The graph in the image shows a line which we need to identify based on its equation from the provided options.

First, let's find the slope of the line by using two points from the graph. We can take the points where the line crosses the x-axis and y-axis because they are usually easier to identify and ensure accuracy.

Looking at the graph:

• The line crosses the y-axis (where x=0) at y=1.
• The line crosses the x-axis (where y=0) at x=0.5.

Now, let's calculate the slope $m$ using the formula: $m = \frac{\text{change in y}}{\text{change in x}} = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the points (0,1) and (0.5,0): $m = \frac{0 - 1}{0.5 - 0} = \frac{-1}{0.5} = -2$

Thus, the slope of the line is -2.

Given the slope and the y-intercept (which is where the line crosses the y-axis, and we identified as 1), the equation of the line in slope-intercept form $y = mx + b$ is: $y = -2x + 1$

Among the options:

• A: $y = -2x + 1$ (Correct)
• B: $y = \frac{1}{2}x + 1$
• C: $y = -2x - 1$
• D: $y = 2x - 1$

Option A is the correct choice, $y = -2x + 1$, which matches our calculated equation.

Would you like more details on this or do you have any questions?

Here are some further questions to explore:

1. How would the graph change if the slope were positive?
2. What happens to the y-intercept if the line moves parallel to its current position but upwards?
3. Can you predict where this line will intersect another line with a slope of 1?
4. How does changing the y-intercept but keeping the same slope affect the line's position?
5. What are the implications of having a negative slope on the graph's direction?

Tip: When you identify the slope from a graph, choose points that are exactly on the grid lines to minimize errors in slope calculation.