Exponential Functions - AP Precalculus

1. What is the initial value in $f(x) = 15 \times 1.3^x$?

   $15$. The coefficient 15 is the value when $x = 0$.

2. What is the y-intercept of $f(x) = 1.5 \times 10^x$?

   $1.5$. Y-intercept occurs when $x = 0$: $f(0) = 1.5 \times 10^0 = 1.5$.

3. Find the reflection of $f(x) = 2^x$ across the x-axis.

   $f(x) = -2^x$. Multiplying by -1 reflects the function across the x-axis.

4. Identify the transformation: $f(x) = 2 \times 3^{x-1}$.

   Right shift by 1 unit. The $(x-1)$ in the exponent shifts the graph right by 1.

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

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

6. Determine the asymptote of $f(x) = 100 \times (0.99)^x$.

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

7. What is the effect of a negative exponent in $f(x) = 3 \times b^{-x}$?

   Reflection across y-axis. Negative exponent transforms $b^{-x}$ to $(1/b)^x$, reflecting over y-axis.

8. What is the decay rate in $f(x) = 12 \times (0.8)^x$?

   $0.2$ or 20\text{%}. Since $0.8 = 1 - 0.2$, the decay rate is $0.2$.

9. If $f(x) = 5 \times (1.2)^x$, is this function growth or decay?

   Growth. Since $1.2 > 1$, the function increases as $x$ increases.

10. What happens to $f(x) = a \times b^x$ as $x \to -\text{∞}$ when $0 < b < 1$?

    $f(x) \to \text{∞}$. When $0 < b < 1$ and $x \to -\infty$, $b^x \to \infty$.

11. What is the value of $f(0)$ for $f(x) = 7 \times 3^x$?

    $7$. Substitute $x = 0$: $f(0) = 7 \times 3^0 = 7 \times 1 = 7$.

12. Determine if $f(x) = 8 \times (1.01)^x$ models growth or decay.

    Growth. Since $1.01 > 1$, the function shows exponential growth.

13. What is $f(x) = 0$ for $f(x) = a \times b^x$?

    Never. Exponential functions have horizontal asymptote at $y = 0$, never equal 0.

14. Identify the horizontal shift in $f(x) = 5 \times 2^{x+3}$.

    Left shift by 3 units. The $(x+3)$ in the exponent shifts the graph left by 3.

15. Identify the growth rate in $f(x) = 10 \times (1.05)^x$.

    $0.05$ or 5\text{%}. From $(1 + r) = 1.05$, so $r = 0.05$.

16. If $b = 1$, what type of function is $f(x) = a \times b^x$?

    Constant function. When $b = 1$, $f(x) = a \times 1^x = a$ for all $x$.

17. State the exponential growth formula.

    $f(x) = a \times (1 + r)^t$. Where $a$ is initial value, $r$ is growth rate, $t$ is time.

18. Identify the base in $f(x) = 6 \times e^{0.5x}$.

    $e$. The base is the constant $e$ being raised to the power.

19. State the exponential decay formula.

    $f(x) = a \times (1 - r)^t$. Where $a$ is initial value, $r$ is decay rate, $t$ is time.

20. For $f(x) = 7 \times 1.2^x$, identify the initial value.

    $7$. The coefficient 7 is the initial value when $x = 0$.

21. What does $0 < b < 1$ indicate in an exponential function $f(x) = a \times b^x$?

    Exponential decay. Base between 0 and 1 means output decreases as $x$ increases.

22. Find $f(2)$ for $f(x) = 4 \times (0.5)^x$.

    $1$. Substitute $x = 2$: $f(2) = 4 \times (0.5)^2 = 4 \times 0.25 = 1$.

23. Express $f(x) = 3 \times (0.9)^x$ in terms of decay rate.

    Decay rate is $0.1$ or 10\text{%}. Since $0.9 = 1 - 0.1$, the decay rate is $0.1$.

24. Identify the base in the exponential function $f(x) = 3 \times 2^x$.

    $2$. The base is the constant being raised to the power $x$.

25. Find the base for the function $f(x) = 3 \times 6^x$.

    $6$. The base is the constant 6 being raised to the power $x$.

26. What does $b > 1$ indicate in an exponential function $f(x) = a \times b^x$?

    Exponential growth. Base greater than 1 means output increases as $x$ increases.

27. What is the y-intercept of the function $f(x) = 4 \times 3^x$?

    $4$. Y-intercept occurs when $x = 0$: $f(0) = 4 \times 3^0 = 4$.

28. Find $f(1)$ for $f(x) = 7 \times (1.1)^x$.

    $7.7$. Substitute $x = 1$: $f(1) = 7 \times 1.1^1 = 7 \times 1.1 = 7.7$.

29. Determine the effect of $f(x) = 4 \times 2^{-x}$ on the graph.

    Reflection over y-axis. The negative exponent $2^{-x}$ reflects the graph over y-axis.

30. Identify the vertical stretch in $f(x) = 10 \times 2^x$.

    Factor of 10. The coefficient 10 stretches the graph vertically by factor 10.

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

    Initial value or y-intercept. When $x = 0$, $f(0) = a \times b^0 = a$.

32. What is the horizontal asymptote of $f(x) = 5 \times 0.8^x$?

    $y = 0$. Exponential decay functions approach 0 as $x$ approaches infinity.

33. Calculate $f(-1)$ for $f(x) = 3 \times 4^x$.

    $\frac{3}{4}$. Substitute $x = -1$: $f(-1) = 3 \times 4^{-1} = 3 \times \frac{1}{4} = \frac{3}{4}$.

34. Determine if $f(x) = 2 \times (0.7)^x$ shows growth or decay.

    Decay. Since $0.7 < 1$, the function decreases as $x$ increases.

35. Find $f(3)$ for $f(x) = 2 \times 5^x$.

    $250$. Substitute $x = 3$: $f(3) = 2 \times 5^3 = 2 \times 125 = 250$.

36. Find the decay factor for $f(x) = 9 \times (0.75)^x$.

    $0.75$. The decay factor is the base 0.75 in the exponential function.

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