Question

1. susie was saving for a new ipod. her dad gave her $20 for her birthday. after that she earned $5 per week for helping to clean the kitchen.

is the relationship between time and her money proportional? yes / no
explain:

1. sammi loves candy bars. she ate 1 candy bar on jan. 1st, 2 on jan. 2nd, and then 3 candy bars a day every day for the rest of the month.

is the relationship between the date and her number of eaten candy bars proportional? yes / no
explain:
for each graph decide whether it is proportional or not. then explain why.

1. yes / no

explain

1. yes / no

explain

1. yes / no

explain

1. yes / no

explain

Explanation:

Problem 2

Step1: Recall proportional relationship

A proportional relationship has the form \( y = kx \), meaning it passes through the origin \((0,0)\) and has a constant ratio \( \frac{y}{x}=k \). Susie starts with \$20 (when time \( x = 0 \), money \( y = 20 \)), so the graph would not pass through the origin.

Step2: Analyze the equation

Let \( x \) be weeks, \( y \) be money. The equation is \( y = 5x + 20 \). For proportionality, \( y/x \) should be constant, but here \( y/x=5 + 20/x \), which is not constant (depends on \( x \)).

Step1: Recall proportional relationship

A proportional relationship has a constant ratio of \( \frac{\text{candy bars}}{\text{date}} \) and passes through \((0,0)\) (or starts with 0 when date is 0, but here on day 1: 1, day 2: 2, then 3. The ratio changes from 1/1 = 1, 2/2 = 1, but then for day 3, it's 3/3 = 1? Wait, no, wait: she ate 1 on Jan 1st (date 1), 2 on Jan 2nd (date 2), then 3 every day after. So for date \( x \): if \( x = 1 \), \( y = 1 \); \( x = 2 \), \( y = 2 \); \( x\geq3 \), \( y=3+(x - 2)\times3=3x - 3 \)? Wait, no, cumulative? Wait, the problem says "the relationship between the date and her number of eaten candy bars". Wait, maybe cumulative? Wait, no, the question is about the number of eaten candy bars (per day? Or cumulative?). Wait, the wording: "the relationship between the date and her number of eaten candy bars". Let's assume it's the number of candy bars eaten on that date. So on date 1: 1, date 2: 2, date \( x\geq3 \): 3. So the ratio \( y/x \) is 1/1 = 1, 2/2 = 1, 3/3 = 1? Wait, no, date 3: 3, so 3/3 = 1. Wait, but wait, the first two days have ratio 1, then from day 3, it's 3 per day. Wait, no, if it's the number of candy bars eaten on the date (not cumulative), then on day 1:1, day 2:2, day 3:3, etc. Wait, but the problem says "then 3 candy bars a day every day for the rest of the month". So day 1:1, day 2:2, day 3:3, day 4:3, etc. Wait, no, that's not. Wait, maybe cumulative. Let's re - read: "She ate 1 candy bar on Jan. 1st, 2 on Jan. 2nd, and then 3 candy bars a day every day for the rest of the month." So cumulative number: day 1:1, day 2:1 + 2 = 3, day 3:3+3 = 6, day 4:6 + 3 = 9, etc. Then the ratio of cumulative candy bars to date: day 1:1/1 = 1, day 2:3/2 = 1.5, day 3:6/3 = 2, day 4:9/4 = 2.25. The ratio is not constant. Also, the initial values: if we consider the relationship as a function, for \( x = 1 \), \( y = 1 \); \( x = 2 \), \( y = 3 \) (cumulative? No, the problem says "number of eaten candy bars" – maybe per day. Wait, the problem is a bit ambiguous, but the key is: a proportional relationship has a constant rate and passes through the origin (or has \( y = kx \)). Here, the number of candy bars eaten on day \( x \) is \( y=

$$\begin{cases}x, & x = 1,2\\3, & x\geq3\end{cases}$$

\). So the rate (slope) changes. For \( x = 1 \), slope 1; \( x = 2 \), slope 2 - 1 = 1 (wait, no, per day, the change from day 1 to 2 is 1, day 2 to 3 is 1 (from 2 to 3). Wait, no, day 1:1, day 2:2 (change of 1), day 3:3 (change of 1). Wait, then it's a linear relationship with slope 1? But wait, the problem is about proportionality. A proportional relationship is a linear relationship that passes through the origin (\( y = kx \)). Here, when \( x = 0 \) (date 0), \( y = 0 \) (assuming), but on date 1, \( y = 1 \), date 2, \( y = 2 \), date 3, \( y = 3 \). So \( y = x \) for all days? But the problem says "then 3 candy bars a day every day for the rest of the month". Wait, maybe I misread: she ate 1 on Jan 1st, 2 on Jan 2nd, and then 3 candy bars a day every day for the rest of the month. So cumulative: Jan 1st:1, Jan 2nd:1 + 2 = 3, Jan 3rd:3+3 = 6, Jan 4th:6 + 3 = 9, etc. So the relationship between date \( x \) (1 - based) and cumulative candy bars \( y \): \( y=

$$\begin{cases}1, & x = 1\\3, & x = 2\\3+3(x - 2), & x\geq3\end{cases}$$

\). For \( x = 1 \), \( y = 1 \), ratio \( y/x=1/1 = 1 \); \( x = 2 \), \( y = 3 \), ratio \( 3/2 = 1.5 \); \( x = 3 \), \( y = 6 \), ratio \( 6/3 = 2 \). The ratio is not constant, so not proportional.

Step2: Conclusion

The ratio of candy bars to date is not constant (changes fr…

Step1: Recall proportional graph properties

A proportional relationship's graph is a straight line passing through the origin \((0,0)\). This graph has a y - intercept (starts above the origin), so it does not pass through \((0,0)\).

Step2: Check the ratio

For a proportional graph, \( y/x \) should be constant. Since it has a y - intercept, when \( x = 0 \), \( y
eq0 \), so the ratio \( y/x \) is not defined (or infinite) at \( x = 0 \) and changes as \( x \) increases, so it's not proportional.

Answer:

no; because the relationship has a y - intercept (starts with \$20) and the ratio of money to time is not constant.

Problem 3