To solve this, we need to identify the point-slope form equation that matches the graph shown in the image.

Step 1: Analyzing the graph

1. Slope: The line on the graph slopes downward, which means the slope is negative.
2. Two points from the graph:
   • The line passes through two key points:
     • Point 1: $(-1, 1)$
     • Point 2: $(1, 0)$

To find the slope, use the formula for the slope between two points, $(x_1, y_1)$ and $(x_2, y_2)$: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the values: $m = \frac{0 - 1}{1 - (-1)} = \frac{-1}{2} = -\frac{1}{2}$

Step 2: Writing the point-slope equation

The point-slope form of an equation is: $y - y_1 = m(x - x_1)$

Using the slope $m = -\frac{1}{2}$ and the point $(-1, 1)$, we substitute into the formula: $y - 1 = -\frac{1}{2}(x + 1)$

Step 3: Matching with the options

The correct equation is: $y - 1 = -\frac{1}{2}(x + 1)$

Conclusion:

The correct answer is the first option.

Would you like more details on any step?

Here are 5 related questions you can explore:

1. How do you convert between point-slope form and slope-intercept form?
2. What is the general form of a linear equation, and how can we derive it from point-slope form?
3. How do we interpret the slope geometrically in a real-world context?
4. Can you find the equation of a line parallel to this one but passing through a different point?
5. How would you find the equation of the perpendicular line to the one given?

Tip: When finding the slope from a graph, always choose clear points where the line crosses the grid to avoid calculation errors!