What is the point-slope form of a linear equation?

The point-slope form of a linear equation is a way to represent a straight line using two points or a point and a slope. It's particularly useful when you want to find the equation of a line given a point it passes through and its slope. Here's how it's derived and used:

  1. Starting with two points: Given two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) on the line, the slope (mm) of the line can be calculated using the formula:
m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
  1. Using point-slope form: Once you have the slope, you can use the point-slope form of a linear equation to write the equation of the line. The point-slope form is given by:
y−y1=m(x−x1)y - y_1 = m(x - x_1)

where (x1,y1)(x_1, y_1) is a point on the line (called the "point" in point-slope form), and mm is the slope of the line.

  1. Finding the equation of a tangent: In the context of calculus, you might use the point-slope form to find the equation of the tangent line to a curve at a specific point. To do this, you'll first need to find the slope of the tangent (which is the derivative of the function at that point) and then use one of the points on the curve as (x1,y1)(x_1, y_1) in the point-slope form.