1. For each of the following mathematical relations, state what happens to the value of [tex]y[/tex] when the following changes are made. ([tex]k[/tex] is a constant) and state the proportionality.

a. [tex]y = kx[/tex], [tex]x[/tex] is tripled.
b. [tex]y = 3 + kx[/tex], [tex]x[/tex] is tripled.
c. [tex]y = \frac{k}{x}[/tex], [tex]x[/tex] is doubled.
d. [tex]y = kx^2[/tex], [tex]x[/tex] is doubled.
e. [tex]y = kx^2[/tex], [tex]x[/tex] is tripled.



Answer :

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])