1

$y=-2x^{2}+120x-1000$

VERTEX = BEST

$(30,\,800)$: sell 30 units for the most profit, $800.

X-INTERCEPTS = BREAK-EVEN

$(10,\,0)$ and $(50,\,0)$: profit is exactly 0.

Y-INTERCEPT = SELL NOTHING

$(0,\,-1000)$: with no sales the company loses $1000.

READ

Zoom so the whole story fits

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What does the point (5, 0) mean?

SAT-style · your turn: 45 sec, then GO

A model rocket is launched upward from the top of a platform. Its height above the ground, in meters, $t$ seconds after launch is modeled by $h(t)=-5t^2+20t+25$. When $y=h(t)$ is graphed in the $ty$-plane, it crosses the $t$-axis at $(5,\,0)$. What is the best interpretation of this point in context?

AThe rocket reaches the ground 5 seconds after launch.✓

BThe rocket reaches its maximum height 5 seconds after launch.that is the vertex

CThe rocket was launched from a height of 5 meters.that is the y-intercept

DThe maximum height of the rocket is 5 meters.mixes axes

TYPE

1

$y=-5x^{2}+20x+25$

READ

$(5,\,0)$ is an $x$-intercept: height 0 at 5 seconds. The rocket is on the ground: A.

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Which coordinate is the answer?

SAT-style · your turn: 45 sec, then GO

$P(x)=-4x^2+120x-500$ A food truck models its daily profit $P(x)$, in dollars, when it charges $x$ dollars for a meal. According to the model, what price, in dollars, gives the greatest daily profit?

A5an x-intercept

B15✓

C400the profit, not the price

D−500the y-intercept

TYPE

1

$y=-4x^{2}+120x-500$

READ

The vertex is $(15,\,400)$. The question asks for the price (the $x$), so 15. 400 is the profit.

READ

Zoom so the whole story fits

Desmos is a trademark of Desmos Studio PBC. No affiliation.

What does the 10 mean?

SAT-style · your turn: 45 sec, then GO

A diver jumps from a platform into a pool. The diver's height above the water, in meters, $t$ seconds after the dive begins is modeled by $h(t)=-4.9t^2+9.8t+10$. What does the value 10 represent in this context?

AThe dive lasts a total of 10 seconds before the diver enters the water.an x-intercept

BThe diver reaches a maximum height of 10 meters above the platform.the vertex

CThe diver was 10 meters above the water at the moment the dive began.✓

DThe diver enters the water at a speed of 10 meters per second.not on the graph

TYPE

1

$y=-4.9x^{2}+9.8x+10$

READ

The constant is the $y$-intercept $(0,\,10)$: the height at $t=0$, when the dive begins. C.

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Skills we learned today

Zoom out

Find a crossing that is off the screen.

$\tfrac{2}{3}+\tfrac{3}{4}=\tfrac{17}{12}$

Fraction toggle

Turn a long decimal into an exact grid-in.

Solve any equation

Type it as written; read the vertical line.

Graph both sides

Only real crossings count: no fake roots.

Line meets parabola

Tap the crossing points.

Circle in disguise

Type it as is; read the edges.

Shaded region

Plot the choices; find the overlap.

Slide k

Parallel, touching, or the same line.

Match the curve

The equivalent choice lands on top.

Shift and flip f

Type f once, then f(x+3), −f(x).

Absolute values

Graph both sides; read the crossings.

Graphs in context

Zoom to fit; say what each point means.

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

Download the Bluebook app to take free, full-length practice SATs from the College Board.

FREE · EVERY WEEK

Scan for practice + answers

highperformancetutoring.com

LINKS IN THE CHAT

GREAT WORK TODAY

One question
before you go?

NEXT WEEK

Desmos 2: Regressions & Statistics

Thursday, Nov 5 · 5:00 PM PT

Turn a table into an exact equation, and a list into answers.

REJOIN ANY SESSION zoom.us/j/89849937335 Bring a friend next week.

Desmos 1.

Recall these three tricks in Desmos

Set up the screen.

Radians or degrees?

Zooming in and out for clarity

Rename the letters.

Long decimal? Toggle to a fraction.

Solve by seeing.

Solve any equation

Another method for solving equations

Where does a line meet a parabola?

A circle in disguise.

Which point is in the shaded region?

Systems.

Slide k until the lines never meet.

Slide k until the lines match.

Slide k until the parabola just touches.

The answer is past 10.

Fraction choices? Type each one.

Count the points inside the region.

Match the curve.

Which choice lands exactly on top?

Slide p until the curves match.

Transformations.

Define f once. Reuse it.

Inside: sideways. Outside: vertical.

Where does the minus sign go?

Shift right 6: which equation?

Shift, then read the new x-intercepts.

Absolute Values.

Solve an absolute value equation

Read both crossings, then add.

All solutions: read the band.

Graphs in context.

Three gray points, three answers.

Break-even, best, and zero sales.

What does the point (5, 0) mean?

Which coordinate is the answer?

What does the 10 mean?

Skills we learned today

Practice this week.

One questionbefore you go?

One question
before you go?