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Describe each transformation as it relates to the graph of the parent function

The given function is $ c(x) = 4.5 + 0.32x $, where $ c(x) $ represents the cost of a salad, and $ x $ is the weight of the salad in ounces.

### Interpretation of the Function:

1. **Parent Function**: 
   The parent function of this linear equation is $ f(x) = x $, which is the simplest linear function representing a direct proportionality between $ x $ (input) and $ f(x) $ (output).

2. **Transformation from the Parent Function**:
   The given function can be seen as a transformation of the parent function in the following ways:
   - **Vertical Stretch**: The term $ 0.32x $ indicates that the slope of the graph has been modified. It suggests that for every increase of 1 ounce in weight, the cost of the salad increases by $0.32. This is a vertical stretch of the graph by a factor of 0.32.
   - **Vertical Shift (Translation)**: The constant term $ 4.5 $ indicates a vertical shift upwards by 4.5 units. This represents a base cost of $4.50, which is added regardless of the salad's weight.

3. **Interpretation in Context**:
   - The function $ c(x) = 4.5 + 0.32x $ represents the cost of a salad where there is a fixed cost of $4.50 (perhaps for the container or preparation) and an additional cost of $0.32 per ounce of salad weight.
   - As $ x $, the salad's weight, increases, the total cost $ c(x) $ increases linearly at a rate of $0.32 per ounce.

### Summary of Transformations:
- **Slope (0.32)**: This modifies the steepness of the line, indicating how fast the cost rises as the weight increases. It stretches the graph vertically compared to the parent function $ f(x) = x $.
- **Constant (4.5)**: This shifts the graph upwards by 4.5 units, representing the base cost of the salad before any weight is added.

Would you like more details or have any questions?

### Related Questions:
1. How would the graph look if the constant was 0 instead of 4.5?
2. How does the rate of $0.32 per ounce compare to a different rate, such as $0.50 per ounce?
3. What happens to the graph if the base cost increases to $6.00?
4. Can you describe the impact of reducing the rate per ounce to $0.20 on the graph?
5. What would the function look like if there was no base cost, and you were only charged per ounce?

### Tip:
Always remember that in a linear function, the slope represents the rate of change, and the constant represents the starting value or initial condition.

The cost for a salad depends on its weight, x, in ounces, and is described by c(x) = 4.5 + 0.32x. Describe how c(x) is related to its parent function and interpret the function in the context of the situation.

Explore how the linear function c(x) = 4.5 + 0.32x describes the cost of a salad depending on its weight. Learn how this function relates to its parent function and the role of transformations such as vertical stretch and vertical shift in the context of the problem. Suitable for students in Grades 8-10.

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