X and Y Intercept Calculator

How to Calculate X and Y Intercepts

The x and y intercepts are the points where a line crosses the x-axis and y-axis, respectively. They are important in understanding the behavior of linear functions.

Intercept Formulas

For a line in slope-intercept form \( y = mx + b \):

• Y-intercept: \( b \) (where the line crosses the y-axis)
• X-intercept: \( -\frac{b}{m} \) (where the line crosses the x-axis)

Where:

• \( m \) is the slope of the line
• \( b \) is the y-intercept

Calculation Steps

1. If given slope and y-intercept:
   • The y-intercept is already given as \( b \)
   • Calculate x-intercept using \( -\frac{b}{m} \)
2. If given two points:
   • Calculate the slope using \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
   • Use point-slope form to find y-intercept: \( y - y_1 = m(x - x_1) \)
   • Rearrange to slope-intercept form \( y = mx + b \) to find \( b \)
   • Calculate x-intercept using \( -\frac{b}{m} \)

Example Calculation

Let's find the x and y intercepts of the line passing through points (1, 3) and (4, 9).

1. Calculate the slope: \[ m = \frac{9 - 3}{4 - 1} = \frac{6}{3} = 2 \]
2. Use point-slope form with (1, 3): \[ y - 3 = 2(x - 1) \]
3. Rearrange to slope-intercept form: \[ y = 2x - 2 + 3 = 2x + 1 \]
4. Y-intercept is 1
5. Calculate x-intercept: \[ x = -\frac{1}{2} = -0.5 \]

Therefore, the y-intercept is (0, 1) and the x-intercept is (-0.5, 0).

Visual Representation

A visual representation can help understand these concepts better. Here's a diagram showing the line y = 2x + 1 with its x and y intercepts:

This diagram illustrates the line y = 2x + 1, showing its x-intercept at (-0.5, 0) and y-intercept at (0, 1).