Back to Exercise: Interpret exponential function properties

Exercises: Interpret Exponential Function Properties

Work through each section in order. For each exponential function, identify the initial value, classify as growth or decay, and determine the percent rate before interpreting in context.

Grade 9·19 problems·~30 min·Common Core Math - HS Functions·standard·hsf-if-c-8b

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Recall / Warm-Up

Simplify: $(1.05)^{12t}$ using the exponent rule $(a^m)^n = a^{mn}$ .

$((1.05)^{12})^t$

$(1.05)^{12} \cdot t$

$(1.05)^{t} + 12$

$12 \cdot (1.05)^t$

For $f(t) = 3000 \cdot (1.07)^t$ , what is the initial value (the value when $t = 0$ )?

$1.07$

$1$

$3000$

$0.07$

Which base value indicates exponential decay?

$b = 1.5$

$b = 1$

$b = 0.75$

$b = -0.5$

Fluency Practice

Identify the initial value, classify as growth or decay, and determine the percent rate for $y = 500 \cdot (0.88)^t$ .

Initial value: 500; decay; rate: 88% per period

Initial value: 500; decay; rate: 12% per period

Initial value: 0.88; growth; rate: 88% per period

Initial value: 500; growth; rate: 12% per period

For $f(t) = 2000 \cdot (1.035)^t$ , what is the annual percent growth rate? Enter as a percent (e.g., enter 3.5 for 3.5%).

Which exponential function represents the fastest growth?

$y = (1.05)^t$

$y = (0.92)^t$

$y = (1.18)^t$

$y = (1.10)^t$

A savings account earns 0.5% interest per month, modeled by $A(t) = P \cdot (1.005)^{12t}$ , where $t$ is in years. Rewrite as $A(t) = P \cdot b^t$ and find the annual growth factor $b$ . Round to 4 decimal places.

The function $y = 1000 \cdot (1.2)^{t/10}$ models growth over decades (each unit of $t$ is 10 years). Rewrite it in the form $y = 1000 \cdot b^t$ where $t$ is in years. Which is $b$ ?

$b = 1.2$

$b = (1.2)^{10} \approx 6.19$

$b = (1.2)^{1/10} \approx 1.0184$

$b = 0.2$

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