Lesson 2 Floating Down the River Solidify Understanding

Learning Focus

Identify key features of functions.

Use key features of functions to analyze tables and graphs.

How do I describe key features of a function?

Open Up the Math: Launch, Explore, Discuss

Part I:

Alonzo, Maria, and Sierra were floating in inner tubes down a river, enjoying their day. Alonzo noticed that sometimes the water was much deeper in some places than in others. Maria noticed there were times they seemed to be moving faster than at other times. Sierra laughed and said “Math is everywhere!” To learn more about the river, Alonzo and Maria collected data throughout the trip. Here is Alonzo’s data table about the depth of the water as they floated along.

1.

Use the data that Alonzo collected in the table to interpret the key features of this relationship.

Maria created a graph by collecting data on a GPS unit that provided the distance she had traveled over a period of time.

A continuous graph on a coordinate grid with the horizontal axis labeled “time (in minutes.)” The vertical axis is labeled “Distance (in feet).

2.

Describe the key features of the relationship shown in Maria’s graph. Include the intervals of increase and decrease, domain, range, maximum, minimum, and intercepts.

3.

Part II: Interpreting data

Sierra looked at the data collected by her two friends and made several of her own observations. Explain why you either agree or disagree with each observation made.

a.

The depth of the water increases and decreases throughout the $120$ minutes of floating down the river.

Agree

Disagree

Reason:

b.

The distance traveled is always increasing.

Agree

Disagree

Reason:

c.

The distance traveled is a function of time.

Agree

Disagree

Reason:

d.

The distance traveled is greatest during the last ten minutes of the trip than during any other ten-minute interval of time.

Agree

Disagree

Reason:

e.

The domain of the distance/time graph is all real numbers.

Agree

Disagree

Reason:

f.

The $y$ -intercept of the depth of water over time function is $(0, 0)$ .

Agree

Disagree

Reason:

g.

The distance traveled increases and decreases over time.

Agree

Disagree

Reason:

h.

The depth of the water is never $11$ feet.

Agree

Disagree

Reason:

i.

The range of the distance/time graph is from $[0, 15000]$ .

Agree

Disagree

Reason:

j.

The domain of the depth of water with respect to time is from $[0, 120]$ .

Agree

Disagree

Reason:

k.

The range of the depth of water over time is from $[4, 5]$ .

Agree

Disagree

Reason:

l.

The distance/time graph has no maximum value.

Agree

Disagree

Reason:

m.

The depth of water reached a maximum at $30$ minutes.

Agree

Disagree

Reason:

Ready for More?

Use the data given in the table and the graph to graph the relationship between distance (independent variable) and depth (output variable) during the float trip and interpret the meaning. Create statements like Sierra’s related to features of the function. Make three statements that are true and one statement that is false.

Statements:

Takeaways

Strategies for Finding Features of Functions:

Adding Notation, Vocabulary, and Conventions

In words:	As an inequality:	In interval notation:

Vocabulary

interval notation
set
Bold terms are new in this lesson.

Lesson Summary

In the lesson we learned how to find features of functions in a table and clarified what each feature describes. We learned to use interval notation to write domains, ranges, and intervals of increase and decrease for continuous functions.

Retrieval

1.

Graph the two equations and determine the point where they intersect.

$f (x) = - 5 x + 4 g (x) = 2 x - 3$

Point of intersection:

2.

Find the explicit and recursive equations for the table.

$n$	$f (n)$
$1$	$64$
$2$	$61$
$3$	$58$
$4$	$55$
$5$	$52$

3.

Find the explicit and recursive equations for the table.

$n$	$g (n)$
$1$	$64$
$2$	$32$
$3$	$16$
$4$	$8$
$5$	$4$