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Algebra 2 9-12 / Week 08 / Garden Day
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Week 08 · The Function Toolkit

Garden Day

Garden Day: string graphs
// Build, move, compare, cross
⏱ about 20 min

Garden Day: String Graphs

Saturday a sheet of graph paper is taped to the kitchen table, and two lengths of string lie across it.

"Each card has two rules," Comet says. "Lay one string along each graph. Where do they cross?"

The first card reads y = x² and y = 2x + 3. Someone calls out "3!" before the string is down.

"How did you know?" Wren asks. "I noticed 3 squared is nine, and two times three plus three is nine too."

Nova hovers above the paper, her light a soft hint. "Would you like to check the other crossing?"

"Minus one," comes the answer. "One equals negative two plus three."

"Next card," Wren says. "This one has a table instead of a rule. Can you still find the crossing?"

  1. Build a function by adding, multiplying or composing two others. In f(g(x)), g runs first.
  2. Four moves: f(x) + k slides up, f(x + k) slides left, kf(x) stretches, -f(x) flips.
  3. Average rate of change is output change over input change. A line has one rate; a curve's rate changes.
  4. A crossing of y = f(x) and y = g(x) has an x that solves f(x) = g(x). Find it by graph, table or algebra.
  5. Compare two functions by asking each the same question in its own language.
◇ FAMILY STRING GRAPHS
Write four cards, each with two rules such as y = x² and y = x + 6. Keep the numbers small.
Plot each rule with dots on graph paper, then lay a string along each. Read where the strings cross.
Check every crossing by algebra: set the rules equal and solve. Do the string and the algebra agree?
For a harder round, give one rule as a table of five rows and find the crossing from the table.
For a grown-up
This is a paper and string activity. No ladder, no tools, nothing heated.
A wrong crossing is a good moment: put the x into both rules and see which side does not match.
The rules on the cards are made up for the game, not measurements of anything real.
GARDEN DAY CHECK
  • Read the question.
  • Tap your answer.
A card reads y = x² and y = x + 6. Where do the strings cross?
A card reads L(x) = x + 6 and P(x) = x². What is L(P(2))?
A card reads y = x² and y = 2x + 3. What is the larger x where the strings cross? Type the number.
WHY THIS EXERCISEEvery card is the same equation shape. The string shows the crossing and the algebra proves it.
Draw two of the family cards with their rules, and mark each crossing with its coordinates.
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