Week 02 · Friday: Polynomial Functions (Algebra 2 9-12)

Friday: Key Features in the Build

Comet pins the finished cut-3 box to the Open House board with a sketch of V(x) beside it.

"Visitors will ask what the curve means," Wren says. "Say every feature in the words of the box."

"Zero at x = 0: no cut, no height, no box," Comet says. "Zero at 9: the width is gone."

"Rising from 0 to about 3: a deeper box holds more. Falling after: the base shrinks too fast."

Nova projects the curve with the peak glowing. "Would you like a hint? Say what the peak is, in cubic centimeters."

"About 650," Comet says. "The most soil a box from this sheet can hold."

"And the end behavior?" Wren asks. "Rises forever on the right."

"True of the function, not of the box," Comet says. "Past 9, there is no box. What do you notice? Math and build are not the same."

Reading a graph in the build's words

Feature	On the graph	In the build
zero at x = 0	V(0) = 0	no cut, no height, no box
zero at x = 9	V(9) = 0	the width is cut away
rising on 0 to about 3	the curve climbs	a deeper box holds more
falling on about 3 to 9	the curve drops	the base shrinks too fast
turning point	near x = 3, V about 648	the cut that holds the most
end behavior	up on the right forever	not part of the build: no box past 9

A polynomial function covers every real x. A build covers only the x values that make sense.

When you read a graph, say each feature twice: once for the function, once for the thing it models.

The week in one table

Skill	Where to look	Example
zeros	the factors	(x + 2)(x - 1)(x - 3): -2, 1, 3
end behavior	the leading term	x⁴ - 5x² + 4: up on the left, up on the right
multiplicity	repeated factors	(x - 2)²(x + 1): touches at 2, crosses at -1
turning points	between crossings	the box: one peak near x = 3
evaluate	either form	V(3) = 648

MIXED REVIEW

Read the question.
Tap your answer.

What are the zeros of p(x) = x⁴ - 5x² + 4?

How does the graph of y = x³ - 4x behave at the far left and far right?

What is the multiplicity of the zero 3 in x(x - 3)²(x + 2)?

If p(x) = x³ - 4x, what is p(-1)?

SAY IT IN THE BUILD

Read the question.
Tap your answer.

V(9) = 0. What does this zero mean for the crew's box?

V(x) rises on cuts from 0 to about 3. What does that mean for the box?

The graph of V(x) rises forever on the right. What does the crew say about the build?

For p(x) = (x - 2)²(x + 1), the zero at x = 2 has multiplicity ____. Type the number.

V(x) = 4x³ - 84x² + 432x has degree ____. Type the number.

At a zero of odd multiplicity the graph does this to the x-axis. Type one word.

Statement	True or false?
A polynomial's end behavior is set by its leading term.	?
Every feature of a graph has meaning in the build it models.	?
4 × 27 - 84 × 9 + 432 × 3 = 648	?
The crew's box function has exactly one turning point between 0 and 9.	?
A degree-4 polynomial always has four zeros.	?

WHY THIS EXERCISEDeciding which features belong to the build is the last step of reading any graph.

Try it

Pick a sheet size of your own, say 30 by 20. Write V(x), find its zeros and the largest sensible cut.
Work out V(x) for whole-number cuts and find the peak. Write the peak in the words of the box.

Draw V(x) for the crew's sheet and label every feature twice: once as math, once as the box.

Glass House Log: write the five key features of the box graph in the words of the build.Which features of the function are not part of the build? Write why.What was the hardest idea this week? Write one sentence about how you got it.

A full week. Tomorrow is Garden Day: you build a box at home and show your family the curve behind it.