What is the Y-Intercept?
The y-intercept is the point on a coordinate plane where a line crosses (intercepts) the vertical y-axis.
Because the y-axis is positioned at an x-coordinate of 0, the coordinates of the y-intercept are always formatted as (0, b), where b is the y-value.
Method 1: Finding Y-Intercept in Slope-Intercept Form (y = mx + b)
If your linear equation is in slope-intercept form:
- •m is the slope (steepness of the line).
- •\mathbf{b} is the y-intercept!
You don't need to do any algebra: simply look at the constant number added or subtracted at the end.
- •For y = 3x + 7, the y-intercept is 7, or (0, 7).
- •For y = -2x - 4, the y-intercept is -4, or (0, -4).
- •For y = 12x, the y-intercept is 0, or (0, 0) (passes through the origin).
Worked Example 1: Slope-Intercept Form & Graph
Problem:
Find the y-intercept of the line given by the equation:
y = -5x + 9
Step 1: Identify the form
The equation is already in slope-intercept form: y = mx + b.
Step 2: Extract the constant b
The slope is m = -5.
The constant term is b = 9.
Step 3: Write as a coordinate point
The line crosses the y-axis at y = 9, so the coordinate point is (0, 9).
Method 2: Finding Y-Intercept in Standard Form (Ax + By = C)
When an equation is in standard form, x and y are on the same side of the equal sign:
To find the y-intercept, use the Cover-Up Method (Set x = 0):
- Replace x with 0: A(0) + By = C.
- The x term vanishes: By = C.
- Divide both sides by B: y = CB.
Worked Example 2: Standard Form with Negatives
Problem:
Find the y-intercept of the equation:
6x - 4y = 28
Step 1: Substitute x = 0
Step 2: Solve for y
Don't drop the minus sign in front of 4!
Divide both sides by -4:
Step 3: Write the ordered pair
The y-intercept is -7, or (0, -7).
Method 3: Finding the Y-Intercept on a Graph
If you have a graph without an equation:
- Locate the vertical axis (the vertical line running up and down through 0).
- Trace the straight line until it crosses the y-axis.
- Read the number on the vertical axis at that exact intersection.
- If it crosses at 3, the y-intercept is (0, 3).
Quick Reference Summary
| Equation Format | Example | How to Find Y-Intercept | Y-Intercept Point |
|---|---|---|---|
| Slope-Intercept | y = 2x - 5 | Read constant b directly | (0, -5) |
| Standard Form | 3x + 4y = 12 | Set x=0 ⟹ 4y=12 | (0, 3) |
| Point-Slope Form | y - 2 = 3(x - 1) | Set x=0 ⟹ y - 2 = -3 | (0, -1) |
| Coordinate Graph | Line crossing (0, -2) | Visual intersection on y-axis | (0, -2) |
Common Trap: Swapping X-Intercept and Y-Intercept!
Y-intercept: line crosses the y-axis* (x = 0) ⟹ (0, b).
X-intercept: line crosses the x-axis* (y = 0) ⟹ (a, 0).
Always double check that your zero is in the first slot for the y-intercept!