Thirty-Six

Unit 5 · FunctionsF3 · Graph Features · Guidance

Domain, Range & Graph Features

Read intercepts, extrema, intervals, domain, and range from formulas and graphs.

  • 30 min
  • Unlocks: Trajectory Display
  • Desmos way

Ready when you are

7 steps · about 30 minutes

Start with the warm-up.

Start

Step 1 of 7Warm-upDone

Warm-up

Finish the previous step to continue, or use “Show everything” above to explore.

Key terms

domain
all allowed input values

Step 2 of 7Try it firstDone

Try it first

Finish the previous step to continue, or use “Show everything” above to explore.

Try this before the explanation. Commit to a setup, even if you are unsure.

Step 3 of 7LearnDone

The big idea

Finish the previous step to continue, or use “Show everything” above to explore.

The decision that matters

Check denominators and even roots before trusting a graph window. Write that decision before calculating. It turns a crowded ACT stem into a familiar structure and makes the likely distractors easier to spot.

Key idea

Structure before arithmetic

Check denominators and even roots before trusting a graph window. Keep exact values through the setup; round only when the stem asks.
Core relationship
$\sqrt{A}$ needs $A\ge0$; a denominator cannot equal 0.

Use the formula only after matching every symbol to the diagram or equation.

Algebra or Desmos?

Derive domain restrictions from the formula before graphing. Desmos helps display a radical endpoint or a parabola minimum, but a window edge is never evidence that a domain or range stops there.

Explore in Desmos

Locate a radical endpoint and parabola minimum

Zoom enough to see the radical endpoint and parabola minimum; do not treat window edges as domain endpoints.

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

Locate a radical endpoint and parabola minimum

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

Loading Desmos…

Trap

The main trap

A window edge is not a true endpoint unless the formula or stated interval makes it one.

Step 4 of 7Worked examplesDone

Worked examples

Finish the previous step to continue, or use “Show everything” above to explore.

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

Problem

Find the domain of f(x)=2x−6f(x)=\sqrt{2x-6}.
Require 2x−6≥02x-6\ge0.

Which inequality makes the radicand valid?

The domain is [3,∞)[3,\infty).

Why is the endpoint 3 included?

Worked example

Example 2: ACT twist

Problem

Find the domain of g(x)=x+1x2−9g(x)=\frac{x+1}{x^2-9}.
Factor the denominator: (x−3)(x+3)(x-3)(x+3).

Which inputs make the denominator zero?

All other real inputs are allowed.

Why must both −3-3 and 33 be excluded?

Faded example · you finish it

Example 3: You finish it

Problem

Find the range of y=(x−2)2−5y=(x-2)^2-5.
A square is at least 0.

Use the minimum value at the vertex.

Desmos way

Read domain and range features from the graphs

Zoom enough to see the radical endpoint and parabola minimum; do not treat window edges as domain endpoints.

Read domain and range features from the graphs

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

Loading Desmos…

Step 5 of 7PracticeDone

Practice

Finish the previous step to continue, or use “Show everything” above to explore.

Step 6 of 7Spot the mistakeDone

Spot the mistake

Finish the previous step to continue, or use “Show everything” above to explore.

Spot the mistake

Where did it go wrong?

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

Step 7 of 7Exit ticketDone

Exit ticket

Finish the previous step to continue, or use “Show everything” above to explore.
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.