Algebra 2 practice worksheet with exponential functions and compound interest problems.

Here are the solutions to the problems on the worksheet, broken down by section.

Simplify the expression

*Rule: When multiplying terms with the same base (like $e$), you add their exponents.*

1. $e^{-5} \cdot e^2$
Add exponents: $-5 + 2 = -3$.
Answer: $e^{-3}$

2. $e^3 \cdot e^{-3}$
Add exponents: $3 + (-3) = 0$. Since $e^0 = 1$.
Answer: $1$

3. $(e^4)^{-3}$
Multiply exponents: $4 \cdot -3 = -12$.
Answer: $e^{-12}$

4. $(2e^3)^2$
Square the number and multiply the exponent by 2: $2^2 = 4$ and $3 \cdot 2 = 6$.
Answer: $4e^6$

5. $\frac{(3e^3)^2}{(6e^2)}$
Simplify top: $(3e^3)^2 = 9e^6$.
Divide: $\frac{9e^6}{6e^2}$. Numbers: $\frac{9}{6} = 1.5$. Exponents: $6 - 2 = 4$.
Answer: $1.5e^4$ (or $\frac{3}{2}e^4$)

6. $\frac{8e^{2x-1}}{2e^5}$
Divide numbers: $\frac{8}{2} = 4$.
Subtract exponents: $(2x - 1) - 5 = 2x - 6$.
Answer: $4e^{2x-6}$

7. $3e^x \cdot 2e^{4x}$
Multiply numbers: $3 \cdot 2 = 6$.
Add exponents: $x + 4x = 5x$.
Answer: $6e^{5x}$

8. $\sqrt{9e^4} \cdot 2e^{-3}$
Square root of $9e^4$ is $3e^2$.
Multiply: $3e^2 \cdot 2e^{-3}$. Numbers: $3 \cdot 2 = 6$. Exponents: $2 + (-3) = -1$.
Answer: $6e^{-1}$ (or $\frac{6}{e}$)

9. $\frac{e^3}{e^{x+3}}$
Subtract exponents: $3 - (x + 3) = 3 - x - 3 = -x$.
Answer: $e^{-x}$

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Use a calculator to evaluate the expression

*(Rounded to three decimal places)*

10. $e^7$
Calculation: $1096.6331...$
Answer: $1096.633$

11. $e^{-3/2}$
Calculation: $e^{-1.5} \approx 0.2231...$
Answer: $0.223$

12. $e^{0.6}$
Calculation: $1.8221...$
Answer: $1.822$

13. $e^{\sqrt{3}}$
Calculation: $\sqrt{3} \approx 1.732$. $e^{1.732} \approx 5.6522...$
Answer: $5.652$

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Tell whether the function is exponential growth or decay

*Rule: Look at the exponent. If it is positive ($kx$), it's growth. If it is negative ($-kx$), it's decay.*

14. $f(x) = 4e^{2x}$ $\rightarrow$ Positive exponent $\rightarrow$ Growth
15. $f(x) = e^{-5x}$ $\rightarrow$ Negative exponent $\rightarrow$ Decay
16. $f(x) = 6e^{-x}$ $\rightarrow$ Negative exponent $\rightarrow$ Decay
17. $f(x) = \frac{1}{4}e^{4x}$ $\rightarrow$ Positive exponent $\rightarrow$ Growth
18. $f(x) = \frac{1}{8}e^{-x}$ $\rightarrow$ Negative exponent $\rightarrow$ Decay
19. $f(x) = -e^{-x/2}$ $\rightarrow$ Negative exponent $\rightarrow$ Decay

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Graph the function. State the domain and range.

*Note: For all these graphs, the Domain is always All Real Numbers $(-\infty, \infty)$.*

20. $f(x) = 3e^x$
* Domain: All real numbers
* Range: $y > 0$
* Graph: Passes through $(0, 3)$. Goes up very fast to the right. Approaches the x-axis on the left.

21. $f(x) = 3e^{-x}$
* Domain: All real numbers
* Range: $y > 0$
* Graph: Passes through $(0, 3)$. Goes down towards the x-axis on the right. Goes up very high on the left.

22. $f(x) = -e^x + 3$
* Domain: All real numbers
* Range: $y < 3$
* Graph: Flipped upside down. Horizontal line (asymptote) at $y=3$. Y-intercept at $(0, 2)$. Curve goes down to the right.

23. $f(x) = 2e^{x-1} + 1$
* Domain: All real numbers
* Range: $y > 1$
* Graph: Shifted right 1 and up 1. Asymptote at $y=1$. Passes through $(1, 3)$.

24. $f(x) = \frac{1}{2}e^{x-2} - 3$
* Domain: All real numbers
* Range: $y > -3$
* Graph: Shifted right 2 and down 3. Asymptote at $y=-3$. Passes through $(2, -2.5)$.

25. $f(x) = e^{2x+1} + 2$
* Domain: All real numbers
* Range: $y > 2$
* Graph: Shifted left 0.5 and up 2. Asymptote at $y=2$. Steep growth. Passes through $(0, e+2 \approx 4.7)$.

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Finance Word Problems

*Formula for Compound Interest:* $A = P(1 + \frac{r}{n})^{nt}$
* $P = 2200$ (Principal)
* $r = 0.03$ (3% interest rate)
* $t = 15$ (Years)

26. Compounded Quarterly ($n=4$)
$$A = 2200(1 + \frac{0.03}{4})^{4 \cdot 15}$$
$$A = 2200(1.0075)^{60}$$
$$A \approx 2200(1.56568)$$
$$A \approx 3444.50$$
Answer: $3,444.50

27. Compounded Continuously
*Formula:* $A = Pe^{rt}$
$$A = 2200 \cdot e^{0.03 \cdot 15}$$
$$A = 2200 \cdot e^{0.45}$$
$$A \approx 2200 \cdot 1.56831$$
$$A \approx 3450.29$$
Answer: $3,450.29