Let’s work on the problems provided in the image step-by-step.

The equation for the exponential function is:

$f(x) = b \cdot a^x$

To find $b$ and $a$, we substitute the given points into the equation and solve.

Problem 25: Points (0, 2) and (1, 5)

1. Substituting $(0, 2)$: $2 = b \cdot a^0 \implies b = 2$

2. Substituting $(1, 5)$: $5 = 2 \cdot a^1 \implies a = \frac{5}{2}$

Thus, the equation is: $f(x) = 2 \cdot \left(\frac{5}{2}\right)^x$

Problem 26: Points $(-2, 32)$ and $(2, 8)$

1. Substituting $(-2, 32)$: $32 = b \cdot a^{-2} \implies b = 32 \cdot a^2$

2. Substituting $(2, 8)$: $8 = b \cdot a^2$

   Substitute $b = 32 \cdot a^2$: $8 = (32 \cdot a^2) \cdot a^2 \implies 8 = 32 \cdot a^4$ $a^4 = \frac{8}{32} = \frac{1}{4} \implies a^2 = \frac{1}{2} \implies a = \frac{1}{\sqrt{2}}$

3. Substitute $a = \frac{1}{\sqrt{2}}$ into $b = 32 \cdot a^2$: $b = 32 \cdot \frac{1}{2} = 16$

Thus, the equation is: $f(x) = 16 \cdot \left(\frac{1}{\sqrt{2}}\right)^x$

Problem 27: Points $(\frac{1}{2}, 3)$ and $(1, 1)$

1. Substituting $(\frac{1}{2}, 3)$: $3 = b \cdot a^{1/2}$

2. Substituting $(1, 1)$: $1 = b \cdot a$

   From the second equation: $b = \frac{1}{a}$

3. Substitute $b = \frac{1}{a}$ into $3 = b \cdot a^{1/2}$: $3 = \frac{1}{a} \cdot a^{1/2} \implies 3 = a^{-1 + 1/2} = a^{-1/2}$ $a^{1/2} = \frac{1}{3} \implies a = \frac{1}{9}$

4. Substitute $a = \frac{1}{9}$ into $b = \frac{1}{a}$: $b = 9$

Thus, the equation is: $f(x) = 9 \cdot \left(\frac{1}{9}\right)^x$

Problem 28: Points $(-2, \frac{1}{2})$ and $(2, 2)$

1. Substituting $(-2, \frac{1}{2})$: $\frac{1}{2} = b \cdot a^{-2} \implies b = \frac{1}{2} \cdot a^2$

2. Substituting $(2, 2)$: $2 = b \cdot a^2$

   Substitute $b = \frac{1}{2} \cdot a^2$: $2 = \left(\frac{1}{2} \cdot a^2\right) \cdot a^2 \implies 2 = \frac{1}{2} \cdot a^4$ $a^4 = 4 \implies a = \sqrt{2}$

3. Substitute $a = \sqrt{2}$ into $b = \frac{1}{2} \cdot a^2$: $b = \frac{1}{2} \cdot 2 = 1$

Thus, the equation is: $f(x) = 1 \cdot (\sqrt{2})^x = (\sqrt{2})^x$

Let me know if you have questions or would like more details! Here are related questions:

1. How do transformations affect the graph of an exponential function?
2. What happens to an exponential function when $a > 1$ vs. $0 < a < 1$?
3. Can you derive exponential equations from data points in a real-world context?
4. How does the base $a$ influence the growth rate of the function?
5. What is the difference between solving exponential equations and logarithmic equations?

Tip: Always use logarithms if solving for the base or exponent becomes complex!