Y-Intercept and X-Intercept - Algebra-2

1. Fundamental Concepts

Y-Intercept: The point where a line crosses the y-axis. It is represented as \((0, b)\), where b is the value on the y-axis at the intersection (i.e., the y-coordinate when the x-coordinate is 0).
X-Intercept: The point where a line crosses the x-axis. It is represented as \((a, 0)\), where a is the value on the x-axis at the intersection (i.e., the x-coordinate when the y-coordinate is 0).

Graphical Features:
- The y-intercept is the single point where the line intersects the y-axis (vertical axis), with an x-coordinate always equal to 0.
- The x-intercept is the single point where the line intersects the x-axis (horizontal axis), with a y-coordinate always equal to 0.
Connection to Equations:
- In the slope-intercept form \(y = mx + b\), b is the y-coordinate of the y-intercept \((0, b)\).
- To find the x-intercept algebraically, set \(y = 0\) in the linear equation and solve for x; the result is the x-coordinate of \((a, 0)\).

Simple (Graph Description): A line crosses the y-axis at \((0, 3)\) and the x-axis at \((-1, 0)\). Identify the y-intercept and x-intercept.
- Y-intercept: \((0, 3)\)
- X-intercept: \((-1, 0)\)
Medium (Graph Description): A line passes through the origin \((0,0)\) and slopes downward to the right. What are its intercepts?
- Analysis: Since the line crosses both axes at \((0,0)\), this point is both the y-intercept and x-intercept.
- Y-intercept: \((0, 0)\)
- X-intercept: \((0, 0)\)
Hard: Find the y-intercept and x-intercept of the line \(3x - 6y = 12\).
- Y-intercept: Substitute \(x = 0\): \(3(0) - 6y = 12 \implies y = -2\), so the y-intercept is \((0, -2)\).
- X-intercept: Substitute \(y = 0\): \(3x - 6(0) = 12 \implies x = 4\), so the x-intercept is \((4, 0)\).

From a Graph:
1. Locate where the line crosses the y-axis: the coordinates of this point are the y-intercept \((0, b)\).
2. Locate where the line crosses the x-axis: the coordinates of this point are the x-intercept \((a, 0)\).
3. Special cases:
  - Horizontal lines (e.g., \(y = k\)) have a y-intercept \((0, k)\) but no x-intercept (unless \(k = 0\), in which case the x-intercept is all points on the x-axis).
  - Vertical lines (e.g., \(x = h\)) have an x-intercept \((h, 0)\) but no y-intercept (unless \(h = 0\), in which case the y-intercept is all points on the y-axis).
From an Equation:
1. For y-intercept: Set \(x = 0\) and solve for y; the result gives \((0, y)\).
2. For x-intercept: Set \(y = 0\) and solve for x; the result gives \((x, 0)\).