# Tangent line to graph

## Summary and examples

We want to find an equation for the tangent line at a point of the graph of a function .
Before we go into this in more detail, we will first discuss the following general case: what is the equation of the line through a point with coordinates with slope .
The general equation of a line is: Because the slope is equal to is, we have: This line has to go through the point , thus we write: from which it follows: Now the formula for the line is: or: It is most convenient to just remember this formula and this is not difficult given the simple, symmetric shape.

Now we return to the equation of a tangent line at a point of the graph of the function . When we want to determine the tangent line at the point of the graph of the function , the slope can be found by: and thus de formula for the tangent line is: ##### Example 1

Determine the tangent line at the point of the graph of the function: First we calculate the derivative of the function: In the point we have and this is the slope of the tangent line at .

According to the formula given above the equation of the tangent line is: or: ##### Example 2

Where does the tangent line at the point of the graph of the function: intersect the -axis.

First we determine the equation of the tangent line in the point .

The derivative of the function is: In the point the derivative has the value , so the slope of the tangent line is . The tangent line goes through the point and thus the equation of the tangent line is:  The tangent line intersects the -axis in the point .

##### Example 3

Given the function: Determine the equation of the tangent line to the graph of this function, parallel to the line: The slope of the tangent line to the graph of the given function is found by differentiation: In a point  with the slope of the tangent line is: The line is parallel to the given line, so: or: From this we may conclude: or The corresponding tangent points are: or The tangent lines through these points are, respectively: or or: or 0