To determine the equation of the linear function represented by this graph, let's analyze the line’s slope and y-intercept:

1. Identify the y-intercept: The graph intersects the y-axis at $y = -1$. This indicates the y-intercept $b$ of the line is -1.

2. Calculate the slope (m):

   • Pick two points on the line to calculate the slope. One clear pair is $(0, -1)$ (the y-intercept) and another point is $(2, -4)$.
   • Using the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$: $m = \frac{-4 - (-1)}{2 - 0} = \frac{-3}{2}$

3. Formulate the equation: With a slope $m = -\frac{3}{2}$ and y-intercept $b = -1$, the equation of the line is: $y = -\frac{3}{2}x - 1$

4. Compare with the options: The correct option that matches $y = -\frac{3}{2}x - 1$ is the second choice.

Answer: The correct answer is: $y = -\frac{3}{2}x - 1$

Would you like a further breakdown of any step?

Here are 5 related questions to expand your understanding:

1. How do you determine the slope from a graph if the points are not clearly labeled?
2. What is the significance of the y-intercept in the context of linear functions?
3. How can you determine if a line is increasing or decreasing based on its slope?
4. How would you write the equation if the slope were positive but the y-intercept remained the same?
5. How can the slope-intercept form help in quickly graphing a linear function?

Tip: The slope tells you how steep the line is and in which direction it moves (upwards or downwards) as $x$ increases.