To find the value of $a$ in the exponential function $y = a \cdot b^x$, given that the graph passes through the points $(1, 1)$ and $(2, 3.5)$, we'll substitute the coordinates into the equation and solve for $a$ and $b$.

Step 1: Substituting the first point $(1, 1)$

$1 = a \cdot b^1 \quad \text{(1)}$ $a \cdot b = 1 \quad \text{(2)}$

Step 2: Substituting the second point $(2, 3.5)$

$3.5 = a \cdot b^2 \quad \text{(3)}$

Step 3: Solving for $b$

From equation (2), we have: $a = \frac{1}{b}$

Substituting this into equation (3): $3.5 = \left(\frac{1}{b}\right) \cdot b^2$ $3.5 = b$

Step 4: Solving for $a$

Now that we know $b = 3.5$, substitute this back into equation (2): $a \cdot 3.5 = 1$ $a = \frac{1}{3.5}$ $a = \frac{2}{7}$

Conclusion

The value of $a$ is $\frac{2}{7}$.

Would you like further details on the steps or have any questions?

Here are some related questions:

1. How does changing $b$ affect the graph of the function?
2. What would happen if $a$ were negative in the equation $y = a \cdot b^x$?
3. Can you find the exact value of $y$ when $x = 0$?
4. How would the graph look if the function was $y = a \cdot b^{-x}$?
5. What is the significance of the base $b$ in an exponential function?

Tip: When solving exponential equations, identifying key points and using substitution can simplify the process of finding unknown parameters.