Question

a graph is a straight line through the point (0, 5). can the graph represent a proportional relationship? explain.
options:

• yes; the constant of proportionality is 5.
• yes; the equation of the line can be written in the form ( y = kx ), where ( k = 5 ).
• yes; the graph is a straight line that crosses the ( y )-axis at (0, 5).
• no; the graph of a proportional relationship must pass through (0, 0).

Explanation:

A proportional relationship has the form \( y = kx \), where \( k \) is the constant of proportionality. The graph of \( y = kx \) is a straight line that passes through the origin \((0, 0)\) because when \( x = 0 \), \( y = 0 \). The given graph passes through \((0, 5)\), not \((0, 0)\), so it cannot represent a proportional relationship. Let's analyze each option:

• The first option is wrong because for a proportional relationship, when \( x = 0 \), \( y \) must be \( 0 \), not \( 5 \), so the constant of proportionality can't be \( 5 \) in the context of a proportional relationship.
• The second option is wrong because the equation \( y = kx \) passes through \((0, 0)\), not \((0, 5)\), so it can't be written as \( y = 5x \) and represent a proportional relationship.
• The third option is wrong because just being a straight line crossing the \( y \)-axis at \((0, 5)\) doesn't make it a proportional relationship; proportional relationships must cross at \((0, 0)\).
• The fourth option is correct as it states the key property of proportional relationships (passing through \((0, 0)\)) and correctly identifies that the given graph doesn't meet this.

Answer:

D. No; the graph of a proportional relationship must pass through (0, 0).