Desmos Power Moves II — Bounds, Domains & False Answers

Control the graphing window, restrict a model, and reject roots that do not fit the original problem.

29 min
Desmos way

Ready when you are

7 steps · about 29 minutes

Start with the warm-up.

Start

Step 1 of 7Warm-upDone

Warm-up

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Key terms

Domain: All input values allowed by the mathematics and the context.
Extraneous solution: A candidate created by algebra that fails the original condition.
Window bounds: The left, right, bottom, and top values visible on a graph.

Step 2 of 7Try it firstDone

Try it first

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Take a shot before the lesson. Write one relationship from the stem, even if you do not finish.

Step 3 of 7LearnDone

The big idea

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See the structure

A graphing window is part of the solution method. If an intersection is missing, open the wrench and enter bounds that include values allowed by the problem.

Key idea

Name quantities before calculating

Write what each number measures. A correct calculation with the wrong quantity or unit still misses the question.

Build a reliable method

A model has a mathematical domain and often a smaller contextual domain. Type a restriction after the expression, such as y=3x+8{0<=x<=20}.

Explore in Desmos

See an extraneous root disappear

The squared equation shows candidate lines at x=0 and x=5. The original radical equation shows only x=5. Substitute x=0 to explain why it disappears.

The calculator is live: change anything and see what happens.

See an extraneous root disappear

Degree mode

Loading Desmos…

Stretch to a 36

Desmos reports solutions to what you typed. Check each candidate in the original equation and reject negative time, impossible counts, zero denominators, and roots introduced by squaring.

Trap

Squaring can create an answer

If $a=b$ , then $a^2=b^2$ . The reverse is not guaranteed because $a^2=b^2$ also allows $a=-b$ . Every candidate from a squared equation must go back into the original.

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Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1: Restrict before you read

Problem

For

0\le x\le6

, what is the maximum value of

y=(x-4)^2

Type y=(x-4)^2{0<=x<=6}. The braces prevent Desmos from showing irrelevant points outside the interval.

The vertex is $(4,0)$ , a minimum. Where can the interval maximum occur?

Which candidate respects the domain and original equation?

Evaluate the endpoints:

y(0)=(-4)^2=16

and

y(6)=2^2=4

The maximum is $16$ at $x=0$ .

Which substitution or endpoint check proves the result?

Worked example

Example 2: Reject the root squaring created

Problem

Which real number solves

\sqrt{x+4}=x-2

The right side must be nonnegative, so any solution must satisfy

x\ge2

Square both sides: $x+4=(x-2)^2=x^2-4x+4$ .

Which candidate respects the domain and original equation?

Factor:

x(x-5)=0

, giving candidates

0

and

5

Check the original. At $x=0$ , $2=-2$ is false; at $x=5$ , $3=3$ is true. Only $5$ remains.

Which substitution or endpoint check proves the result?

Faded example · you finish it

Example 3: You finish it

Problem

The model

y=3x+8

is valid for

0\le x\le20

. What is ==

y

when

x=12

==?

Apply the stated domain before accepting any output or root.

Which calculation matches the relationship?

The computed answer is

Desmos way

Compare candidates with the original

Type the original equation first. Desmos shows the valid solution line at

x=5

Type the squared equation second. It adds

x=0

. The extra line is visual evidence that squaring created an extraneous candidate.

Compare candidates with the original

Degree mode

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Practice

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Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.