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).

Would you like further details on any part of the explanation?

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.