Final answer:
The parallel line to y = x + 4 through (3, -2) is y = x - 1, having the same slope of 1. The perpendicular line's equation is y = -x + 1, with a slope of -1 (the negative reciprocal of the original slope).
Step-by-step explanation:
The student is provided with a point (3, -2) and an existing line equation y = x + 4. To write an equation for a line that is parallel to the given line, we must use the same slope as the provided equation. The slope (m) for the line y = x + 4 is 1, because the coefficient of x is the slope. Since parallel lines have identical slopes, our new equation will also have a slope of 1. Using the point-slope form, y - y1 = m(x - x1), where (x1, y1) is our given point (3, -2), we substitute m = 1 to get y - (-2) = 1(x - 3), which simplifies to y = x - 1. This equation is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
To write an equation for a line that is perpendicular to the given line, we need to determine the negative reciprocal of the original line's slope. A slope of 1 means the perpendicular slope is -1. Again, using the point-slope form, y - y1 = m(x - x1), with our given point (3, -2) and our new slope m = -1, we get y - (-2) = -1(x - 3), which simplifies to y = -x + 1.
For a line represented by an equation such as y = mx + b or y = a + bx, the slope and the y-intercept are essential in defining its shape. The slope, rise over run, indicates the steepness of a line, and the y-intercept is the point where the line crosses the y-axis. For instance, in Figure A1, the slope is 3, indicating a rise of 3 for every unit increase along the x-axis. The line intersects the y-axis at 9, which is the y-intercept.