Let's analyze each relationship to determine whether or not it represents a direct proportion. Recall that two variables [tex]\(y\)[/tex] and [tex]\(x\)[/tex] are directly proportional if the relationship between them can be described by [tex]\(y = kx\)[/tex], where [tex]\(k\)[/tex] is a constant. Here’s the step-by-step solution for each given relationship:
a. [tex]\(y = 3x\)[/tex]
- This relationship is of the form [tex]\(y = kx\)[/tex], where [tex]\(k = 3\)[/tex].
- Since it fits the direct proportion formula, it means that [tex]\(y\)[/tex] is directly proportional to [tex]\(x\)[/tex].
- Conclusion: Circle this relationship.
b. [tex]\(y = ax + b\)[/tex]
- This relationship includes a constant term [tex]\(b\)[/tex] in addition to [tex]\(ax\)[/tex].
- This is a linear relationship, but not a direct proportion because it is not solely of the form [tex]\(y = kx\)[/tex].
- Conclusion: This is a linear relationship with an intercept, not a direct proportion.
c. [tex]\(y = x\)[/tex]
- This is the same as [tex]\(y = 1x\)[/tex], where [tex]\(k = 1\)[/tex].
- It exactly matches the form [tex]\(y = kx\)[/tex], so it is a direct proportion.
- Conclusion: Circle this relationship.
d. [tex]\(y = ax^2\)[/tex]
- This represents a quadratic relationship, where [tex]\(y\)[/tex] depends on the square of [tex]\(x\)[/tex].
- It does not match the form [tex]\(y = kx\)[/tex] as there’s a power of 2 involved.
- Conclusion: This is a quadratic relationship, not a direct proportion.
e. [tex]\(y = \frac{a}{x}\)[/tex]
- This represents an inverse proportionality, where [tex]\(y\)[/tex] is inversely proportional to [tex]\(x\)[/tex].
- It is described by [tex]\(y = \frac{a}{x}\)[/tex] instead of [tex]\(y = kx\)[/tex].
- Conclusion: This is an inverse proportion, not a direct proportion.
f. [tex]\(y = ax\)[/tex]
- This fits the form [tex]\(y = kx\)[/tex], where [tex]\(k = a\)[/tex].
- Hence, [tex]\(y\)[/tex] is directly proportional to [tex]\(x\)[/tex].
- Conclusion: Circle this relationship.
g. [tex]\(y = \frac{1}{x}\)[/tex]
- This is another case of inverse proportionality.
- It fits the form [tex]\(y = \frac{a}{x}\)[/tex] where [tex]\(a = 1\)[/tex].
- Conclusion: This is an inverse proportion, not a direct proportion.
h. [tex]\(y = \frac{a}{x^2}\)[/tex]
- This relationship shows an inverse square proportion.
- It describes [tex]\(y\)[/tex] as inversely proportional to the square of [tex]\(x\)[/tex], not just [tex]\(x\)[/tex].
- Conclusion: This is an inverse square proportion, not a direct proportion.
Final Answer:
The relationships that are direct proportions are:
- [tex]\( \mathbf{y = 3x} \)[/tex]
- [tex]\( \mathbf{y = x} \)[/tex]
- [tex]\( \mathbf{y = ax} \)[/tex]
So, the circled relationships are:
[tex]\[ a. \ y = 3x \][/tex]
[tex]\[ c. \ y = x \][/tex]
[tex]\[ f. \ y = ax \][/tex]
