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Algebra 2 9-12 / Week 08 / Friday
5/6
Week 08 · The Function Toolkit

Friday

Comparing functions
// Build, move, compare, cross
⏱ about 20 min

Friday: Comparing Functions

"The trellis arch," Comet says, sketching. "Its height rule is -x² + 6x in the crew's units."

"And the hoop house arch only came as a table," Wren says, pinning it up. "No rule, just readings."

"Which arch is taller at its peak?" Comet asks. "One is a rule and one is a table. Can we even compare?"

"Of course," Wren says. "Find the peak of each in its own language. What does the evidence say?"

Nova highlights the rule's vertex. "Would you like a hint? A parabola's peak sits halfway between its zeros."

"Zeros at 0 and 6, so the peak is at x = 3. Height 9," Comet says. "The table's biggest row is 7."

"So the trellis is taller," Wren says. "A rule and a table, compared fairly."

Compare in each one's language

A function can arrive as a rule, a table, a graph or a sentence. To compare two, ask the same question of each.

Peak height: for a rule, find the vertex. For a table, find the biggest row. For a graph, read the top.

Rate of change: for a rule, compute f(b) - f(a) over b - a. For a table, subtract rows.

QuestionTrellis, rule f(x) = -x² + 6xHoop house, table g
Peak height9 at x = 37 at x = 3
Height at x = 286
Average rate, x = 0 to 243
Zerosx = 0 and x = 6x = 0 and x = 6

The hoop house table, every reading the crew's own tape measurement:

x0123456
g(x)0467640

Rule against table

Last week's duckweed count came as a table doubling each day. The walkway rule 2x + 5 is a line.

At small x the line is ahead. The doubling table catches up and passes it. Compare row by row to see where.

COMPARE TWO FUNCTIONS
  • Read the question.
  • Tap your answer.
Trellis f(x) = -x² + 6x. Hoop house table peaks at 7. Which arch is taller at its peak?
Trellis f(x) = -x² + 6x. Hoop house table: g(1) = 4, g(2) = 6, g(3) = 7. At x = 2, which arch is taller?
Walkway rule f(x) = 2x + 5. Duckweed table: g(0) = 1, g(1) = 2, g(2) = 4, g(3) = 8, g(4) = 16, g(5) = 32. At x = 4, which is larger?
The same two: f(x) = 2x + 5 and the duckweed table. At x = 1, which is larger?
MIXED SET
  • Read the question.
  • Tap your answer.
Trellis f(x) = -x² + 6x. What is its average rate of change from x = 0 to x = 2?
A second trellis is g(x) = f(x) + 2 where f(x) = -x² + 6x. What is g(3)?
If f(x) = 2x + 1 and g(x) = x², what is f(g(3))?
Where do y = x² and y = x + 6 meet?
The trellis rule is -x² + 6x. What is its peak height? Type the number.
The hoop house table's biggest reading is its peak. Type that number.
For the trellis, what is the average rate of change from x = 0 to x = 2? Type the number.
StatementTrue or false?
Two functions can be compared even when one is a rule and the other is a table.?
0 - 3 × 3 + 6 × 3 = 9?
A table always has a bigger peak than a rule.?
The average rate of change of a table over an interval is found by subtracting rows.?
A doubling table stays below the line 2x + 5 forever.?
WHY THIS EXERCISEThe week in five lines: build, move, measure the rate, find the crossing, and compare fairly.
Draw the trellis y = -x² + 6x and the hoop house table's dots on one grid. Mark both peaks.

A full toolkit. Tomorrow is Garden Day, and the family finds crossings with string on the kitchen table.

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