Exponential Function Context and Data Modeling - AP Precalculus

1. State the formula for continuous exponential growth.

   $A = P e^{rt}$. Uses natural base $e$ for continuous compounding.

2. What is the doubling time formula in exponential growth?

   $T_d = \frac{\text{ln}(2)}{\text{ln}(b)}$. Time required for quantity to double in size.

3. Define an exponential decay function.

   $f(x) = a \times b^x$, $0 < b < 1$. Decay occurs when base is between 0 and 1.

4. Calculate decay rate if $b = 0.9$ in $f(x) = a \times b^x$.

   10%. Decay rate is $(1 - 0.9) = 0.1 = 10\%$.

5. Determine the effect of $f(x) = a \times (b^x) + c$ when $c > 0$.

   Vertical shift up by $c$. Positive $c$ shifts the entire graph upward.

6. For $f(x) = 6 \times 0.5^x$, what is the decay factor?

   0.5. The base represents the decay factor.

7. Which parameter affects horizontal shifts in $f(x) = a \times b^{x+c}$?

   $c$. Parameter $c$ controls horizontal translation.

8. What is the inverse of an exponential function?

   Logarithmic function. Exponential and logarithmic functions are inverses.

9. Is $f(x) = 1 \times (0.5)^x$ growth or decay?

   Decay. Base 0.5 is less than 1, indicating decay.

10. For $f(x) = 9 \times 1.2^x$, what is the growth rate?

    20%. Growth rate is $(b - 1) \times 100\%$.

11. What is the effect of a negative exponent in $f(x) = a \times b^{-x}$?

    Reflect across y-axis. Negative exponent reverses the function horizontally.

12. Calculate the future value $A$ for $P = 1000, r = 0.08, t = 2$.

    $A = 1000 e^{0.16}$. $A = 1000e^{0.08 \times 2} = 1000e^{0.16}$.

13. State the half-life formula for exponential decay.

    $N(t) = N_0 \times (\frac{1}{2})^{\frac{t}{h}}$. Models quantity halving every $h$ time units.

14. Determine the growth factor for $f(x) = 3 \times 1.05^x$.

    1.05. Growth factor is the base when $b > 1$.

15. Calculate $f(3)$ for $f(x) = 4 \times (0.5)^x$.

    0.5. $f(3) = 4 \times (0.5)^3 = 4 \times 0.125 = 0.5$.

16. Identify the range of $f(x) = 8 \times (1.5)^x$.

    $(0, \text{inf})$. All exponential functions have range $(0, \infty)$.

17. Identify the base in the exponential function $f(x) = 5 \times 2^x$.

    2. The base is the number being raised to the power.

18. State the domain of any exponential function $f(x) = a \times b^x$.

    All real numbers. Exponential functions accept all real number inputs.

19. State the natural base $e$ approximately.

    2.718. Euler's number, base of natural logarithm.

20. What is the horizontal asymptote of $f(x) = 4 \times (0.75)^x$?

    $y = 0$. Exponential functions approach but never reach zero.

21. What property does the base $b$ have in an exponential function $f(x) = a \times b^x$?

    $b > 0$ and $b \neq 1$. Base must be positive and not equal to 1.

22. For $f(x) = 5 \times (0.8)^x$, what is the percentage decay rate?

    20%. Decay rate is $(1 - b) \times 100\%$.

23. What is the formula for finding the rate $r$ in $A = P e^{rt}$?

    $r = \frac{1}{t} \text{ln}(\frac{A}{P})$. Solved from $A = Pe^{rt}$ for rate $r$.

24. What transformation occurs when $f(x) = a \times b^{x+c}$?

    Horizontal shift left by $c$. Adding $c$ to exponent shifts graph leftward.

25. Determine the y-intercept of $f(x) = 2 \times 3^x$.

    2. Y-intercept occurs when $x = 0$.

26. What transformation is applied in $f(x) = a \times b^{x} + d$?

    Vertical shift by $d$. Adding $d$ moves the graph up or down.

27. Find the value of $f(2)$ for $f(x) = 2 \times 5^x$.

    50. $f(2) = 2 \times 5^2 = 2 \times 25 = 50$.

28. Identify the effect of $f(x) = -a \times b^x$ on the graph.

    Reflection over x-axis. Negative coefficient flips graph over x-axis.

29. What is the general form of an exponential function?

    $f(x) = a \times b^x$. Standard form with initial value $a$ and base $b$.

30. What is the value of $f(0)$ for $f(x) = a \times b^x$?

    $a$. When $x = 0$, $b^0 = 1$, so $f(0) = a$.

31. Find the initial value of the function $f(x) = 3 \times 4^x$.

    3. Initial value is coefficient $a$ when $x = 0$.

32. What does the variable $a$ represent in $f(x) = a \times b^x$?

    Initial value. The coefficient $a$ is the starting amount.

33. What does the parameter $r$ represent in $A = P e^{rt}$?

    Rate of growth or decay. The exponent controls growth or decay rate.

34. What is the exponential growth factor if $b = 1.1$?

    1.1. Growth factor is the base $b$ when $b > 1$.

35. What is the range for an exponential decay function?

    $(0, a)$ if $a > 0$. Decay functions approach zero from above.

36. What is the effect of $a$ in $f(x) = a \times b^x$ on the graph?

    Vertical stretch/compression. Parameter $a$ scales the function vertically.

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