https://file.aiquickdraw.com/mathbot/file/1726838401153tuzco2y9.jpg

The problem asks to choose the graph that correctly represents the cost of using the internet at a coffee shop, where the cost is proportional to the time used. The given rate is $2 per hour, with $ x $ representing the number of hours and $ y $ representing the cost.

This relationship can be modeled by the equation:

\[
y = 2x
\]

This is a linear equation with a slope of 2 and a y-intercept of 0, meaning the graph should pass through the origin (0,0) and have a positive slope.

### Analysis of the graphs:
1. **First graph**: A line starting from (0, 0) and increasing with a slope of 2. This graph correctly represents $ y = 2x $.
2. **Second graph**: The line starts from (0,0) but the slope is steeper than 2, making it incorrect.
3. **Third graph**: The line has a negative slope, which doesn't fit the proportional relationship.
4. **Fourth graph**: This also has a negative slope, making it incorrect.

### Conclusion:
The **first graph** is the correct representation of the relationship, as it has a slope of 2 and passes through the origin.

Would you like any further clarification or more details? 

Here are 5 related questions to think about:
1. What would the graph look like if the cost per hour were $3 instead of $2?
2. How would the equation change if there were an initial fixed cost, such as a $1 starting fee?
3. What would the graph look like if the cost decreased with time, instead of increasing?
4. How can you calculate the total cost if you used the internet for 5 hours?
5. How would the graph change if the rate were charged in fractions of an hour?

**Tip**: Whenever a linear relationship is described, always check the slope and y-intercept to identify the correct graph!

At the coffee shop, the cost for using the internet is proportional to the time it's used. It costs $2 per hour to use the internet. Select the graph drawn to show this relationship, where x represents the number of hours and y represents the cost.

This problem explores the linear relationship between time and cost at a coffee shop, where the cost is $2 per hour for internet usage. Learn how to interpret the graph, which shows the proportional increase in cost over time. Suitable for students in Grades 6-8.

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