Question

1. which of the following does not represent a proportional relationship?

a. the equation $y = 4x$
b. the equation $y = x + 3$
c. a graph of a straight line through the origin
d. a table with points (1, 5), (2, 10), (3, 15)

Explanation:

Step1: Recall proportional relationship definition

A proportional relationship has the form \( y = kx \) (where \( k \) is a constant), passes through the origin \((0,0)\), and has a constant ratio \(\frac{y}{x}\).

Step2: Analyze option a

For \( y = 4x \), it is in the form \( y = kx \) with \( k = 4 \), so it is proportional.

Step3: Analyze option b

For \( y = x + 3 \), when \( x = 0 \), \( y = 3
eq 0 \), so it does not pass through the origin. Also, it is in the form \( y = mx + b \) with \( b = 3
eq 0 \), so it is not proportional.

Step4: Analyze option c

A straight line through the origin has the form \( y = kx \) (since when \( x = 0 \), \( y = 0 \)), so it is proportional.

Step5: Analyze option d

For the points \((1,5)\), \(\frac{5}{1}=5\); \((2,10)\), \(\frac{10}{2}=5\); \((3,15)\), \(\frac{15}{3}=5\). The ratio \(\frac{y}{x}\) is constant (\( k = 5 \)), so it is proportional.

Answer:

b. The equation \( y = x + 3 \)