To analyze the graph and its corresponding equation choices in point-slope form, let's break it down step-by-step.

Step 1: Identify the slope and key point

The red line has a negative slope, meaning it's decreasing from left to right.

1. Let's use two clear points on the line to determine the slope:
   • Point A: $(0, 1)$
   • Point B: $(-2, 5)$

The slope (m) is calculated using the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the points: $m = \frac{5 - 1}{-2 - 0} = \frac{4}{-2} = -2$ So, the slope of the line is $-2$.

Step 2: Use point-slope form

Point-slope form is given by: $y - y_1 = m(x - x_1)$ We already know the slope is $-2$. Now, using the point $(0, 1)$ from the graph: $y - 1 = -2(x - 0)$ Simplifying this gives: $y = -2x + 1$

Step 3: Check the provided answer options

We need to find any equations that match this form or equivalent forms.

• $y = -2(x - 0) + 1$ → This matches exactly one of the provided options.
• $y = -2x + 1$ is also present as another option.

Both of these equations are valid representations of the line in point-slope form.

Correct options:

• $y = -2(x - 0) + 1$
• $y = -2x + 1$

Would you like more details or have any further questions?

Here are five related questions to expand your understanding:

1. How is the point-slope form different from the slope-intercept form?
2. What are the steps to derive slope from two given points?
3. How would the equation change if the slope were positive?
4. Can a vertical or horizontal line be represented using point-slope form?
5. How do you convert point-slope form to standard form?

Tip: In point-slope form, the point you select to substitute doesn’t have to be an intercept; any point on the line works!