Point Slope Form Calculator

What is Point Slope Form?

Point slope form is a method of expressing the equation of a linear function using a point on the line and its slope. This form is particularly useful in algebra and calculus when analyzing linear relationships, especially when given a specific point and the rate of change (slope) of the line.

How to Calculate the Point Slope Form

To determine the equation of a line using point slope form, follow these steps:

1. Identify a point \((x_1, y_1)\) that the line passes through
2. Determine the slope \(m\) of the line
3. Apply these values to the point slope form equation
4. Simplify the equation if necessary

Formula

The point slope form of a line is expressed as:

\[ y - y_1 = m(x - x_1) \]

Where:

• \((x_1, y_1)\) is a known point on the line
• \(m\) is the slope of the line
• \(x\) and \(y\) are variables representing any point on the line

Calculation Steps

Let's walk through an example to illustrate the process:

Given: A line passes through the point (2, 5) and has a slope of 3.

1. Identify the point: \((x_1, y_1) = (2, 5)\)
2. Note the slope: \(m = 3\)
3. Substitute these values into the point slope form equation: \[ y - 5 = 3(x - 2) \]
4. This equation is already in point slope form, but we can expand it if desired: \[ y - 5 = 3x - 6 \] \[ y = 3x - 1 \]

Example and Visual Representation

Let's visualize the line \(y - 5 = 3(x - 2)\) on a coordinate plane:

In this graph:

• The blue line represents the equation \(y - 5 = 3(x - 2)\)
• The red point (2, 5) is the point we used in our equation
• The green triangle illustrates the slope of 3, showing a rise of 3 units for every 1 unit run

This visual representation helps us understand how the point and slope define the line in point slope form.