Question
29
Which of the following exponential equations shows a horizontal stretch of the function y= 3^x
?
A
y=5⋅3xy=5\cdot3^xy=5⋅3
x
B
y=35xy=3^{5x}y=3
5x
C
y=315xy=3^{\frac{1}{5}x}y=3
5
1
x
D
y=15⋅3xy=\frac{1}{5}\cdot3^xy=
5
1
⋅3
x
Which of the following exponential equations shows a horizontal stretch of the function y= 3^x
?
A
y=5⋅3xy=5\cdot3^xy=5⋅3
x
B
y=35xy=3^{5x}y=3
5x
C
y=315xy=3^{\frac{1}{5}x}y=3
5
1
x
D
y=15⋅3xy=\frac{1}{5}\cdot3^xy=
5
1
⋅3
x
Answers
GPT-4o mini
To determine which of the given exponential equations shows a horizontal stretch of the function \( y = 3^x \), we need to look for a modification in the exponent that will affect the rate of growth of the function.
A horizontal stretch occurs when the exponent is multiplied by a fraction less than 1. If we have an exponential function of the form \( y = a^{bx} \), then if \( b < 1 \), there is a horizontal stretch.
Now, let's examine the options provided:
A. \( y = 5 \cdot 3^x \)
This does not modify the exponent, so it does not represent a horizontal stretch.
B. \( y = 3^{5x} \)
In this case, the exponent is \( 5x \) which represents a vertical compression (or faster growth), thus no horizontal stretch.
C. \( y = 3^{\frac{1}{5}x} \)
The exponent is \( \frac{1}{5}x \), which means it is stretched horizontally by a factor of 5 (since it’s multiplied by \( \frac{1}{5} \), which is less than 1). Therefore, this represents a horizontal stretch.
D. \( y = \frac{1}{5} \cdot 3^x \)
Similar to option A, this again does not modify the exponent and thus does not represent a horizontal stretch.
Based on this analysis, the correct answer is:
**C. \( y = 3^{\frac{1}{5}x} \)**
A horizontal stretch occurs when the exponent is multiplied by a fraction less than 1. If we have an exponential function of the form \( y = a^{bx} \), then if \( b < 1 \), there is a horizontal stretch.
Now, let's examine the options provided:
A. \( y = 5 \cdot 3^x \)
This does not modify the exponent, so it does not represent a horizontal stretch.
B. \( y = 3^{5x} \)
In this case, the exponent is \( 5x \) which represents a vertical compression (or faster growth), thus no horizontal stretch.
C. \( y = 3^{\frac{1}{5}x} \)
The exponent is \( \frac{1}{5}x \), which means it is stretched horizontally by a factor of 5 (since it’s multiplied by \( \frac{1}{5} \), which is less than 1). Therefore, this represents a horizontal stretch.
D. \( y = \frac{1}{5} \cdot 3^x \)
Similar to option A, this again does not modify the exponent and thus does not represent a horizontal stretch.
Based on this analysis, the correct answer is:
**C. \( y = 3^{\frac{1}{5}x} \)**