Let's work through each part of the problem using the graph shown.

Part (a): Write a point-slope form of the equation for the line shown.

The point-slope form of a line is given by the formula: $y - y_1 = m(x - x_1)$ Where:

• $(x_1, y_1)$ is a point on the line.
• $m$ is the slope of the line.

From the graph, we can see two points on the line:

• Point 1: $(-5, 4)$
• Point 2: $(5, 1)$

First, calculate the slope $m$ using the formula: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 4}{5 - (-5)} = \frac{-3}{10}$ So, the slope $m = -\frac{3}{10}$.

Using the point $(-5, 4)$, the point-slope form is: $y - 4 = -\frac{3}{10}(x + 5)$

Part (b): Estimate the value of the y-intercept of the line.

To estimate the y-intercept, look at where the line crosses the y-axis. The line appears to cross around $y \approx 2.5$.

Part (c): Use proportional reasoning to support your conjecture about the value of the y-intercept.

We can use the slope to calculate the y-intercept more precisely. From the point-slope form of the equation in part (a): $y - 4 = -\frac{3}{10}(x + 5)$ To convert this to slope-intercept form ($y = mx + b$), we simplify: $y - 4 = -\frac{3}{10}x - \frac{3}{2}$ $y = -\frac{3}{10}x + 4 - \frac{3}{2}$ $y = -\frac{3}{10}x + \frac{8}{2} - \frac{3}{2} = -\frac{3}{10}x + \frac{5}{2}$ Thus, the y-intercept $b = \frac{5}{2} = 2.5$, which agrees with the estimate.

Part (d): Rewrite the point-slope form of the equation in slope-intercept form.

We already converted it in part (c): $y = -\frac{3}{10}x + 2.5$

Would you like more details on any part?

Related Questions:

1. How do you calculate the slope of a line from two points?
2. How can we derive the y-intercept from the equation of a line?
3. How does the point-slope form relate to the slope-intercept form?
4. What are the benefits of using the point-slope form in certain cases?
5. How would the equation change if the slope were positive?

Tip: Remember that converting between forms of a line equation (like point-slope and slope-intercept) can give different insights, such as the slope and y-intercept of the line!