Master SAT Desmos

The Desmos calculator is built into the Bluebook app, so you have it for the whole SAT Math section. Open it with the calculator icon at the top of the screen. It solves equations and systems, marks a graph's intercepts, vertex and intersection points, and fits a model to a table of values. Strong Desmos skills come down to three habits: pick the method that fits, type it exactly as given, and answer what the question asks.

Basic Desmos skills for the SAT

• Parentheses: type each equation into Desmos as written, with parentheses around negatives and around any part with more than one term, like (2x+3)/(x-4)^2.
• Exponents: ^ starts a superscript, and a letter, a digit or / stays up in it. To divide $3^x$ by $2$, press Tab after 3^x, then type /2. A + or - comes back down on its own.
• Functions: define a function once, like f(x)=(x^2+7)/(x-2), then type f(5) on a new line.
• Exact values: when a decimal should be exact, click the fraction button next to the result.
• Settings: set degrees or radians to match a trigonometry question, and clear old work with Delete All before a new problem.

Many SAT Math questions give you one equation, a system of equations or a set of inequalities, with the algebra buried under fractions, parentheses or awkward coefficients. That's your cue to open Desmos, where each intersection point gives you a solution. When the same expressions repeat, rename them to get a smaller system first.

How to solve equations, systems and inequalities in Desmos

1. One equation: put each side on its own line. For $5(x-4)+7=3(x+2)+5$, type y=5(x-4)+7 and y=3(x+2)+5. The graphs meet where $x=12$, the solution. Quadratic equations work the same way, but their graphs can cross twice, so look for a second solution.
2. A system: type each equation as it's written, like 2x+3y=13 and 5x-y=7, without solving for $y$ first. The lines cross at the solution, $(2,3)$.
3. Inequalities: type them as written, with <= for $\le$ and >= for $\ge$, and plot each answer choice as a point. The answer sits where all the shadings overlap.
4. Counting solutions: to find how many solutions there are, count the intersections. Two equations that draw the same line, like y=2x+1 and 2y=4x+2, have infinitely many.
5. Answer the question: it may want the positive solution, only $y$, or a value like $x+y$.

Mistakes to avoid with intersections and shading

• Trusting the first window. A second intersection can hide off the screen, so zoom out before you pick or count. When two graphs only seem to touch, check that value with algebra.
• Picking a point on a dotted line. Desmos draws $<$ and $>$ as dotted lines, and points on them don't count.

Many SAT Math questions give you a function and ask about one point on its graph: a zero, a $y$-intercept, an intersection or the vertex of a parabola. Others ask for a value like $f(-2)$, an equivalent expression or a factor. If the answer is a point or a value, graph the function and click it.

How to find intercepts, the vertex and function values in Desmos

1. Type the function. Use $x$ for any input letter like $t$: C(x)=-2x^2+24x+10. To limit the inputs, add curly braces at the end: y=-(x-3)^2+9{0<=x<=5}.
2. Click the curve. Desmos shows its gray Desmos points of interest: intercepts, intersection points and the max or min. Pan or zoom if one is off the screen.
3. Take the coordinate the question asks for. Roots and zeros are the $x$-coordinates of $x$-intercepts.
4. Evaluate on a new line. Once the function is defined, type f(-2), or h(j(5)) for a composition. For a factor test, use the input that makes the factor $0$, and watch the sign: $x+4$ goes with $-4$, and $2x-5$ with $\frac52$. So p(5/2) shows $0$ exactly when $2x-5$ is a factor, and p(2) is the remainder when you divide by $x-2$.
5. Overlap equivalent expressions. Graph the original and one choice. The right one overlaps it everywhere, not at one point. A denominator or a second letter that stays in the choices, like $a$, can fake a match, so check those with algebra. If the question asks you to find that letter, fit it instead.

Mistakes to avoid when reading intercepts and maximums

• The wrong coordinate. For a maximum at $(6,82)$, for what value of $t$ wants $6$, and the greatest value wants $82$.
• A point the problem rules out. A negative time, or an input outside the restrictions, can't be the answer. Neither can an input that makes the original denominator $0$, even if that factor cancels.

Some SAT Math questions hide a number behind a letter, with wording like where $k$ is a constant, the graph passes through the point or $a$ is a positive integer. That constant decides where the graph sits. With a Desmos slider, you adjust it until the graph meets the condition, with no rearranging. Two or more constants, each with its own fact, are a job for custom regression.

How to find unknown constants in Desmos

1. Type the function. Enter it as written, like f(x)=a(x-4)^2-3, and add the slider Desmos offers for $a$.
2. Plot the condition. Type the point, like (2,9), or graph the other line in the system.
3. Find the value. Drag the slider to see which way the graph moves, then type exact values until it meets the condition. If $a$ must be positive or an integer, set the slider's bounds and step to match. Check it: at a=3, f(2) shows $9$.
4. Answer the question. Type what it asks for, like f(0).
5. Use test values for free constants. If rules like $a>b>87$ leave the constants free, type values that follow every rule, like b=88 and a=89, then the function and the expression. A second pair that follows the rules should give the same result, and algebra proves it. If it changes, recheck your typing.

When the answer choices hold a letter, like for each real number $r$, graph the equations, then type each choice with $t$ for $r$, like (t,(4t-5)/3). The choice whose path lies right on the line is the answer, and the lesson on parameterized answer choices shows how to set the range of $t$.

Mistakes to avoid with sliders and test values

• Answering with the constant. The question often asks for something else, like $f(0)$ or $2k$, and the constant itself sits in the choices as a trap.
• Stopping at the first value that fits. For $g(x)=|x-c|+2$ through $(1,7)$, both $c=-4$ and $c=6$ fit, but only $c=6$ meets $c>0$.

Some SAT Math questions hand you a table of values and ask for an equation, a coefficient or a prediction. When the question names a model or asks for a best fit, that's a regression job.

Practice it: Desmos regression practice: 68 questions, from fitting a table to custom regression, circle regression and equivalent expressions, each with a step-by-step Desmos solution.

Put the data in a Desmos table

Put the inputs in x1 and the outputs in y1, one pair per row. Then type a model, or select Add Regression. The tilde, ~, means fit to the table, not equals. If the question gives the rule and asks which table matches it, skip regression: change the y1 header to f(x1), and Desmos fills in the outputs. The patterns below need evenly spaced inputs.

Desmos linear regression

Type y1 ~ mx1 + b for a named linear model, or when the outputs change by about the same amount each step. Desmos linear regression reports the slope $m$ and the intercept $b$.

Desmos quadratic regression

Type y1 ~ ax1^2 + bx1 + c when the jumps between outputs change by a steady amount: $1$, $4$, $9$, $16$ jump by $3$, $5$, $7$. Desmos quadratic regression reports $a$, $b$ and $c$, with $b$ in front of $x$, not $x^2$.

Desmos exponential regression

Type y1 ~ ab^x1 when each output is about the same multiple of the last, like $2$, $6$, $18$, $54$. Desmos exponential regression reports the starting value $a$ and the multiplier $b$, so $b=0.8$ means a $20\%$ drop each time the input goes up by $1$.

Exact fit vs best fit

Read the wording to tell exact fit vs best fit. Function values, or a graph through given points, need a model that hits every point, so check every row. Measured or rounded data only need a model that follows the trend.

Mistakes to avoid with Desmos regression

• Retyping the decimals Desmos shows. It keeps more digits than it displays, so type the letters, like a*b^5, and round once, at the end.
• Choosing the model that hits the most points. On a small table, a curve with more constants can bend through every point. Use the model the question names or the table's pattern supports.

Some SAT Math questions give you a function with unknown constants, like $f(x)=(p-3)x+(2p+4)$, and exact facts that pin them down: $f(4)=16$, a point on the graph or a table of values. When a constant shows up twice, or one function is built from another, custom regression lets Desmos do the algebra. A slider suits one constant you can watch on a graph.

How to solve for an unknown constant in Desmos

1. Count the unknowns and the facts. Each unknown needs its own fact, one that says something new. Delete any slider line like a=3 first, or Desmos treats $a$ as fixed and won't fit it.
2. Type each fact with a tilde. Type the known output, ~ where the equals sign would go, then the function at the known input. For $f(4)=16$, that's 16~(p-3)*4+(2p+4), and Desmos finds $p=4$.
3. Fit several facts at once. Put points in a table and type the model, like y1~a(x1-2)^2+b. If any fact isn't a point, put every fact on one line as two lists in matching order, known values on the left. For $g(x)=a(2x)+b$ with $g(3)=5$ and $a+b=0$, that's [5,0]~[a(2*3)+b,a+b].
4. Fit an identity over a list. When two expressions are equivalent for all values of $x$, or an equation has infinitely many solutions, type x1=[1...10], then both sides with a tilde between them, like (2x1+p)(x1-3)-(x1+q)(x1+1)~x1^2-10x1-11. Desmos finds the constants in equivalent expressions, here $p=2$ and $q=5$.
5. Answer the question. Type the function, then what the question asks for: f(x)=(p-3)x+(2p+4), then f(9). Desmos uses $p=4$ and shows $21$.

Mistakes to avoid with custom regression

• Changing the model. Keep the function exactly as given. Swapping in the built-in y1~ax1^2+bx1+c adds unknowns your facts can't pin down. For a hidden function inside a quotient, fit the whole quotient, like y1~(mx1+b)/(x1+6).
• Reading the wrong function. A fact about $g(x)=f(2x+1)+4$ goes into all of $g$, not $f$.
• Fitting an identity without the list. Leave out x1=[1...10], and Desmos treats x1 as one more unknown, so its constants match at one point instead of every $x$.
• Trusting one decimal. $-1.5$ only suggests $-\frac32$ until it makes every original fact true. And when a constant is squared, Desmos may find one value and miss another, so plug the result back in.

Practice it: the SAT Desmos regression practice set has custom regression questions with one and several unknown constants, and constants in equivalent expressions, each with a step-by-step Desmos solution.

Some SAT Math questions are mostly careful arithmetic: a mean or median, a frequency table, a percent hidden in a sentence, or a rate with several steps. Reach for Desmos when that arithmetic is easy to slip on, like out-of-order values or a discount followed by tax.

How to find mean, median and percents in Desmos

1. Lists: type raw data once as a named list, like A=[6,10,13,2,15,22,10,4,4], then reuse the name: mean(A) or median(A).
2. Spread: reason first. Adding the same number to every value leaves the range and standard deviation alone, and multiplying every value by $k$ multiplies both by $|k|$. Use stdevp(A) only to check.
3. Frequency tables: put the values in x1 and the frequencies in y1, then type total(x1*y1)/total(y1) for the weighted mean.
4. Percents: type the sentence as it reads, with the base after of. $72$ is $120\%$ of what number? becomes 120% of x = 72, and an $18\%$ increase makes 118% of x.
5. Rates: give each step its own line. Two machines making $20$ items in $8$ minutes and $18$ in $12$ minutes, for an hour, become a=20/8, b=18/12, t=60 and n=(a+b)t. Write down what each letter means and its unit, since Desmos won't.

Mistakes to avoid with data and percents

• Counting rows instead of observations. A table with $4$ rows can hold $10$ values, so divide by total(y1).
• Using a part-to-part ratio as a fraction. In a $4:9$ mix of fruit to grain, fruit is $\frac4{13}$ of the mix, not $\frac49$.
• Combining percents that come one after another. A $30\%$ discount and then $8\%$ tax is 108% of (70% of x), not a $22\%$ decrease, because the tax acts on the lower price.

SAT Math circle questions give you an expanded equation like $x^2+y^2+8x-6y-75=0$, the ends of a diameter, or a few points on the circle, and ask for the center, the radius, the equation or a point on it. If you can't see the center at a glance, open Desmos.

How to find a circle's center and radius in Desmos

1. An expanded equation: type it as written, like x^2+y^2-4x+2y-20=0, with no y= in front. Select it, and Desmos marks its highest and lowest points: the center is halfway between them, and the radius is half the gap.
2. The ends of a diameter: name them, like A=(-2,1) and B=(4,9), and use distance and midpoint: midpoint(A,B) is the center, and distance(A,B)/2 is the radius.
3. Three points: with no diameter, put them in a table as x1 and y1 and fit a circle regression: x1^2+y1^2~ux1+vy1+w. The center $(h,k)$ is $\left(\frac u2,\frac v2\right)$, and $r^2=w+h^2+k^2$.
4. A point on the circle: add the clue as a second graph, like y=9 or x=5, and click where it meets the circle. A line usually meets it twice, so keep the point the question asks for.

Mistakes to avoid with circles in Desmos

• Radius or diameter? Halve distance(A,B) only for the ends of a diameter. From the center to a point on the circle is already one radius. And the right side of center-radius form is $r^2$, not $r$.
• Rounding too early. Name the radius, like R=distance(C,P), and type R^2. Use a capital: a lowercase r draws a polar graph. Rounding a radius of $8.485\ldots$ to $8.49$ first gives $72.0801$, not $72$. If the choices are exact, like $3+\sqrt7$, check the one near your decimal by hand.

Getting the right number in Desmos doesn't earn the point yet. The question might want an exact fraction, a value to the nearest tenth, or a percent without the $\%$ sign. So on test day, start every SAT Math question by reading its last sentence.

A test-day routine for Desmos and Bluebook

1. Read the last sentence first. If you have no way in after about $30$ to $45$ seconds, get as far as you can, mark the question and move on.
2. Keep every digit, and round once, at the end. Type 5/7, not 0.714, and let each line use the results above it, like t=v/f, instead of retyping rounded decimals. Then round to the place the question names: to the nearest hundredth, $0.656\ldots$ becomes 0.66.
3. Translate before you type. Say Desmos shows ___, but the question asks for ___. A proportion of $0.6$ goes in as 60 for a percent.
4. Enter it the way Bluebook takes it. With no rounding instruction, enter the fraction if it fits, like 12/73. If it doesn't, fill the box with the decimal, like .3322 for $\frac{100}{301}$. The box takes up to $5$ characters for a positive answer and $6$ for a negative one, counting the decimal point or fraction bar. It takes no mixed number, percent sign, dollar sign or comma.
5. Check it. Confirm the quantity, any restriction like $x>y$, the units, and whether the size makes sense.

Mistakes to avoid on test day

• Rounding partway through. If a pool drains in $21.22$ minutes, rounding the pipe's flow rate to $0.4$ along the way gives $20.83$ instead, which is $20.8$, not $21.2$, to the nearest tenth. A rounded label can't give the exact fraction back either: $0.667$ is $\frac{667}{1000}$, not $\frac23$.
• Entering the wrong number from the screen, like the hourly rate, $25$, when the question wants the total cost, $370$.