This problem asks you to create a graph and write the equation of a function based on the provided information. Let's break it down:

Given Information:

• The beginning value (y-intercept) is $-1$.
• The change (rate of change) is $\frac{1}{3}$ units at each stage.

Equation Type:

• Since the problem describes a "constant rate of change," this indicates the function is linear.
• The general form of a linear equation is: $f(x) = mx + b$ where:
  • $m$ is the slope (rate of change),
  • $b$ is the y-intercept (initial value).

Plugging in the Values:

• The slope $m = \frac{1}{3}$.
• The y-intercept $b = -1$.

Thus, the equation of the function becomes: $f(x) = \frac{1}{3}x - 1$

This equation represents the linear function where the initial value is $-1$, and it increases by $\frac{1}{3}$ for each additional unit of $x$.

Graph:

To graph this function:

• Start at $y = -1$ (the y-intercept) on the vertical axis.
• For each increase of 1 unit in $x$, the function rises by $\frac{1}{3}$ units.

Would you like more details on graphing, or have any other questions?

Here are 5 related questions to expand on this:

1. What is the slope of a line, and how does it affect the graph?
2. How can you find the y-intercept from a given equation?
3. How do linear and exponential functions differ?
4. Can the equation of a linear function be written in other forms besides $y = mx + b$?
5. How would the graph change if the slope was negative?

Tip: The slope determines how steep a line is—positive slopes slant upward, while negative slopes slant downward.