Which value of a in the exponential function below would cause the function to stretch?
[tex]f(x)=a(\frac{1}{3})^x[/tex]
A. 0.3
B. 0.9
C. 1.0
D. 1.5
Answer (2)
A vertical stretch occurs when the absolute value of 'a' is greater than 1.
Check the absolute value of each option.
∣0.3∣ = 0.3 < 1 (compression)
∣0.9∣ = 0.9 < 1 (compression)
∣1.0∣ = 1 (no stretch or compression)
1"> ∣1.5∣ = 1.5 > 1 (stretch)
The value of a that causes a stretch is 1.5 .
Explanation
Understanding the Problem The problem asks us to identify which value of 'a' in the function f ( x ) = a ( 3 1 ) x would cause the function to stretch. A vertical stretch occurs when the absolute value of 'a' is greater than 1. We need to check each of the given options to see which one satisfies this condition.
Analyzing Each Option Let's analyze each option:
0.3: The absolute value of 0.3 is ∣0.3∣ = 0.3 , which is less than 1. So, this would cause a compression, not a stretch.
0.9: The absolute value of 0.9 is ∣0.9∣ = 0.9 , which is less than 1. So, this would also cause a compression.
1.0: The absolute value of 1.0 is ∣1.0∣ = 1.0 , which is equal to 1. This value of 'a' would neither stretch nor compress the function; it would simply be the standard exponential function ( 3 1 ) x .
1.5: The absolute value of 1.5 is ∣1.5∣ = 1.5 , which is greater than 1. This value of 'a' would cause a vertical stretch.
Conclusion Therefore, the value of 'a' that would cause the function to stretch is 1.5.
Examples
Imagine you are adjusting the volume on a sound equalizer. The 'a' value in our function is like the volume control for a specific frequency. If 'a' is greater than 1, you are amplifying that frequency (stretching it). If 'a' is less than 1, you are reducing that frequency (compressing it). Understanding how 'a' affects the function helps you fine-tune the sound to your liking.
The value of 'a' that causes the function f ( x ) = a ( 3 1 ) x to stretch is 1.5, as its absolute value is greater than 1. Therefore, the correct answer is option D. This results in a vertical stretch of the graph of the function.
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