To determine which value of [tex]\( a \)[/tex] in the given exponential function [tex]\( f(x) = a \left(\frac{1}{3}\right)^x \)[/tex] causes the function to stretch, we need to understand the effect of the coefficient [tex]\( a \)[/tex] on the function's behavior.
A function "stretches" vertically if the coefficient [tex]\( a \)[/tex] is greater than 1. This means that for a given input [tex]\( x \)[/tex], the output [tex]\( f(x) \)[/tex] is multiplied by a factor greater than 1, causing the graph of the function to elongate vertically.
Let's analyze the given values of [tex]\( a \)[/tex]:
1. [tex]\( a = 0.3 \)[/tex]:
[tex]\[
f(x) = 0.3 \left(\frac{1}{3}\right)^x
\][/tex]
Here, [tex]\( a \)[/tex] is less than 1. This causes the graph to shrink rather than stretch.
2. [tex]\( a = 0.9 \)[/tex]:
[tex]\[
f(x) = 0.9 \left(\frac{1}{3}\right)^x
\][/tex]
In this case, [tex]\( a \)[/tex] is still less than 1. Thus, the graph will shrink.
3. [tex]\( a = 1.0 \)[/tex]:
[tex]\[
f(x) = 1.0 \left(\frac{1}{3}\right)^x
\][/tex]
When [tex]\( a \)[/tex] equals 1, there is no vertical transformation. The function remains as is, with no stretching or shrinking effect.
4. [tex]\( a = 1.5 \)[/tex]:
[tex]\[
f(x) = 1.5 \left(\frac{1}{3}\right)^x
\][/tex]
Since [tex]\( a \)[/tex] is greater than 1, this will cause a vertical stretch of the graph of the function by a factor of 1.5. Each point on the graph is pulled farther away from the x-axis.
Given these points, the value of [tex]\( a \)[/tex] that causes the function to stretch is:
[tex]\[
\boxed{1.5}
\][/tex]
