Y intercept

The y-intercept of a line is the point where the line crosses the y-axis on a graph. At this point, the x-coordinate is always 0.

Definition in a Formula:

For a linear equation in the slope-intercept form: y=mx+by = mx + b

• mm is the slope of the line.
• bb is the y-intercept.

The value of bb represents the y-coordinate of the point where the line crosses the y-axis.

Example:

1. For the equation y=2x+3y = 2x + 3:
   • The y-intercept is 33, so the line crosses the y-axis at the point (0,3)(0, 3).
2. For y=−4x−2y = -4x – 2:
   • The y-intercept is −2-2, so the line crosses the y-axis at (0,−2)(0, -2).

To find the y-intercept of a line, follow these steps depending on the information or equation provided:

How to Find Y-Intercept

1. From the Equation of a Line

For a linear equation in the general form y=mx+by = mx + b:

• The y-intercept is the constant bb.
  • Example: In y=2x+5y = 2x + 5, the y-intercept is 55 (0,50, 5).

For a linear equation in standard form Ax+By=CAx + By = C:

1. Set x=0x = 0 in the equation.
2. Solve for yy.
   • Example: In 2x+3y=62x + 3y = 6:
     • Set x=0x = 0: 3y=63y = 6
     • y=2y = 2
     • The y-intercept is (0,2)(0, 2).

2. From a Graph

1. Locate where the line crosses the y-axis (vertical axis).
2. The y-coordinate of this point is the y-intercept.
   • Example: If the line crosses the y-axis at (0,−3)(0, -3), the y-intercept is −3-3.

3. From Two Points

If you have two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2):

1. Find the slope mm: m=y2−y1x2−x1m = \frac{y_2 – y_1}{x_2 – x_1}
2. Use the point-slope form: y−y1=m(x−x1)y – y_1 = m(x – x_1)
3. Solve for y=mx+by = mx + b, where bb is the y-intercept.
   • Example: Points: (1,2)(1, 2) and (3,6)(3, 6)
     1. Find the slope: m=6−23−1=42=2m = \frac{6 – 2}{3 – 1} = \frac{4}{2} = 2
     2. Use point-slope form with (1,2)(1, 2): y−2=2(x−1)y – 2 = 2(x – 1)
     3. Simplify: y=2x−2+2  ⟹  y=2xy = 2x – 2 + 2 \implies y = 2x Here, the y-intercept is 00 ((0,0)(0, 0)).

1. To determine the y-intercept of a linear function using the slope and a given point:

1. Identify the Slope and Point: Start by noting the slope (mm) and a specific point on the line ((x1,y1)(x_1, y_1)).
2. Write the Linear Equation: Use the slope-intercept form of the equation: y=mx+by = mx + b
3. Substitute the Known Values: Replace xx, yy, and mm with the values from the given point and slope.
4. Solve for the Y-Intercept (bb): Rearrange the equation to isolate bb.

Example:

Given the point (2,17)(2, 17) and a slope of 3:

1. Start with the equation y=mx+by = mx + b.
2. Substitute the slope (m=3m = 3) and the point (x=2,y=17x = 2, y = 17): 17=3(2)+b17 = 3(2) + b
3. Solve for bb: 17=6+b17 = 6 + b b=11b = 11

The y-intercept is 11, so the line crosses the y-axis at (0,11)(0, 11).

2. To find the y-intercept of a linear function using two points from a table or graph:

1. Identify Two Points: Record the coordinates of the two points as (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2).
2. Calculate the Slope (mm): Use the formula: m=y2−y1x2−x1m = \frac{y_2 – y_1}{x_2 – x_1} This determines the “rise” (difference in yy-coordinates) divided by the “run” (difference in xx-coordinates).
3. Write the Linear Equation: Start with the slope-intercept form: y=mx+by = mx + b
4. Substitute Known Values: Use one of the points and the calculated slope to replace xx, yy, and mm.
5. Solve for the Y-Intercept (bb): Rearrange the equation to isolate bb.

Example:

Given the points (2,4)(2, 4) and (6,12)(6, 12):

1. Calculate the slope: m=y2−y1x2−x1=12−46−2=84=2m = \frac{y_2 – y_1}{x_2 – x_1} = \frac{12 – 4}{6 – 2} = \frac{8}{4} = 2
2. Write the equation: y=mx+by = mx + b
3. Substitute one point, such as (2,4)(2, 4), and the slope (m=2m = 2): 4=2(2)+b4 = 2(2) + b
4. Solve for bb: 4=4+b4 = 4 + b b=0b = 0

The y-intercept is 0, meaning the line passes through the origin (0,0)(0, 0).