Solving SAT math by graphing in Desmos

When to open the graph

Graphing is rarely faster than algebra. Where it does win, it's mostly on hard nonlinear equations, hard nonlinear questions, and systems of two linear equations. Typing an equation takes about ten seconds before Desmos shows you anything, so graph when the hand work would take longer than that, or when it's algebra you tend to slip on. If you keep dropping minus signs when you clear fractions, those ten seconds of typing will save you points.

Leave the graph closed when:

One or two steps finish it by hand, like $4x-7=29$ , or the equation is already factored, like $(x-2)(x+9)=0$ .
You don't have an equation yet. Desmos can't read a word problem, so write the equation first. Often the rest is one line.
The question asks what a number means in a story, or which equation fits the story.
The choices have a letter other than $x$ with no value given, like $3k+1$ .
It's a figure, a table or a printed graph. For plain arithmetic, type it on a row and skip the graph.

Solving one equation

Here's an equation with two fractions to clear.

\frac{3x-1}{4}=\frac{x+2}{6}+1

What value of $x$ is the solution to the given equation?

A. $\frac{8}{7}$
B. $\frac{25}{14}$
C. $\frac{19}{7}$
D. $\frac{17}{7}$

Type it as printed

Read the last line. It asks for $x$ , so write "x" on your scratch paper.
Type (3x-1)/4 and press the right arrow. After a /, everything you type stays in the bottom of the fraction until you do.
Type =(x+2)/6, press the right arrow again, then type +1.
Desmos draws a vertical line at the solution. Click it, then click the gray dot where it meets the $x$ -axis. Its first number is a little over 2.7.
Type a choice on a new row, such as 19/7. It prints a value starting 2.714, the only choice that matches the dot, so the answer is C.

If you don't see the line, zoom out. Find the zoom controls once on the practice site.

By hand, you'd multiply every term by 12. Forget to multiply the 1, and you get A. Multiply only the $3x$ by 3, and you get D. When you type the equation as printed, Desmos does that step for you.

You can also put each side on its own row: y=(3x-1)/4 on one, and y=(x+2)/6, right arrow, +1 on the next. The solution is the $x$ -coordinate where the graphs cross.

Trap: Reading the wrong coordinate

Choice B, $\frac{25}{14}$ , is about 1.79. That's the second number at the crossing, the value both sides equal there.

What to do: Look at what you wrote on scratch paper. The question asked for x, so read the first number.

Gray dots show rounded decimals and have no fraction button. On a , a long decimal has to fill all five characters, like 2.714 for $\frac{19}{7}$ . If the dot shows fewer digits, finish by hand. Grid-in entry has the rules.

A circle's equation goes in as printed too, with no completing the square. The circles lesson shows how to check its center.

Worked example

$0.5(4x+59.2)-0.2(x+3)=1.4(x+24)$

What value of $x$ is the solution to the given equation?

A
$49.7$
B
$17.5$
C
$11.5$
D
$8.5$

Show the answer

A
$49.7$
This is the common value of both sides when $x=11.5$ . It gives the at the intersection, but the question asks for the value of $x$ .
B
$17.5$
This results from multiplying only the $x$ by $0.2$ and subtracting the $3$ unchanged, so $-0.2(x+3)$ becomes $-0.2x-3$ . The left side is then $1.8x+26.6$ , and $0.4x=7$ gives $x=17.5$ . The $-0.2$ multiplies the $3$ too, giving $-0.6$ .
C
$11.5$
Correct answer
Distributing gives $1.8x+29=1.4x+33.6$ . Subtracting $1.4x$ and $29$ from both sides gives $0.4x=4.6$ , so $x=11.5$ .
D
$8.5$
This results from distributing $-0.2$ to $x$ but treating $-0.2(3)$ as $+0.6$ . The term is $-0.6$ , which makes the left side $1.8x+29$ .

Solution

Distribute each :

$0.5(4x+59.2)-0.2(x+3)=1.4(x+24)$

$2x+29.6-0.2x-0.6=1.4x+33.6$ . Combine :

$1.8x+29=1.4x+33.6$ . Subtract $1.4x$ from both sides and subtract $29$ from both sides:

$0.4x=4.6$ . Divide both sides by $0.4$ :

$x=11.5$

Report a problem

Two equations at once

For a , type each equation on its own row and click where the graphs cross, even for a line and a . Type y=x^2+2x-8 and y=2x+1, then click either graph. Gray dots appear at the two crossings, $(-3,-5)$ and $(3,7)$ . If the question asks for $y$ in the solution where $x$ is positive, the answer is 7.

Desmos needs $x$ and $y$ . If a system uses $r$ and $s$ , type $x$ for every $r$ and $y$ for every $s$ , and write "x is r, y is s" on scratch paper so you read the right coordinate. The systems lesson has more on two lines.

The lowest or highest point

What is the minimum value of $f(x)=x^2+4x-5$ ?

Type y=x^2+4x-5 and click the curve. Gray dots appear at the , $(-2,-9)$ , where it crosses the $x$ -axis, $(-5,0)$ and $(1,0)$ , and at the , $(0,-5)$ . The minimum value is the $y$ -coordinate of the vertex, $-9$ .

Trap: The x-coordinate of the vertex

A choice of $-2$ comes from the same dot. It's where the minimum happens, and the question asked for the minimum itself.

What to do: “Minimum value” or “maximum value” asks for y. “For what value of x” asks for x.

Which expression is equivalent

Which is equivalent to $2x^2-5x-12$ ? Type y=2x^2-5x-12 on the first row and each choice, after y=, on a row below it. The right choice, here y=(2x+3)(x-4), lands exactly on top of the original. A wrong one pulls away somewhere, so look across the whole screen.

Trap: A choice that drew nothing

A choice typed with a slip can draw no graph at all. You still see one curve on the screen, and it looks like a perfect match.

What to do: Hide the original by clicking the colored circle at the left of its row, and check that a curve is still there. Find that circle once on the practice site.

When the question is about form, like which choice shows the $y$ -intercept as a number, graphing can't help. All four choices are equivalent, so their graphs all land on top of each other.

How many solutions

Put each side on its own y= row and count the crossings. When one side is 0, the $x$ -axis is the second graph. Zoom out first, because a crossing off the screen still counts.

What you see	Solutions
Two crossings	2
One touch, where the graphs meet without crossing	1
No meeting	0 real solutions
Two graphs on top of each other	Infinitely many

y=(x-3)^2 touches the $x$ -axis at $x=3$ and turns back up, so $(x-3)^2=0$ has one solution. y=x^2+1 never reaches the $x$ -axis, so $x^2+1=0$ has no real solutions. Something that looks like one line may be two identical lines, so hide a row to check.

Two crossings very close together can look like a touch. When the answer depends on it, check with the , $b^2-4ac$ , which is 0 for a touch. The nonlinear equations lesson has examples.

Finding a constant with a slider

For what value of $k$ does $x^2-4x+k=0$ have exactly one real solution?

Type y=x^2-4x+k, and Desmos offers "add slider: k". Click it and drag the slider. With a small $k$ , the curve crosses the $x$ -axis twice. As $k$ grows, the two crossings slide together into one touch near 4, and then the curve lifts off the axis.

So you need the touch. A number read off a slider is only as exact as where you let go, so finish by hand. A touch means the discriminant is 0.

One solution means $b^2-4ac=0$ . Here $a=1$ , $b=-4$ and $c=k$ .
$(-4)^2-4(1)(k)=0$
$(-4)^2=16$ .
$16-4k=0$
Add $4k$ to both sides.
$16=4k$
Divide both sides by 4.
$k=4$

Worked example

$5x^2-40x+k=0$

The given equation has exactly one real solution. What is the value of the constant $k$ ?

Show the answer

Accepted answer: 80

Solution

A has exactly one real solution when its , $b^2-4ac$ , equals zero. Substituting $a=5$ , $b=-40$ , and $c=k$ gives $(-40)^2-4(5)k=0$ , so $1600-20k=0$ and $k=80$ .

The wrong answer $400$ comes from omitting the $5$ and solving $1600-4k=0$ . The wrong answer $-80$ comes from changing $-20k$ to $20k$ without changing the sign of $-1600$ , producing $20k=-1600$ .

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Inequalities

Desmos shades the points that make an true. Type y<=2x+1 and it shades below the line. With $\le$ or $\ge$ the edge is solid, and points on it count. With $\lt$ or $\gt$ the edge is dashed, and points on it don't. For two inequalities, the solutions are where the two shadings overlap.

When the choices are points, it's usually as fast to put each point's numbers into both inequalities by hand. A point where the two sides come out equal sits on the edge, so it counts only with $\le$ or $\ge$ .

Worked example

Bruna is reviewing a parking garage’s hourly fee $x$ and all-day fee $y$ . An acceptable fee pair satisfies both inequalities.

$y \gt 2x + 1$ $x + y \leq 13$

Which point $(x, y)$ is a solution to the given system of inequalities?

A
$(4, 8)$
B
$(3, 8)$
C
$(3, 7)$
D
$(5, 9)$

Show the answer

A
$(4, 8)$
Substituting $x=4$ and $y=8$ gives $4+8\leq13$ , but $8\gt 2(4)+1$ is false because $8$ is not greater than $9$ . This point satisfies only one .
B
$(3, 8)$
Correct answer
Substituting $x=3$ and $y=8$ gives $8\gt 2(3)+1$ , or $8\gt 7$ , and $3+8\leq13$ , or $11\leq13$ . The point satisfies both .
C
$(3, 7)$
Substituting $x=3$ and $y=7$ gives $7=2(3)+1$ , so the point lies on the boundary $y=2x+1$ . The $y\gt 2x+1$ excludes points on that boundary.
D
$(5, 9)$
This point results from reversing both symbols: $9\lt 2(5)+1$ and $5+9\geq13$ are true. The given system requires $y\gt 2x+1$ and $x+y\leq13$ .

Solution

Substitute each point into both , $y\gt 2x+1$ and $x+y\leq13$ . A solution must make both statements true.

For $(3,8)$ , the first inequality becomes $8\gt 2(3)+1$ , so $8\gt 7$ . The second becomes $3+8\leq13$ , so $11\leq13$ . Both statements are true, so $(3,8)$ is a solution to the system.

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Practice

Before you type anything, decide whether graphing will save time. After you graph, say which coordinate the question wants before you click.

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5 questions on this skill, with every choice explained.

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Related lessons

Solving by graphing in Desmos

When to open the graph

Solving one equation

Two equations at once

The lowest or highest point

Which expression is equivalent

How many solutions

Finding a constant with a slider

Inequalities

Practice