Question

determine whether each of these linear relationships is proportional or non - proportional.
proportional
non - proportional
x | 2 3 5
y | 4 5 7
(y = 2x)
a giraffe at birth is 3 feet tall and grows 161 inches every month for a year.
from their initial step, a student walks at a constant speed of 1 miles per hour.
(y = 20 + 4x) (with a corresponding graph)

Explanation:

Step1: Analyze \( y = 2x \)

A proportional relationship has the form \( y=kx \) (no constant term). For \( y = 2x \), when \( x = 0 \), \( y = 0 \), and \( \frac{y}{x}=2 \) (constant). So it's proportional.

Step2: Analyze the table (\( x: 2, 3, 5 \); \( y: 4, 5, 7 \))

Check \( \frac{y}{x} \): \( \frac{4}{2}=2 \), \( \frac{5}{3}\approx1.67 \), \( \frac{7}{5}=1.4 \). Ratios aren't constant, so non - proportional.

Step3: Analyze giraffe growth

Giraffe at birth: 3 feet (initial value \( b = 3 \) feet, \( 3\) feet \(= 36\) inches). Growth: 161 inches per month? Wait, likely a typo (maybe 16 inches? But as per given, growth with initial value (3 feet) means the relationship is \( y=\text{initial}+\text{growth rate}\times t \), so non - proportional (has a non - zero initial value).

Step4: Analyze student walking

Student walks from initial step (implies initial position, so distance \( d=\text{initial position}+1\times t \), non - proportional (has initial value).

Step5: Analyze \( y = 20 + 4x \)

This is in the form \( y=mx + b \) with \( b = 20
eq0 \), so non - proportional.

Step6: Analyze the graph

The graph starts at \( y\) - intercept (not at origin), so it's a linear relationship with \( b
eq0 \), non - proportional.

Answer:

• \( y = 2x \): Proportional
• Table (\( x:2,3,5 \); \( y:4,5,7 \)): Non - proportional
• Giraffe growth: Non - proportional
• Student walking: Non - proportional
• \( y = 20+4x \): Non - proportional
• Given graph: Non - proportional