Question
How do the graphs of y=1x and y=3x−4 compare?(1 point) Responses When compared to the graph of y=1x , the graph of y=3x−4 is a vertical stretch by a factor of 3 and a translation of 4 units left. When compared to the graph of y is equal to 1 over x, the graph of y is equal to 3 over x minus 4 is a vertical stretch by a factor of 3 and a translation of 4 units left. When compared to the graph of y=1x , the graph of y=3x−4 is a vertical shrink by a factor of 3 and a translation of 4 units up. When compared to the graph of y is equal to 1 over x, the graph of y is equal to 3 over x minus 4 is a vertical shrink by a factor of 3 and a translation of 4 units up. When compared to the graph of y=1x , the graph of y=3x−4 is a vertical stretch by a factor of 3 and a translation of 4 units down. When compared to the graph of y is equal to 1 over x, the graph of y is equal to 3 over x minus 4 is a vertical stretch by a factor of 3 and a translation of 4 units down. When compared to the graph of y=1x , the graph of y=3x−4 is a vertical shrink by a factor of 3 and a translation of 4 units down.
Answers
Answered by
GPT-4o mini
To analyze the comparison of the graphs of \( y = \frac{1}{x} \) and \( y = 3x - 4 \), we first need to interpret what each equation represents.
1. \( y = \frac{1}{x} \) is a hyperbola. Its graph approaches the x-axis and y-axis but never touches them, consisting of two branches located in the first and third quadrants.
2. \( y = 3x - 4 \) is a linear equation. Its graph is a straight line with a slope of 3 and a y-intercept of -4.
Now, let's compare the transformations between these two functions:
### Transformations:
- **Vertical Stretch/Shrink**: This concept generally applies to functions like \( y = ax \) where changing \( a \) alters the steepness of the line. In the case of a hyperbola like \( y = \frac{1}{x} \), there is no vertical stretch or shrink involved because it does not transform into a linear graph with a vertical scaling factor.
- **Translation**: A translation shifts the graph up, down, left, or right without changing its shape. The linear function \( y = 3x - 4 \) moves the hyperbola downwards because of the subtraction involved in the transformation, but it does not translate the hyperbola left or right.
### Conclusion:
The comparison of these two functions does not fit into the described transformations like vertical stretch or translations as suggested in the initial statement. For \( y = \frac{1}{x} \) to reach a form similar to \( y = 3x - 4 \) would include both transformations which aren't applicable.
None of the given responses accurately describes the relationship, as:
- \( y = 3x - 4 \) is not a transformation of \( y = \frac{1}{x} \) through vertical stretches or horizontal translations.
### Final Answer:
The graphs do not transform into one another through the operations outlined in your options, and comparing a hyperbolic graph to a linear graph involves different characteristics that are not directly relatable through the definitions provided.
1. \( y = \frac{1}{x} \) is a hyperbola. Its graph approaches the x-axis and y-axis but never touches them, consisting of two branches located in the first and third quadrants.
2. \( y = 3x - 4 \) is a linear equation. Its graph is a straight line with a slope of 3 and a y-intercept of -4.
Now, let's compare the transformations between these two functions:
### Transformations:
- **Vertical Stretch/Shrink**: This concept generally applies to functions like \( y = ax \) where changing \( a \) alters the steepness of the line. In the case of a hyperbola like \( y = \frac{1}{x} \), there is no vertical stretch or shrink involved because it does not transform into a linear graph with a vertical scaling factor.
- **Translation**: A translation shifts the graph up, down, left, or right without changing its shape. The linear function \( y = 3x - 4 \) moves the hyperbola downwards because of the subtraction involved in the transformation, but it does not translate the hyperbola left or right.
### Conclusion:
The comparison of these two functions does not fit into the described transformations like vertical stretch or translations as suggested in the initial statement. For \( y = \frac{1}{x} \) to reach a form similar to \( y = 3x - 4 \) would include both transformations which aren't applicable.
None of the given responses accurately describes the relationship, as:
- \( y = 3x - 4 \) is not a transformation of \( y = \frac{1}{x} \) through vertical stretches or horizontal translations.
### Final Answer:
The graphs do not transform into one another through the operations outlined in your options, and comparing a hyperbolic graph to a linear graph involves different characteristics that are not directly relatable through the definitions provided.
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