Question

part d: proportional vs. linear relationships

1. graph comparison

graph a:
graph a image
graph b:
graph b image
10a. which graph represents a proportional relationship?
□ graph a □ graph b
10b. how do you know? (hint: think about the origin)

1. equation analysis

a. ( y = 4x )
b. ( y = 4x + 3 )
11a. which equation represents a proportional relationship? ______
11b. what feature of the equation tells you this? ______

Explanation:

Step1: Identify proportional graph rule

A proportional relationship's graph passes through the origin $(0,0)$.

Step2: Check Graph A

Graph A intersects the y-axis at $(0,4)$, not the origin.

Step3: Check Graph B

Graph B passes through the origin $(0,0)$.

Step4: Identify proportional equation rule

A proportional relationship has the form $y=kx$ (no constant term, $b=0$).

Step5: Check Equation a

Equation $y=4x$ follows $y=kx$ with $k=4$, no constant term.

Step6: Check Equation b

Equation $y=4x+3$ has a constant term $3$, so it is not proportional.

Answer:

10a. Graph B
10b. A proportional relationship's graph must pass through the origin $(0,0)$, and Graph B goes through this point, while Graph A does not.
11a. $y = 4x$
11b. Proportional relationships are represented by equations in the form $y=kx$ (with no constant added term, i.e., the y-intercept is 0). The equation $y=4x$ fits this form, while $y=4x+3$ has a non-zero constant term.