https://file.aiquickdraw.com/mathbot/file/1730489372178ho1qmi8t.jpg

Let's examine each part of the question in the image to determine if the functions are proportional or non-proportional.

1. **Equation $ y = \frac{3}{4}x + 7 $**:
   - This is in the form $ y = mx + b $, where $ b = 7 $.
   - Since there is a non-zero y-intercept (7), this function is **non-proportional**. 
   - A proportional relationship would require that the line passes through the origin (0,0), meaning $ b = 0 $.

2. **Table for "Miles Driven" vs. "Total Cost of the Ride"**:
   - The costs at different miles are:
     - For 1 mile, $2.50.
     - For 5 miles, $12.50.
     - For 10 miles, $25.00.
   - To be proportional, the cost per mile should be constant.
   - If we check the cost per mile:
     \[
     \frac{12.50}{5} = 2.50, \quad \frac{25.00}{10} = 2.50
     \]
   - Since the rate ($2.50 per mile) is constant, this table shows a **proportional** relationship.

3. **Jeremiah's Charges**:
   - Jeremiah charges a $15 flat fee plus $3 per hour.
   - This can be represented by $ y = 3x + 15 $, where 15 is the initial fee.
   - The presence of the flat fee (15) makes it **non-proportional** because a proportional relationship would require that there is no initial fixed fee (it should pass through the origin).

4. **Graph**:
   - The graph shows a line that does not pass through the origin (0,0), which indicates a y-intercept other than 0.
   - Therefore, this line represents a **non-proportional** relationship.

### Summary:
- **Equation $ y = \frac{3}{4}x + 7 $**: Non-proportional
- **Miles Driven vs. Total Cost Table**: Proportional
- **Jeremiah's Lawn Mowing Charges**: Non-proportional
- **Graph**: Non-proportional

Would you like further details or explanations on any of these points?

Here are some related questions for further practice:
1. What are the main differences between proportional and non-proportional relationships?
2. How can you determine if a table represents a proportional relationship?
3. Can a graph of a proportional relationship ever have a y-intercept other than zero?
4. What form does a linear equation need to have to be proportional?
5. How do you determine the constant rate in a proportional table?

**Tip:** A proportional relationship always has a constant ratio and passes through the origin when graphed.

Identify whether each of the following functions is proportional or non-proportional: y = 3/4 x + 7, Miles Driven vs. Total Cost of the Ride table, Jeremiah's flat fee plus $3 per hour, and a graph of a line that does not pass through the origin.

Learn how to identify proportional and non-proportional relationships in linear functions through equations, tables, and graphs. This exercise involves examining y-intercepts, constant ratios, and linear equations in slope-intercept form. Suitable for Grade 8 students.

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