Exponential functions

High School Math 1 · Unit 10 · exponential-growth-decay · Teacher edition

exponential-growth-decay

Name ______________________________Date ______________

1.
Let $f(x) = 800(0.5)^{x}$ . Find $f(3)$ .
Answer: ______________
2.
A car bought for $24,000 loses $15\%$ of its value each year. What is its value after $2$ years, in dollars?
Answer: ______________
3.
A table gives $x = 0, 1, 2, 3$ and $y = 3, 6, 12, 24$ . Which function matches the table?
1. (A)
  $y = 2 \cdot 3^{x}$
2. (B)
  $y = 3x + 3$
3. (C)
  $y = 3 \cdot 2^{x}$
4. (D)
  $y = 6^{x}$
4.
Which situation is modelled by an exponential function?
1. (A)
  A bacteria count doubles every hour
2. (B)
  A plant grows $2$ cm each week
3. (C)
  A taxi charges $2 per mile
4. (D)
  A savings jar gains $2 every month
5.
$2,000 is deposited in an account paying $6\%$ annual interest, compounded semi-annually. What is the balance after $1$ year, in dollars, to the nearest cent?
Answer: ______________

Answer key — Exponential functions

1.
100exponential-growth-decay-01
$f(3) = 800 \cdot \frac{1}{8} = \mathbf{100}$ . Each step halves: $800 \to 400 \to 200 \to 100$ .
2.
17340exponential-growth-decay-02
$V(2) = 24000 \cdot 0.85^2 = 24000 \cdot 0.7225 = \mathbf{17340}$ dollars. (Subtracting $15\%$ of the original twice, $24000 - 2 \cdot 3600 = 16800$ , is linear decay, not exponential.)
3.
(C)
$y = 3 \cdot 2^{x}$
exponential-growth-decay-03
Start value $3$ , ratio $2$ : $\mathbf{y = 3 \cdot 2^{x}}$ . Check $x = 3$ : $3 \cdot 8 = 24$ . ✓ ( $y = 2 \cdot 3^x$ swaps $a$ and $b$ — it gives $2$ at $x = 0$ ; $y = 3x + 3$ adds $3$ each step instead of doubling; $6^x$ gives $1$ at $x = 0$ .)
4.
(A)
A bacteria count doubles every hour
exponential-growth-decay-04
A bacteria count doubles every hour multiplies by a constant factor $2$ , so it is exponential: $N(t) = N_0 \cdot 2^t$ . (The plant, taxi, and jar each add a constant $2$ per step — linear, even though the number $2$ appears.)
5.
2121.8exponential-growth-decay-05
$A = 2000(1.03)^2 = 2000 \cdot 1.0609 = \mathbf{2121.80}$ dollars. (Simple $6\%$ for one year would give $2,120; compounding twice earns the extra $1.80.)