Extremes, End Behavior, and Sketching

In this lesson:

• Identify and interpret relative maximums and minimums
• Describe end behavior and symmetry of a function's graph
• Sketch a graph showing key features from a verbal description

What You Will Learn Today

1. Identify relative maxima and minima as turning points
2. Interpret extremes in context
3. Distinguish relative (local) from absolute (global) extremes
4. Describe end behavior as
5. Sketch a graph from a verbal description

Revisiting the Soccer Ball Height Problem

— ball in air from to .

We found the zeros. Now: what is the highest point, and when?

This is the relative maximum — the turning point where the ball stops rising and starts falling.

Relative Maximum — Peak and Turning Point

A relative maximum is a point greater than all nearby values — where increasing ends and decreasing begins.

For : maximum at

Ball Maximum: Graph and Interpretation

• Relative maximum at : the ball reaches 20 meters at seconds
• Increasing on — ball going up
• Decreasing on — ball coming down

Maximum Connects to Increasing and Decreasing

Relative maxima sit at the boundary between an increasing and a decreasing interval.

Hiking Trail: Both a Max and a Min

Relative Versus Absolute Extremes Compared

The trail peaks at 1,200 m (km 3) — locally highest, but the trail may climb higher elsewhere.

Check: Relative Max Is Not the Global Peak

Trail: relative max of 1,200 m at km 3. Which must be true?

• A. Trail reaches 1,200 m at km 3 — YES
• B. No other point exceeds 1,200 m — NOT NECESSARILY
• C. Trail increases before km 3 and decreases after — YES

Practice: Identify and Interpret Extremes

1. Relative minimum at ; = days, = temperature (°C). Interpret.

2. Graph increases on and decreases on . Where is the relative maximum?

3. Sketch a curve: relative max at , relative min at .

Extremes Practice: Check Your Answers

1. Day 4 is the coldest point (−3°C); temperatures rise afterward.

2. Relative maximum at — increasing ends, decreasing begins.

3. Any smooth curve: rises to , dips to , rises again.

Transition: From Local to Long-Run Behavior

You can now identify and interpret local features — intercepts, increasing/decreasing intervals, and turning points.

Now the big picture: end behavior

What happens to as or ?

This describes the long-run trend — not a specific point, but the overall direction.

Three Types of End Behavior

• Linear : as , ; as ,
• Quadratic : as , (both ends up)
• Exponential decay : as , ; as ,

Contextual End Behavior: Medicine Example

— drug concentration (mg) after hours.

• As : — drug wears off, never fully gone
• As : not meaningful (no backward time)

End behavior tells us the long-term trend, not a specific value.

Even and Odd Symmetry: Mirror Patterns

Even functions: symmetric about the y-axis —

• : same output for opposite inputs, e.g.,

Odd functions: rotational symmetry about the origin —

Symmetry is a shortcut: understand half the graph, know the whole.

Sketching: Translate Words Into Graph Features

Roller Coaster Sketch: Step by Step

Check: Sketch the Cooling Coffee

Coffee starts at 180°F and cools toward room temperature (72°F) but never drops below it.

Sketch key features:

• y-intercept at 180
• Always decreasing; rate slows over time
• Horizontal asymptote at 72°F

(Sketch a decreasing curve from 180 that flattens near 72.)

Synthesis Practice: All Key Features

1. Stock: starts $50, peaks at $80 (week 6), drops to $30 (week 14), then rises without bound. Sketch and label.

2. End behavior of ?

3. is symmetric about the y-axis. . What is ?

All Three Synthesis Answers Revealed

1. Curve: starts at 50, max at , min at , then rises.

2. As : ; as : .

3. — even symmetry: .

All Seven Key Features: Complete Summary

What's Next: Domain and Graph Features

Coming up in HSF.IF.B.5:

• Relate the domain of a function to its graph
• Identify appropriate domains for real-world models
• Connect domain restrictions to the context

The graph analysis skills from this lesson apply directly — domain determines which features are visible and meaningful.