the graph of an exponential function, y = a x b^x, passes though the points (1,1) and (2,3.5). Find the value of a?

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.

Learn how to find the value of 'a' in the exponential function y = a * b^x, given specific points (1, 1) and (2, 3.5). This detailed step-by-step solution involves substitution and algebraic manipulation to determine 'a'. Suitable for Grades 10-12.

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