## 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