Equivalent expressions by graph overlap

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Graph the expression and each answer choice. The choice whose graph lands exactly on top is equivalent. Learn the near misses and domain traps that fool it.

Difficulty: Intermediate
Estimated time: 40 minutes
Domains: Advanced Math

What you’ll learn

Explain why two equivalent expressions in one variable draw the same graph wherever both are defined.
Type the original expression and an answer choice in the same form.
Change the problem's letter to x in every expression when it uses a different one.
Test answer choices by editing a single line.
Tell a full overlap from a single meeting point or a near miss.
Spot when a restriction on the allowed inputs, or a letter that stands for a constant, means overlap isn't enough.
Choose algebra by hand when it's shorter or settles the question for sure.

Prerequisites

Graph and read points of interest0%
Graph expressions and inspect intersections and curve behavior.

Why this matters on the SAT

Find a matching rewrite without expanding everything

Two expressions are equivalent when they give the same value for every $x$ you're allowed to plug in. Showing that by hand can mean distributing, combining like terms, factoring or rewriting powers. But when the expressions are long and the answer choices are already written out, Desmos gives you a shortcut. Graph the original and one answer choice. If the choice is equivalent, it draws exactly the same curve.

SAT example

Which expression is equivalent to

(4x-3)(x+2)-(x-1)(2x+5)?

Fast Desmos solution

Type the original on line 1 and an answer choice on line 2, then swap in the choices one at a time. When line 2 becomes choice D, the curve changes color but doesn't move. The two curves sit on top of each other across the whole window, so the answer is D.

That's the idea to hold on to: equivalent curves don't just meet, they overlap.

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Choice D turns the curve a new color without moving it.

When should you use graph overlap?

Pick the route with less setup and fewer places to slip.

Start with graph overlap when…

The question asks which expression is equivalent, and only one letter in it varies.
The expressions pile up products, powers and minus signs, like $(4x-3)(x+2)-(x-1)(2x+5)$ .
Typing the answer choices would be quicker than expanding everything by hand.
Each answer choice is a complete expression you can type into line 2 in turn.

Work it out by hand when…

Distributing or factoring takes one short line, like turning $2(x-5)$ into $2x-10$ .
The question asks you to write or explain a form yourself instead of picking one.
A denominator, a radical or a piecewise rule limits which inputs are allowed, like $\frac{1}{x-3}$ , which has no value at $x=3$ .
A letter such as $a$ stands for any constant and stays in the answer choices, so one value you pick could make two different forms match by coincidence.

Long answer choices in one variable: graph them. A one-line rewrite, a form you have to write yourself, exact restrictions or a letter that stays in the answer choices: work it out by hand. When the question asks you to find a constant, like the $b$ that makes two expressions equivalent, that's a different job, covered in find constants in equivalent expressions.

Check your understanding:

Which would you graph: rewriting $-5(2x-7)$ , or picking which of four long quadratic expressions, each built from several products, matches the original?

1. Put both expressions in terms of x

Start each expression with y=, so Desmos draws it as a graph:

y = original expression
y = answer choice

With the same y= on both lines, the only thing you're comparing is the right sides. If the problem uses a different letter, like $r$ , $t$ or $z$ , change it to lowercase x in every expression you graph. The letter is only a name, so changing it doesn't change the relationship.

Try it with

7(2z-3)

and

14z+21.

With $z$ changed to x, they're the two lines in the calculator, and they don't overlap. Now change the plus sign in line 2 to a minus sign. Line 2 jumps right onto line 1, because

7(2z-3)=14z-21.

Try it yourself:

After you fix the sign, click the colored icon next to line 2 to hide it, then click again to show it. The line you see shouldn’t move at all. That tells you the original is sitting right underneath.

Check your understanding:

The graphs of $y=x^2+1$ and $y=x^2+2x+1$ meet at $x=0$ . Does that make the expressions equivalent?

Common mistake:

Typing the original with $z$ but changing $z$ to x in only the answer choice. Desmos treats $z$ and x as different letters, so the two lines no longer show the question you were asked. Change the letter everywhere.

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Change the plus sign on line 2 to a minus sign and watch the two lines become one.

Test one answer choice at a time

Line 1 holds the original, and it never changes. Line 2 is where you test the choices, one at a time:

Type the original carefully.
Type choice A on line 2.
Compare the two curves across the whole window.
If they differ, edit line 2 into choice B, then C, then D.
When a curve overlaps completely, reread that choice and the question before you submit.

Try it on

(2x^2-5x+7)-(x^2+4x-3)+2(3x-1).

Choice A, $x^2+3x+8$ , is already on line 2, and it doesn't match. Change only the sign in front of $3x$ . Now the curves overlap, so the equivalent form is

x^2-3x+8.

Check your understanding:

Why test the choices on one line instead of graphing all four at once?

Common mistake:

Stopping because two curves share a vertex or an intercept. That's the same trap as a single meeting point: the expressions are equal at that input, not at every input.

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Leave line 1 alone. Edit line 2 until one choice lands exactly on the original curve.

Look for full overlap, not a close picture

A matching curve sits on the original across the whole window, not only in one spot. So check more than one small region:

Look at both sides of the graph.
Zoom in or out. Curves with nearly equal coefficients can look identical at one scale and split apart at another. For two lines with nearly equal slopes through the same point, pan out to larger inputs or compare outputs.
Hide line 2 and show it again.
If the picture still isn't clear, plug in a convenient $x$ -value and compare the two outputs.

A picture only helps if the window shows the part where the curves differ.

A graph is evidence you can see, not a proof. On a multiple-choice SAT question, that's enough when one choice, and only one, plainly overlaps everywhere. It's the only answer that can work. But if two choices still look identical, don't guess from the screen. Compare the expressions themselves and find the term that's different.

Check your understanding:

Two lines seem to overlap near the origin, but one has slope $2$ and the other has slope $2.01$ . What should you do?

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In this tight window the lines look the same. Pan out to larger inputs, or compare their outputs at a large x-value.

Watch for a hidden hole

Overlap only compares outputs where Desmos draws both curves. It can hide a difference in which inputs are allowed. Take

\frac{(x-3)(x+4)}{x-3}

and

x+4.

They're equal for every $x\ne3$ . But at $x=3$ the fraction would divide by zero, so it has no value there, while $x+4$ equals $7$ . On the graph, that's one missing point, and at a normal zoom it's hard to see. So when a denominator, a radical or a piecewise rule changes which inputs are allowed, don't let overlap be your final check. The next lesson shows you how to handle those.

Common mistake:

Deciding the domains match because you can't see a hole. Curves that line up perfectly don't bring back an input that's excluded, like $x=3$ here.

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The curves line up, but the fraction still has no value at x=3, even though the gap is hard to see.

Watch for letters that stand for constants

Sometimes an expression has a second letter, like $a$ , that stands for any constant. If you give it one convenient value, two different forms can match by coincidence. Values like $0$ and $1$ are the usual culprits, because they can wipe out terms or powers. So one value is a quick test, not a proof.

Compare

y=(x+a)^2

and

y=x^2+a^2.

Expanded, $(x+a)^2$ is $x^2+2ax+a^2$ , so the second expression is missing the $2ax$ term. When $a=0$ , that term is $0$ anyway, and the curves overlap by coincidence. For other values of $a$ , the two expressions aren't the same.

Graph overlap works best when $x$ is the only letter and every other number is fixed. When a letter like $a$ stays in the expressions and the answer choices, compare the algebra instead. Trying more values of $a$ can rule a choice out, but it can't prove two forms are equivalent.

A letter you have to find is a different situation. If a question says $(2x+p)(x-3)-(x+q)(x+1)$ is equivalent to $x^2-10x-11$ and asks for $p+q$ , there's one right pair of values, and Desmos can find it by fitting both sides over a list of inputs. That's the lesson find constants in equivalent expressions.

Try it yourself:

Change $a$ from $0$ to $2$ . The curves split apart, because the missing $2ax$ term is now worth $4x$ .

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At a=0 the curves overlap. Set a to another value, such as 2, and watch them separate.

Example: Rename the variable and find the matching curve

Worked example

Which of the following is equivalent to

(3y-4)^2+5(y+2)-(y^2-16)?

A
$8y^2-29y+42$
B
$8y^2-19y+26$
C
$7y^2-19y+42$
D
$8y^2-19y+42$

Step 1

Change the problem’s y to x

This problem uses $y$ , but Desmos graphs most naturally in x. So change every $y$ in the original to x:

y = (3x - 4)^2 + 5(x + 2) - (x^2 - 16)

Now there are two y's to keep apart. The y= on the left is the graph's output. The problem's $y$ is the x on the right.

Step 2

Test the choices on line 2

Put the choices on line 2, changing their $y$ to x as well. Choices A, B and C each have one coefficient or constant that's off, so each of their curves pulls away from the original somewhere in the window.

Step 3

Check that choice D overlaps everywhere

Choice D becomes

y = 8x^2 - 19x + 42

Its curve covers the original completely. Hide line 2 and show it again, and the parabola doesn't move.

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With every y changed to x, choice D lands exactly on the original.

Step 4

See why with algebra

Expanding each piece shows why the graphs matched:

(3y-4)^2=9y^2-24y+16,

5(y+2)=5y+10,

and

-(y^2-16)=-y^2+16.

The $y^2$ terms give $9y^2-y^2=8y^2$ , the $y$ terms give $-24y+5y=-19y$ , and the constants give $16+10+16=42$ . Together that's

8y^2-19y+42.

So the answer is D.

Common mistake:

Changing $y$ to x in the original but leaving a $y$ in an answer choice. With this letter the slip is easy to miss, because a stray $y$ looks like it belongs next to y=. Rename it in every expression you compare.

Finish the solution

This one has a common factor you can pull out right away, so factoring by hand is quicker than setting up a graph.

Find the matching factored form

Finish the solution

Which expression is equivalent to

(2x-3)(x+5)-(x+5)?

First steps

Spot the $(x+5)$ in both terms.
Factor out $(x+5)$ .
Simplify what's left in the other factor.
Pick the choice that matches.

Finish it

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You can leave this blank. If you want to check your factoring, graph the original and your answer.

Practice problems

Now it's your turn. Before you type anything, take a moment to decide: is there a quick bit of algebra here, or is this a job for the graph?

Recognize a perfect square

Practice problem

Which expression is equivalent to

x^2+6x+9?

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Faster by hand if you spot the pattern. Graphing works too.

Compare polynomial products

Practice problem

Which expression is equivalent to

(5x-2)(x+3)-(2x+1)(x-4)?

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Type the original once, then test the choices on line 2.

Find the missing coefficient

Practice problem

The expression

3(x+4)^2-5

is equivalent to

3x^2+bx+43,

where $b$ is a constant. What is the value of $b$ ?

Your answer

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You need one exact coefficient, so expanding the square by hand is the direct way in.

Rename the variable and test

Practice problem

Which expression is equivalent to

(1+r)(1-r+r^2-r^3+r^4)?

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Change r to x in every expression before you graph.

Finish the lesson

5 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Equivalent expressions have the same value at every allowed input, so their graphs overlap wherever both are defined.
Keep the original on line 1 and test the choices one at a time on line 2.
If the problem uses another letter, like $r$ or $z$ , change it to lowercase x in every expression you compare.
Meeting at one point isn't matching. Equivalent curves overlap everywhere.
When a match isn't clear, hide and show line 2, zoom, pan, or compare a value.
Graph long one-variable answer choices, and use algebra when a line or two settles it.
A hidden hole or a letter standing for a constant can fake a match. Keep track of restrictions, and don't treat one value of the constant as proof.

Next lesson

Restrictions, piecewise functions, and rational expressions

Keep track of excluded inputs, graph restricted rules, and compare rational expressions without losing domain information.

Start next lesson

Practice

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553 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

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