Let's analyze how the graph of the exponential function changes when the initial value, $a$, is decreased while remaining greater than zero.

The given function is:

$f(x) = a \cdot 5^x$

where $a = 2$.

If we decrease $a$ but keep it greater than zero, let's consider what each option suggests:

1. The graph will increase at a slower rate.

   This statement is incorrect. The rate at which the graph increases is determined by the base of the exponent, which is 5 in this case. The value of $a$ affects only the starting point (the y-intercept) but does not affect the rate of increase.

2. The graph will show a decreasing, rather than increasing, function.

   This statement is incorrect. The function remains exponential with a positive base (5), so it continues to increase. The decrease in $a$ only affects the initial value, not the direction of the function.

3. The graph will show an initial value that is lower on the $y$-axis.

   This statement is correct. Decreasing the value of $a$ means that the y-intercept of the graph (where $x = 0$) is lower. The function will start at a smaller $y$-value when $x = 0$.

4. The graph will increase at a constant additive rate, rather than a multiplicative rate.

   This statement is incorrect. Exponential functions always increase at a multiplicative rate defined by the base (5 in this case). Changing $a$ does not alter this fundamental characteristic.

Thus, the correct change when the $a$ value is decreased, while still positive, is that the graph will show an initial value that is lower on the $y$-axis.

Summary:

• Initial Value Decrease: The initial value (the y-intercept) is lower.
• Graph Behavior: The function remains an increasing exponential function with the same rate of change (determined by the base 5).

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Do you have any other questions or need further clarification? Here are some related topics you might be interested in:

1. How does changing the base of an exponential function affect its graph?
2. What is the effect of a negative initial value on an exponential function's graph?
3. How do horizontal shifts affect the graph of an exponential function?
4. Can exponential functions have a zero initial value?
5. What is the difference between exponential and linear growth?
6. How does changing the sign of the exponent affect an exponential function?
7. How do you identify the y-intercept of an exponential graph visually?
8. What happens if the base of the exponent is less than 1?

Tip: In exponential functions, the initial value $a$ sets the y-intercept, while the base $b$ determines how quickly the function grows or decays.