Question

linear relationships in tables
name: _______________________
period: ___ date: ________

learning goals:

• determine the rate of change of a linear relationship by reading (x, y) values from a table.
• determine whether a table of values represents a linear proportional or linear non - proportional relationship.

ron has a player’s card for the arcade at the mall. his player’s card keeps track of the number of credits he earns as he wins games. each winning game earns the same number of credits, and those credits can be redeemed for various prizes. ron has been saving his credits to collect a prize worth 500 credits.

(there is a graph on the left: the x - axis is number of games won today with values from 0 to 40, and the y - axis is number of credits on ron’s player’s card with values from 0 to 800. there is a table on the right:

number of games ron won today	number of credits on ron’s player’s card
12	216
18	264
25	320
40	440

1. is this relationship proportional or non - proportional? explain how you know.
2. explain the meaning of the ordered pair (0, 120) listed in the table.

Explanation:

Question 1

Step 1: Recall Proportional Relationship Rule

A proportional relationship has the form \( y = kx \), meaning when \( x = 0 \), \( y = 0 \) (the graph passes through the origin), and \( \frac{y}{x} \) is constant for all \( x
eq0 \).

Step 2: Check \( x = 0 \) Case

From the table, when \( x = 0 \) (number of games won is 0), \( y = 120 \) (credits on the card). In a proportional relationship, \( y \) should be 0 when \( x = 0 \), so this fails the origin test.

Step 3: Verify Constant Ratio (Optional)

Calculate \( \frac{y}{x} \) for other points: For \( x = 12 \), \( y = 216 \), \( \frac{216}{12}=18 \); for \( x = 18 \), \( y = 264 \), \( \frac{264}{18}\approx14.67 \) (not equal to 18). So the ratio isn't constant.

In the context of the problem, the \( x \) - value in the ordered pair \( (0,120) \) represents the number of games Ron won today, and the \( y \) - value represents the number of credits on his player's card. So when \( x = 0 \) (Ron won 0 games today), \( y = 120 \) means Ron had 120 credits on his player's card before winning any games today (or when he won 0 games on this particular day).

Answer:

The relationship is non - proportional. A proportional relationship requires \( y = 0 \) when \( x = 0 \) (passes through the origin) and a constant \( \frac{y}{x} \) ratio. Here, when \( x = 0 \) (0 games won), \( y = 120 \) (non - zero credits), and the \( \frac{y}{x} \) ratio isn't constant (e.g., \( \frac{216}{12}=18 \) vs \( \frac{264}{18}\approx14.67 \)), so it's non - proportional.

Question 2