Domain from Graphs and Discrete vs. Continuous

Deck 2 of 2: Reading Graphs and Synthesizing Perspectives

In this deck:

• Read domain from any graph using the vertical scan
• Distinguish discrete from continuous domains

Review: What We Know So Far

From Deck 1:

• Natural domain: algebraic restrictions (denominators, radicands)
• Contextual domain: real-world constraints narrow the domain further
• Context wins in modeling

In Deck 2 we add:

• Graphical domain: read from the graph directly

Reading Domain with the Vertical Scan

Sweep a vertical line from left to right across the graph.

Every -value where the line touches the graph is in the domain.

Graph 1: Line Extending Both Directions

A line without endpoints extends forever in both directions.

• Vertical scan touches the graph at every -value
• No restrictions

Domain: — all real numbers

Graph 2: Ray Starting at a Point

A ray starts at with a solid dot and extends to the right.

• Solid dot at : the endpoint is included
• Vertical scan touches the graph for all

Domain:

Graph 3: Bounded Curve with Closed Endpoints

A curve runs from to with solid dots at both ends.

• Both endpoints included (solid dots)
• Vertical scan only touches graph between and

Domain:

Graph 4: Curve with a Hole

Open circle at : that single point is excluded.

Domain:

• At , , , : function is defined
• Only exactly is missing

Graph 5: Function Has a Vertical Asymptote

A curve approaches but never touches the vertical line .

• is excluded — the function blows up there
• Domain is all values except

Domain:

Graph 6: Only Isolated Dots Appear

The graph shows isolated dots at , , , , .

• No curve — just individual, unconnected points
• Vertical scan only touches at exactly those -values

Domain: — set-roster notation

Check: Match Graph to Domain

Match each mini-graph to its domain:

• Graph A: parabola on , solid endpoints
• Graph B: curve, open circle at
• Graph C: dots at only

Write each domain before the next slide.

Matching Answers and Dot Convention Summary

• Graph A: — solid endpoints, both included
• Graph B: — open circle removes
• Graph C: — four isolated dots

Dot rule: Solid (●) = included · Open (○) = excluded

Graph and Formula Should Always Agree

When a function is given both ways, the domains should agree.

Graph: vertical asymptote at

Both methods agree: domain is

Continuous vs. Discrete: The Distinction

Continuous domain: all real values in an interval

• Examples: time, temperature, distance
• Graphed with a solid, connected curve

Discrete domain: only isolated, individual values

• Examples: people, tickets, correct answers
• Graphed with isolated dots

Bacteria Growth Shows a Continuous Domain

• Time flows continuously — fractional hours are valid
• Domain: , continuous
• Graph: solid curve starting at

Discrete: Bonus Points per Correct Answer

• Answers are integers:
• Domain is discrete — you can't get 2.5 correct
• Graph: isolated dots, NOT a connected line

Capstone: Water Taxi — All Four Perspectives

Check: What Is the Candle Domain?

Is this domain discrete or continuous?
What is the contextual domain?

Write your answer before the next slide.

Candle Burns Continuously from Zero to Twenty-Four

— candle length in inches after hours

• Time flows continuously — fractional hours are valid
• Minimum:
• Candle burns out:

Contextual domain: — closed interval, continuous

Three Ways to Identify a Function's Domain

In modeling: contextual domain takes priority.

Key Takeaways from Deck Two

✓ Vertical scan: every touched belongs to the domain

✓ Solid ● = included; open ○ = one excluded point

✓ Continuous → curve; Discrete → dots (never connect)

✓ Contextual domain takes priority in modeling

Up Next: Average Rate of Change

HSF.IF.B.6 — Average Rate of Change

With domain mastered, you can specify intervals precisely.

• Average rate of change over requires knowing are in the domain
• Contextual domain determines which intervals are meaningful

Domain is the foundation for everything that follows.