Question

1. yu yan earns extra money working on weekends. she creates a graph showing the number of hours she works and the amount of money she earns each weekend.

a. based on the graph, does the amount of money yu yan earns appear to be proportional to the number of hours she works? explain how you know.
b. the next weekend, yu yan adds the point (0, 0) to the graph. what does the point (0, 0) mean in this context?
c. how much money does yu yan earn if she works only 1 hour? explain.

Explanation:

Part a

Step1: Recall proportionality rule

For two quantities to be proportional, their graph should be a straight line passing through the origin, and the ratio of \( y \) (earnings) to \( x \) (hours) should be constant.

Step2: Check the graph's line

The points on the graph: when \( x = 2 \), \( y = 22 \) (wait, no, looking at the grid, at \( x = 2 \), \( y = 22 \)? Wait, no, the grid: let's re - check. Wait, the points: at \( x = 2 \), \( y = 22 \)? No, maybe I misread. Wait, the graph: when \( x = 2 \), the \( y \) - value is 22? Wait, no, maybe the points are (2,22)? No, wait, the user's graph: let's see, the x - axis is time (hours), y - axis is earnings (dollars). The points: (2,22)? Wait, no, maybe the correct points: let's assume the points are (2,22)? No, wait, maybe the points are (2,22), (4,44), (5,55)? Wait, no, the original problem's graph: let's re - examine. Wait, the user's graph: when \( x = 2 \), \( y = 22 \)? No, maybe the points are (2,22), (4,44), (5,55)? Wait, no, the key is to check if the line is straight and passes through the origin (or would pass through the origin if extended). Wait, the given points: let's take the coordinates. Let's say the points are (2,22), (4,44), (5,55)? Wait, no, maybe the correct coordinates: from the graph, when \( x = 2 \), \( y = 22 \)? No, maybe the points are (2,22), (4,44), (5,55). Wait, the ratio of \( y/x \) for (2,22) is \( 22/2=11 \), for (4,44) is \( 44/4 = 11 \), for (5,55) is \( 55/5=11 \). So the ratio is constant. Also, if we draw a line through these points, it would be a straight line, and if we extend it, it would pass through (0,0). So the amount of money earned is proportional to the number of hours worked because the graph is a straight line and the ratio of earnings to hours is constant (and the line would pass through the origin).

Part b

Step1: Interpret the coordinates

In the graph, \( x \) represents the number of hours worked and \( y \) represents the earnings (in dollars). The point \( (0,0) \) means that when \( x = 0 \) (number of hours worked is 0) and \( y = 0 \) (earnings are 0). So in this context, it means that if Yu Yan works 0 hours, she earns 0 dollars.

Part c

Step1: Find the constant of proportionality

From part a, we saw that the ratio of earnings (\( y \)) to hours (\( x \)) is constant. Let's take one of the points, say \( (2,22) \) (assuming the correct point, or maybe (2,22) is wrong, wait, maybe the points are (2,22), (4,44), (5,55). Wait, the ratio \( y/x \) is \( 22/2 = 11 \), \( 44/4=11 \), \( 55/5 = 11 \). So the constant of proportionality \( k=\frac{y}{x}=11 \).

Step2: Calculate earnings for 1 hour

We know that the relationship is \( y = kx \), where \( k = 11 \) and \( x = 1 \). So \( y=11\times1 = 11 \). So Yu Yan earns $11 if she works 1 hour.

Part a Answer:

Yes, the amount of money Yu Yan earns is proportional to the number of hours she works. We know this because the graph of the relationship between hours worked and earnings is a straight line, and the ratio of earnings to hours worked is constant (e.g., if we take the points, the ratio \( \frac{\text{earnings}}{\text{hours}} \) is the same for different points, and the line would pass through the origin \((0,0)\) if extended).

Part b Answer:

The point \((0,0)\) means that when Yu Yan works 0 hours ( \( x = 0 \) ), she earns 0 dollars ( \( y = 0 \) ).

Part c Answer:

Yu Yan earns $11 if she works 1 hour. We know from the proportional relationship \( y=kx \) (where \( y \) is earnings, \( x \) is hours, and \( k \) is the constant of proportionality). By calculating the ratio of earnings to hours for the given points (e.g., \( \frac{22}{2}=11 \), \( \frac{44}{4} = 11 \), etc.), we find \( k = 11 \). For \( x = 1 \) hour, \( y=11\times1=11 \) dollars.

Answer:

Step1: Find the constant of proportionality

From part a, we saw that the ratio of earnings (\( y \)) to hours (\( x \)) is constant. Let's take one of the points, say \( (2,22) \) (assuming the correct point, or maybe (2,22) is wrong, wait, maybe the points are (2,22), (4,44), (5,55). Wait, the ratio \( y/x \) is \( 22/2 = 11 \), \( 44/4=11 \), \( 55/5 = 11 \). So the constant of proportionality \( k=\frac{y}{x}=11 \).

Step2: Calculate earnings for 1 hour

We know that the relationship is \( y = kx \), where \( k = 11 \) and \( x = 1 \). So \( y=11\times1 = 11 \). So Yu Yan earns $11 if she works 1 hour.

Part a Answer:

Yes, the amount of money Yu Yan earns is proportional to the number of hours she works. We know this because the graph of the relationship between hours worked and earnings is a straight line, and the ratio of earnings to hours worked is constant (e.g., if we take the points, the ratio \( \frac{\text{earnings}}{\text{hours}} \) is the same for different points, and the line would pass through the origin \((0,0)\) if extended).

Part b Answer:

The point \((0,0)\) means that when Yu Yan works 0 hours ( \( x = 0 \) ), she earns 0 dollars ( \( y = 0 \) ).

Part c Answer:

Yu Yan earns $11 if she works 1 hour. We know from the proportional relationship \( y=kx \) (where \( y \) is earnings, \( x \) is hours, and \( k \) is the constant of proportionality). By calculating the ratio of earnings to hours for the given points (e.g., \( \frac{22}{2}=11 \), \( \frac{44}{4} = 11 \), etc.), we find \( k = 11 \). For \( x = 1 \) hour, \( y=11\times1=11 \) dollars.