Let's analyze each function in detail.

1. Function A: This is represented by the graph. The line in the graph has a positive slope, indicating a positive rate of change. By observing two points, $(0, 1)$ and $(4, 5)$, the slope is $\frac{5 - 1}{4 - 0} = \frac{4}{4} = 1$. So, the equation for Function A is: $y = x + 1$ with a rate of change (slope) of $1$ and a starting value (y-intercept) of $1$.

2. Function B: The equation is given as: $y = -2x - 1$ Here, the rate of change (slope) is $-2$ and the starting value (y-intercept) is $-1$.

3. Function C: The description says, "y is equal to the sum of two and the product of three and x." This translates to the equation: $y = 3x + 2$ with a rate of change (slope) of $3$ and a starting value (y-intercept) of $2$.

Analyzing the Questions

1. The function with the greatest rate of change: Function C has the highest slope of $3$.
2. The function with the smallest starting value: Function B has the smallest starting value of $-1$.
3. The function with the smallest rate of change: Function A has the smallest slope of $1$.

Answers:

• The function with the greatest rate of change is Function C.
• The function with the smallest starting value is Function B.
• The function with the smallest rate of change is Function A, with a value of 1.

Would you like more details on any of these steps, or have any questions?

Here are five related questions to expand your understanding:

1. How does the rate of change impact the steepness of a line?
2. What does the starting value (y-intercept) represent in real-world scenarios?
3. Can you determine the slope by only looking at a graph?
4. What would happen if the slopes of two functions were the same?
5. How can you find the point of intersection between two functions?

Tip: The slope indicates how fast the function changes; a higher slope means a steeper line.