Let's break down and work through this worksheet.

Part 1: Information Based on the Graph

The first part of the worksheet asks for the slope, y-intercept, and the equation of the line from the given graph. Here’s how we analyze it:

1. Finding the slope: The line on the graph slopes downward, meaning the slope ($m$) is negative. We can calculate the slope using two points on the line. From the graph, we see:

   • The line passes through the point (0, 4) (the y-intercept) and another point like (2, 0) (x-intercept).
   • Slope $m = \frac{\text{rise}}{\text{run}} = \frac{4 - 0}{0 - 2} = -2$.

   So, the slope $m = -2$.

2. The y-intercept: The y-intercept ($b$) is the point where the line crosses the y-axis. From the graph, this is at the point (0, 4), so $b = 4$.

3. The x-intercept: The x-intercept is the point where the line crosses the x-axis. From the graph, this is at the point (2, 0), so the x-intercept is 2.

4. Equation of the line: Using the slope-intercept form $y = mx + b$, where $m = -2$ and $b = 4$, the equation of the line is: $y = -2x + 4$

Part 2: Determining the Slope and Y-Intercept from the Equations

1. Equation: $y = 2x + 5$

   • Slope ($m$) = 2
   • Y-intercept ($b$) = 5

2. Equation: $y = -x - 3$

   • Slope ($m$) = -1 (since the coefficient of $x$ is -1)
   • Y-intercept ($b$) = -3

3. Equation: $y = 7$

   • This is a horizontal line, so the slope ($m$) is 0 (because there's no $x$-term).
   • Y-intercept ($b$) = 7

4. Equation: $y = -x$

   • Slope ($m$) = -1
   • Y-intercept ($b$) = 0 (since there is no constant term, $b = 0$)

Summary:

Part 1 (From the graph):

• Slope ($m$): $-2$
• Y-intercept ($b$): $4$
• X-intercept: $2$
• Equation: $y = -2x + 4$

Part 2 (From the equations):

1. $y = 2x + 5$: $m = 2$, $b = 5$
2. $y = -x - 3$: $m = -1$, $b = -3$
3. $y = 7$: $m = 0$, $b = 7$
4. $y = -x$: $m = -1$, $b = 0$

Would you like further details on any of these steps, or do you have any questions? Here are some related questions:

1. What does a positive slope indicate about the direction of a line?
2. How does the y-intercept affect the position of a line on a graph?
3. What is the difference between slope-intercept form and point-slope form?
4. How can you determine the x-intercept of a line if you know the equation?
5. How do horizontal lines differ in their equations compared to vertical lines?

Tip: When identifying the slope from a graph, always pick two points and apply the formula $\frac{\text{rise}}{\text{run}}$.