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Algebra 1 9-12 / Week 07 / Friday
5/6
Week 07 · Exponents and Exponential Change

Friday

Reading the rate in the rule
// Linear change adds, exponential change multiplies
⏱ about 20 min

Friday: Reading the Rate in the Rule

Wren measures the seedling on the Loft windowsill. "Twenty centimeters. Last week 19. About 5% taller."

"So its rule is 20 times 1.05 to the power t," Comet says. "Growth factor 1.05, 5% a week. Our own numbers, not a plant fact."

"And the ball was times 0.8," Wren says. "That is a 20% loss. The percent hides inside the factor."

Nova projects a rule that reads 4 to the power t over 2. "Would you like a hint? What is 4 to the one half?"

"Two," Comet says. "So 4 to the t over 2 is 2 to the t. It doubles every step, and the rule was hiding it."

"Rewrite the rule, read the rate," Wren says. "What do you notice about 1.21 to the t over 2?"

"Our engineer," Comet says, "find the hidden factor, and then let us review the week."

Percent changes become factors

Growing 5% each step means multiplying by 1 + 0.05 = 1.05. Shrinking 20% means multiplying by 1 - 0.2 = 0.8.

A rule like 20 × 1.05t says: start at 20 and grow 5% every step. The start and the rate are both in plain view.

A factor above 1 grows. A factor below 1 decays. A factor of exactly 1 is a flat line.

Change each stepFactorRule from a start of 50
grows 5%1.0550 × 1.05^t
grows 10%1.150 × 1.1^t
shrinks 20%0.850 × 0.8^t
doubles250 × 2^t

The hidden rate

Nova's rule 4 to the t/2 is the same as (4 to the 1/2) to the t, by the power rule. And 4 to the 1/2 is 2.

So 4 to the t/2 equals 2 to the t: the quantity doubles each step. Rewriting the expression revealed the rate.

In the same way 1.21 to the t/2 equals 1.1 to the t, because 1.1 squared is 1.21. That is 10% growth each step.

Choosing a form that shows the rate is a skill you will use again with the launch chart in week 11.

READ THE FACTOR
  • Read the question.
  • Tap your answer.
The crew's seedling grows about 5% each week. By what factor is its height multiplied each week?
A quantity shrinks 20% each drop. By what factor is it multiplied each drop?
A quantity grows 10% each week. By what factor is it multiplied each week?
A quantity halves each step, shrinking 50%. By what factor is it multiplied each step?
HIDDEN RATES
  • Read the question.
  • Tap your answer.
Rewrite 4 to the t/2 as one factor to the power t. What is the factor?
Rewrite 1.21 to the t/2 as one factor to the power t. What is the factor?
Rewrite 9 to the t/2 as one factor to the power t. What is the factor?
WRITE THE RULE
  • Read the question.
  • Tap your answer.
A quantity starts at 50 and grows 20% each week. Which rule fits?
A quantity starts at 80 and loses a quarter of its value each step. Which rule fits?
Which rule is linear, not exponential?
StatementTrue or false?
A growth factor of 1.05 means a 5% increase each step.?
A growth factor of 0.8 means an 80% decrease each step.?
4^(1/2) = 2?
1.1 × 1.1 = 1.21?
An exponential function eventually passes any linear function.?
WHY THIS EXERCISEThese lines are the week in short. Next week the crew lays out cardboard tiles for the launch pad and meets polynomials.
Try it
Pick something in your home you can measure weekly, like a plant or a stack of recycling.
Measure it today and write two rules: one that adds a fixed amount each week and one that multiplies by a factor.
Next week, measure again and see which rule came closer.
Draw two curves from a start of 50: one with factor 1.2 and one with factor 0.75, over 5 steps.

Terrific week. You can tell adding from multiplying, read a rate off a rule, and turn roots into powers. Tomorrow is Loft Day.

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