https://file.aiquickdraw.com/mathbot/file/1728711264266j7qnzw3i.jpg

Let's analyze the image step by step to answer the questions:

### Given Functions:
1. **Function 1**: Graph shown with a negative slope (the line is slanting downwards from left to right).
2. **Function 2**: Given in a table format, where \( x \) and \( y \) values are listed for different points.
3. **Function 3**: Equation \( y = 4x - 4 \) (linear function with slope = 4 and y-intercept = -4).
4. **Function 4**: Described with a slope of 1 and a y-intercept of 3.

### Questions:

#### (a) Which function has the graph with a **y-intercept** closest to 0?

- **Function 1**: The graph's y-intercept appears to be around -2.
- **Function 2**: To find the y-intercept, we observe the table and see that when \( x = 0 \), \( y = 1 \), so the y-intercept is 1.
- **Function 3**: The equation \( y = 4x - 4 \) shows a y-intercept of -4.
- **Function 4**: The y-intercept is directly given as 3.

**Conclusion for (a)**: **Function 2** has the y-intercept of 1, which is the closest to 0.

#### (b) Which function has the graph with the greatest **slope**?

- **Function 1**: The slope is negative (we estimate it to be around -2 or so from the graph).
- **Function 2**: We can calculate the slope using two points from the table, such as:
  \[
  \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 11}{-1 - (-2)} = \frac{-5}{1} = -5
  \]
  So the slope is -5.
- **Function 3**: The slope is directly given as 4 from the equation \( y = 4x - 4 \).
- **Function 4**: The slope is directly given as 1.

**Conclusion for (b)**: **Function 3** has the greatest slope, which is 4.

### Final Answers:
- (a) **Function 2** (y-intercept closest to 0).
- (b) **Function 3** (greatest slope).

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### Related questions:
1. How is the slope calculated between two points on a graph?
2. How does the y-intercept affect the graph's position?
3. Why does a negative slope cause the line to fall?
4. Can a slope be zero? What does that imply about the graph?
5. How would you graph a function with a slope of 0 and a y-intercept of 5?

**Tip**: To calculate the slope between any two points, use the formula \( \frac{y_2 - y_1}{x_2 - x_1} \). This measures how much the line rises or falls between the points.

Which function has the graph with a y-intercept closest to 0? Which function has the graph with the greatest slope?

This problem asks you to analyze four different functions and determine which has the graph with the y-intercept closest to 0 and the greatest slope. Learn about key concepts like y-intercepts and slopes in linear functions. Ideal for students in Grades 7-9.

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