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 (1)
Vertical stretch occurs when the absolute value of 'a' is greater than 1.
Check the absolute value of each given option: 0.3, 0.9, 1.0, and 1.5.
Only 1.5 has an absolute value greater than 1.
Therefore, the value of a that causes a vertical stretch is 1.5 .
Explanation
Understanding Vertical Stretch We are given the exponential function f ( x ) = a ( 3 1 ) x and asked to find which value of a would cause the function to stretch. A vertical stretch occurs when 1"> ∣ a ∣ > 1 . We need to check each of the given values of a to see if its absolute value is greater than 1.
Checking Each Value The possible values for a are 0.3, 0.9, 1.0, and 1.5. Let's examine each one:
If a = 0.3 , then ∣ a ∣ = ∣0.3∣ = 0.3 , which is not greater than 1.
If a = 0.9 , then ∣ a ∣ = ∣0.9∣ = 0.9 , which is not greater than 1.
If a = 1.0 , then ∣ a ∣ = ∣1.0∣ = 1.0 , which is not greater than 1 (it is equal to 1, which corresponds to no stretch or compression).
If a = 1.5 , then ∣ a ∣ = ∣1.5∣ = 1.5 , which is greater than 1.
Conclusion Therefore, the value of a that would cause the function to stretch is 1.5.
Examples
In real life, understanding stretches and compressions of functions is crucial in fields like signal processing and image manipulation. For instance, if you're adjusting the contrast of an image, you're essentially stretching or compressing the intensity values of the pixels. Similarly, in audio engineering, you might stretch or compress the amplitude of a sound wave to increase its loudness or dynamic range. The value 'a' in our function acts like a volume knob, amplifying the output when greater than 1 and reducing it when less than 1, providing a practical way to manipulate data.
