When b > 1, as x grows in the negative direction, y __________. When 0 < b < 1, as x grows in the negative direction, y __________.
Understand the Problem
The question involves determining the behavior of a variable 'y' as 'x' grows negatively under different conditions of another variable 'b'. Specifically, it is asking about how 'y' changes with 'x' based on the values of 'b' in two scenarios: when 'b' is greater than 1 and when 'b' is between 0 and 1.
Answer
When $b > 1$, $y$ decreases as $x$ decreases. When $0 < b < 1$, $y$ also decreases as $x$ decreases.
Answer for screen readers
When $b > 1$, as $x$ grows in the negative direction, $y$ decreases.
When $0 < b < 1$, as $x$ grows in the negative direction, $y$ also decreases.
Steps to Solve
- Understanding the Role of 'b' When $b > 1$
When $b$ is greater than 1, the relationship between $y$ and $x$ can be expressed as $y = b \cdot x$. As $x$ grows negatively (i.e., $x$ approaches $-\infty$), since $b$ is positive and greater than 1, the value of $y$ will also become more negative.
- Conclusion for $b > 1$
Thus, as $x$ decreases, $y$ also decreases. This indicates that $y$ will have a negative direction as $x$ becomes more negative.
- Understanding the Role of 'b' When $0 < b < 1$
When $b$ is between 0 and 1, the relationship can still be expressed as $y = b \cdot x$. In this case, even though $b$ is positive, it is less than 1. Thus, as $x$ decreases (grows negatively), $y$ also decreases but at a slower rate compared to when $b$ was greater than 1.
- Conclusion for $0 < b < 1$
Again, as $x$ becomes more negative, $y$ will decrease, but it does so more gradually due to the value of $b$ being less than 1.
When $b > 1$, as $x$ grows in the negative direction, $y$ decreases.
When $0 < b < 1$, as $x$ grows in the negative direction, $y$ also decreases.
More Information
In both scenarios, the pattern shows that $y$ decreases as $x$ becomes more negative. The rate of decrease, however, is influenced by the value of $b$. A value of $b$ greater than 1 leads to a faster decrease in $y$ compared to a value of $b$ between 0 and 1.
Tips
- Mistaking the relationship when $b$ is between 0 and 1. While $y$ does decrease, students might incorrectly assume it could eventually increase.
- Forgetting the effect of the sign of $x$. Negative growth might confuse those who don't track the relationship between $x$ and $y$ properly.
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