Functions
A straight line is defined by two points, or by one point and the slope of the line. There is an ingenious formula that allows you to find the formula for a straight line using one point and the slope of the line:
Formula
Point-Slope Equation
The formula that defines the line with slope through the point is
Solve the equation with respect to and the expression looks like the function for a straight line .
Example 1
Find the function for the line through with a slope of 3
Put the numbers into the point-slope equation and solve for :
Example 2
Find the function for the straight line through the points and
First, you find the slope:
Then, select one of the points in the exercise and put it together with the slope into the point-slope equation:
Since , you know that the line passes through the origin. The function for the straight line is
If you know the function , you can use the point-slope equation to find the equation of the tangent line at a point on the graph of . This is because the slope of the tangent is equal to the value of the derivative of the function at the same point.
Formula
The Equation for an Arbitrary Tangent
where is a point on the tangent (often the point of tangency) and is the slope of the point. When using the formula, you must always solve for — that is, get alone on one side.
Example 3
Given the function , find the equation for the tangent at
To fill in the equation, you need values for and . You know that , so and . We begin by computing :
Now you put this into the equation and get
Example 4
Let . Find the equation for the tangent at .
You need the values of and . First, differentiate the function:
Now you can calculate . Since , you get
You find by putting into :
You put all this into the equation and get