Week 02 · Polynomial Functions

Garden Day

Garden Day: a box from any sheet

// A box whose volume is a cubic

⏱ about 20 min

Garden Day: A Box From Any Sheet

Comet finds a cereal box in the recycling and opens it flat. "Twenty by sixteen. Perfect."

"What are you making?" her brother asks.

"A seedling box, and a cubic," she says. "Cut a square from each corner, side x, and fold."

He frowns. "How big a square?"

"That is the question," Comet says. "V(x) = (20 - 2x)(16 - 2x)x. We try a few and watch the volume."

Nova hovers by the window. "Would you like a hint? Make a table from x = 1 up to x = 7."

Her brother fills the first box with dried beans and counts cups. "Cut 3 is the big one," he says.

"The cubic agrees," Comet says, and shows him the curve.

A polynomial function adds terms with whole-number powers of x. The highest power is its degree.
Zeros come from factors: x - a gives a zero at a. End behavior comes from the leading term.
Even multiplicity touches the x-axis; odd multiplicity crosses it.
Between crossings the graph turns. A turning point is a peak or a valley.
Read each feature in the build's words, and notice which parts of the function the build does not use.

◇ BUILD A BOX TOGETHER

Find a flat piece of cardboard and measure its length L and width W in centimeters.
Together, write V(x) = (L - 2x)(W - 2x)x and say what each factor means.
Cut squares of side 1, 2 and 3 from three sheets with scissors, fold and tape. Fill each with dried beans and count cups.
Work out V(1), V(2) and V(3) from the formula and compare with the cups.
Ask a family member to guess the best cut, then test it with the formula. Nothing here is eaten.

Comet's cereal-box numbers

Comet's sheet is 20 by 16, so V(x) = (20 - 2x)(16 - 2x)x. Expanded, V(x) = 4x³ - 72x² + 320x. Her numbers are made up for this lesson.

Cut x	Base	V(x)
1	18 by 14	252
2	16 by 12	384
3	14 by 10	420
4	12 by 8	384
5	10 by 6	300
6	8 by 4	192
7	6 by 2	84

COMET'S BOX

Read the question.
Tap your answer.

Comet's V(x) = 4x³ - 72x² + 320x. Find V(2).

For Comet's 20 by 16 sheet, at what cut does the width 16 - 2x reach 0?

In Comet's table, which cut gives the largest volume? Type the number.

WHY THIS EXERCISEThe peak of the cubic is the cut worth making. The table finds it to the nearest whole number.

Draw Comet's V(x) from x = 0 to x = 8. Mark the zeros, the peak and the cut you would choose.

Glass House Log: record your sheet size, your V(x), your cup counts and the best cut.Write how you explained a zero and a turning point to your family, in the words of the box.Did the cup counts match the formula? Write one reason they might not.

Well done this week. Next week the seedling trays are shared into rows, and the leftover is a remainder.

Grown-up checked my answers

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