Tangent and Normal
Illustrating some examples to find the tangent or normal line to a function at a given point, and then using GraphFunc utility online to verify the result. You need Java Runtime Environment (JRE) to run GraphFunc applet in this website.
Summary
Finding a Tangent Line to a Graph
To find the tangent line to the curve y = f(x) at the point, we need to determine the slope of the curve. The slope of the curve can be found by taking the derivative, , of the curve and evaluating it at the point. The equation of the tangent line to the curve at the point is
.
Finding a

Example 1 Find
the equation of the tangent line to the curve
at the point x = 3.
Solution
· Given value:then .
· The slope of tangent line is
. =>.
· Substitute the values, and into the expression to obtain the tangent line, namely
or
.
Thus, the equation y = 4x – 8 is the tangent line to the curve, as shown in Figure 0 below.
Figure 0

1. Begin by going to the site http://graph.seriesmathstudy.com, and wait for the applet to be loaded. (Take a few seconds)
2. Enter the expression x^2 –2*x +1 in the text field or the command line with the label marked f(x) = .
3. Click on the Graph It! button to plot the graph.
4. Click on the Tangent button to display the Add Tangent/Normal window. From this window, enter the value x = 3 in the text field and then click on the Find button to find the tangent line as shown in Figure 1. Click on the Plot Tangent button to draw this tangent and close the window.
5. In addition, you can see the dynamic tangent line to be appeared by either clicking with the mouse anywhere on the graph or dragging it.
Figure 1

Example 2 Find
the equation of the normal line to the curve
at the point x = 3.
Solution
· Given value: then.
· The slope of tangent line is
=>.
· Substitute values, and into the expression to obtain the normal line, namely
or
. 
Figure 2 
Example 3 Find
the equation of the tangent line to the curve
at the point x = 1.
Solution
· Given value: then f(1) = 2.
· The slope of tangent line is
.
· Substitute the above values into the expression to obtain the tangent line, namely
or
y = 2. 
Figure 3 
Example 4 Find
the equation of the tangent line to the curve
at the point .
Solution
· We have .
· The slope of tangent line is
.
Therefore, .
· Substitute the above values into the expression to obtain the tangent line, namely
or
. 
Figure 4 
Example 5 Find
the equation of the normal line to the curve
through the point x = 1.
Solution
· Given value:=> f(1) = 1.
· The slope of tangent line is . => .
· Substitute the calculated values into the expression to obtain the normal line, namely
or
. 
Figure 5

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