Let's address each part of the problem step-by-step and identify what happens to [tex]\( y \)[/tex] for the given changes in [tex]\( x \)[/tex]:
### a. [tex]\( y = kx \)[/tex], [tex]\( x \)[/tex] is tripled
If [tex]\( x \)[/tex] is tripled, that means [tex]\( x \rightarrow 3x \)[/tex].
[tex]\[
y = k(3x) = 3kx
\][/tex]
So when [tex]\( x \)[/tex] is tripled, [tex]\( y \)[/tex] becomes three times its original value.
Proportionality:
[tex]\( y \)[/tex] is directly proportional to [tex]\( x \)[/tex]. This is denoted as [tex]\( y \propto x \)[/tex].
### b. [tex]\( y = 3 + kx \)[/tex], [tex]\( x \)[/tex] is tripled
If [tex]\( x \)[/tex] is tripled, that means [tex]\( x \rightarrow 3x \)[/tex].
[tex]\[
y = 3 + k(3x) = 3 + 3kx
\][/tex]
So when [tex]\( x \)[/tex] is tripled, [tex]\( y \)[/tex] becomes [tex]\( 3 + 3kx \)[/tex]. It increases linearly with [tex]\( x \)[/tex], starting from a base value of 3.
Proportionality:
[tex]\( y \)[/tex] increases linearly with [tex]\( x \)[/tex], but includes a constant term. This can be denoted as [tex]\( y \propto x + \text{constant} \)[/tex].
### c. [tex]\( y = k/x \)[/tex], [tex]\( x \)[/tex] is doubled
If [tex]\( x \)[/tex] is doubled, that means [tex]\( x \rightarrow 2x \)[/tex].
[tex]\[
y = \frac{k}{2x}
\][/tex]
So when [tex]\( x \)[/tex] is doubled, [tex]\( y \)[/tex] becomes half its original value.
Proportionality:
[tex]\( y \)[/tex] is inversely proportional to [tex]\( x \)[/tex]. This is denoted as [tex]\( y \propto \frac{1}{x} \)[/tex].
### d. [tex]\( y = klx^2 \)[/tex], [tex]\( x \)[/tex] is doubled
If [tex]\( x \)[/tex] is doubled, that means [tex]\( x \rightarrow 2x \)[/tex].
[tex]\[
y = kl(2x)^2 = kl \cdot 4x^2 = 4klx^2
\][/tex]
So when [tex]\( x \)[/tex] is doubled, [tex]\( y \)[/tex] becomes four times its original value.
Proportionality:
[tex]\( y \)[/tex] is proportional to the square of [tex]\( x \)[/tex]. This is denoted as [tex]\( y \propto x^2 \)[/tex].
### e. [tex]\( y = kx^2 \)[/tex], [tex]\( x \)[/tex] is tripled
If [tex]\( x \)[/tex] is tripled, that means [tex]\( x \rightarrow 3x \)[/tex].
[tex]\[
y = k(3x)^2 = k \cdot 9x^2 = 9kx^2
\][/tex]
So when [tex]\( x \)[/tex] is tripled, [tex]\( y \)[/tex] becomes nine times its original value.
Proportionality:
[tex]\( y \)[/tex] is proportional to the square of [tex]\( x \)[/tex]. This is denoted as [tex]\( y \propto x^2 \)[/tex].
In summary:
a. When [tex]\( x \)[/tex] is tripled in [tex]\( y = kx \)[/tex], [tex]\( y \)[/tex] becomes three times its original value. ([tex]\( y \propto x \)[/tex])
b. When [tex]\( x \)[/tex] is tripled in [tex]\( y = 3 + kx \)[/tex], [tex]\( y \)[/tex] becomes [tex]\( 3 + 3kx \)[/tex]. ([tex]\( y \propto x + \text{constant} \)[/tex])
c. When [tex]\( x \)[/tex] is doubled in [tex]\( y = k / x \)[/tex], [tex]\( y \)[/tex] becomes half its original value. ([tex]\( y \propto \frac{1}{x} \)[/tex])
d. When [tex]\( x \)[/tex] is doubled in [tex]\( y = klx^2 \)[/tex], [tex]\( y \)[/tex] becomes four times its original value. ([tex]\( y \propto x^2 \)[/tex])
e. When [tex]\( x \)[/tex] is tripled in [tex]\( y = kx^2 \)[/tex], [tex]\( y \)[/tex] becomes nine times its original value. ([tex]\( y \propto x^2 \)[/tex])
