Question

select the equations that show a proportional relationship between x and y. y = 2x² y = 4x

Explanation:

Step1: Recall Proportional Relationship

A proportional relationship between \( x \) and \( y \) is of the form \( y = kx \), where \( k \) is a constant (the constant of proportionality), and the graph passes through the origin \((0,0)\). It is a linear relationship with a constant rate of change.

Step2: Analyze \( y = 2x^2 \)

The equation \( y = 2x^2 \) is a quadratic equation (degree 2 in \( x \)). For a proportional relationship, the relationship should be linear (degree 1 in \( x \)). Let's check the ratio \( \frac{y}{x} \): \( \frac{2x^2}{x}=2x \), which is not a constant (it depends on \( x \)). So \( y = 2x^2 \) does not represent a proportional relationship.

Step3: Analyze \( y = 4x \)

The equation \( y = 4x \) is in the form \( y = kx \) with \( k = 4 \) (a constant). The ratio \( \frac{y}{x}=\frac{4x}{x} = 4 \) (constant for \( x
eq0 \)), and when \( x = 0 \), \( y = 0 \), so it passes through the origin. Thus, \( y = 4x \) represents a proportional relationship.

Answer:

The equation that shows a proportional relationship between \( x \) and \( y \) is \( \boldsymbol{y = 4x} \). The equation \( y = 2x^2 \) does not (it is a quadratic relationship).