Let's go through each question step-by-step using the information given for the four functions.

Analysis of Each Function:

1. Function 1: The graph shows a line with a negative slope. We can observe that the slope appears to be less steep than Function 3 (which has a slope of -5).
2. Function 2: Given as a table of points. We can calculate the slope by finding the difference in $y$-values over the difference in $x$-values. For example: $\text{slope} = \frac{-8 - (-12)}{-1 - (-2)} = \frac{4}{1} = 4$ So, the slope of Function 2 is 4.
3. Function 3: Defined by the equation $y = -5x - 2$, where the slope is -5, which is steep and negative.
4. Function 4: Defined with a slope of 2 and a $y$-intercept of 3.

Solving Each Part:

(a) Which function’s graph is the steepest?

• Answer: The steepest slope in terms of magnitude is -5 in Function 3. Thus, Function 3 is the steepest.

(b) Which functions have graphs with $y$-intercepts greater than $-3$?

• Function 1: Observing the graph, the $y$-intercept appears to be at -2.

• Function 2: From the table, when $x = 0$, $y = -4$, which is less than -3.

• Function 3: The equation $y = -5x - 2$ has a $y$-intercept of -2.

• Function 4: The $y$-intercept is 3.

• Answer: Function 1, Function 3, and Function 4 have $y$-intercepts greater than -3.

(c) Which function has the graph with a $y$-intercept farthest from 0?

• $y$-intercepts:
  • Function 1: -2
  • Function 2: -4
  • Function 3: -2
  • Function 4: 3

The $y$-intercept farthest from 0 in absolute value is -4 from Function 2.

Final Answers:

• (a): Function 3
• (b): Functions 1, 3, and 4
• (c): Function 2

Would you like more details on any of these calculations?

Here are some related questions to further your understanding:

1. How do we find the slope between two points on a line?
2. What does the steepness of a line signify in real-world terms?
3. How do we determine a line's intercept from its graph?
4. Why is the absolute value important in determining the farthest distance from 0?
5. What are the characteristics of a line with a positive slope versus a negative slope?

Tip: To identify the steepest graph, always compare the absolute values of the slopes. The higher the absolute value, the steeper the line.