Question
What is the effect on the graph of f(x)=1x
when it is replaced by t(x)=−56x
?
Option #1: It vertically compresses by a factor of 56
.
Option #2: It reflects across the y-axis and vertically stretches by a factor of 65
.
Option #3: It horizontally stretches by a factor of 65
.
Option #4: It reflects across the x-axis and vertically compresses by a factor of 56
.
when it is replaced by t(x)=−56x
?
Option #1: It vertically compresses by a factor of 56
.
Option #2: It reflects across the y-axis and vertically stretches by a factor of 65
.
Option #3: It horizontally stretches by a factor of 65
.
Option #4: It reflects across the x-axis and vertically compresses by a factor of 56
.
Answers
GPT-4o mini
To analyze the effect of replacing \( f(x) = 1x \) (which is just \( f(x) = x \)) with \( t(x) = -56x \), we can break down the transformations involved.
1. The negative sign in \( -56x \) indicates a reflection across the x-axis. This is because every output value of the function \( f(x) = x \) is negated in \( t(x) = -56x \).
2. The coefficient of \( -56 \) means that the function is stretched vertically. The magnitude of the stretch is 56, which implies that for every value of \( x \), the output is 56 times further from the x-axis (in the opposite direction due to the negative sign).
Putting this together:
- The function reflects across the x-axis.
- The function is vertically stretched by a factor of 56.
Therefore, the correct interpretation is:
**Option #4: It reflects across the x-axis and vertically compresses by a factor of 56.**
However, it is important to note that the description in Option #4 has an inconsistency. The factor of stretching actually indicates a vertical stretch by 56, not a compression. If the positioning of the options is open to interpretation regarding whether reflect or compression was the intended term, we see that the closest answer to the transformations derived would be:
**The function reflects across the x-axis and vertically stretches by a factor of 56.**
But since this specific wording isn't an option, we can reason based on choices, and the understanding of vertical transformations lead to an assumption of compression terminology that might have got mixed with descriptions, leading us to point to:
**Select the correct choice: It reflects across the x-axis and could match the expected behavior of describing effect as related to 56 - keeping in mind that in context to expressions, it's about how function scaling is described across transformations.**
1. The negative sign in \( -56x \) indicates a reflection across the x-axis. This is because every output value of the function \( f(x) = x \) is negated in \( t(x) = -56x \).
2. The coefficient of \( -56 \) means that the function is stretched vertically. The magnitude of the stretch is 56, which implies that for every value of \( x \), the output is 56 times further from the x-axis (in the opposite direction due to the negative sign).
Putting this together:
- The function reflects across the x-axis.
- The function is vertically stretched by a factor of 56.
Therefore, the correct interpretation is:
**Option #4: It reflects across the x-axis and vertically compresses by a factor of 56.**
However, it is important to note that the description in Option #4 has an inconsistency. The factor of stretching actually indicates a vertical stretch by 56, not a compression. If the positioning of the options is open to interpretation regarding whether reflect or compression was the intended term, we see that the closest answer to the transformations derived would be:
**The function reflects across the x-axis and vertically stretches by a factor of 56.**
But since this specific wording isn't an option, we can reason based on choices, and the understanding of vertical transformations lead to an assumption of compression terminology that might have got mixed with descriptions, leading us to point to:
**Select the correct choice: It reflects across the x-axis and could match the expected behavior of describing effect as related to 56 - keeping in mind that in context to expressions, it's about how function scaling is described across transformations.**