Tangents and normals
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How to calculate the equations of the normal and tangent lines to a function at a given point.

This topic is part of the TCS FREE high school mathematics 'How-to Library', and will help you to calculate the equations of the normal and tangent lines to a function at a given point.
(See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions.

Theory:

Consider the function: y = f(x), with point (x1,y1) lying on the function graph. The tangent line to the function at  (x1,y1) is the straight line that touches y = f(x) at that point. Both the graph of y = f(x) and the tangent line pass through the point, and the tangent line has the same gradient, 'm',  as the function at that point.

The normal line to function y = f(x) at the point (x1,y1) is the straight line that passes through the point making a 90º angle with the graph. The gradient of the normal line is -1/m, where 'm' is the gradient of the tangent line at the same point.

For example, consider the function y = x2. The tangent and normal lines at the point (1,1) are shown on the diagram below:

The equation of the tangent line to y = f(x) at the point (x1,y1):   
( 'm' is the gradient at (x1,y1) )

The equation of the normal line to y = f(x) at (x1,y1) is:

The derivative of y = f(x) at (x1,y1) gives us the gradient 'm'.

For example: Calculate the tangent and normal lines to the function: y = x2, when x = 1.

SolutionPart 1, tangent line.

At x = 1, y = 12 = 1. So (x1,y1) = (1,1)

The gradient 'm' at x = 1 is found by calculating the derivative of the function at that point. For y = x2, dy/dx = 2x. At x = 1, dy/dx = 2×1 = 2. So 'm' = 2.

The tangent line is given by: 

                                                                         

        Substituting, we have:      y = 2(x - 1) + 1

Expanding and simplifying:      y = 2x - 2 + 1

                                              so:       y = 2x - 1

SolutionPart 2, normal line.

At x = 1, the gradient of the tangent, 'm' = 2. (See above.)

The normal line has a gradient of  -1/m-1/2 , and so has this equation: 

                                                                         

        Substituting, we have:      y = -1/2(x - 1) + 1

Expanding and simplifying:      y = -0.5x + 0.5 + 1

                                              so:       y = -0.5x

Maths Helper Plus is able to find the tangent and normal lines to a function y = f(x) at a given point. It can graph the function and the lines, as well as display the working steps for the calculations.

Download the free support file... We have created a Maths Helper Plus document containing the completed example from this topic. You can use this to practice the steps described below, and as a starting point for solving your own problems.

File name:  'Tangents and normals.mhp'   File size: 4kb
Click here to download the file.

If you choose 'Open this file from its current location', then Maths Helper Plus should open the document immediately. If not, try the other option: 'Save this file to disk', then run Maths Helper Plus and choose the 'Open' command from the 'File' menu. Locate the saved file and open it. If you do not yet have Maths Helper Plus installed on your computer, click here for instructions.

Method:

NOTE: We will use the example function from the 'theory' section above: y = x2, and find the tangent and normal lines at x = 1. To solve other similar problems, substitute your own functions and 'x' values.

Step 1  Start with an empty Maths Helper Plus document

If you have just launched the software then you already have an empty document, otherwise hold down ‘Ctrl’ while you briefly press the ‘N’ key.

Step 2  Graph the function y = f(x)

1. Press the F3 key to activate the 'input box' for typing (see below):

2. Type the function into the input box:

             

3. Press Enter to complete the entry

For more help on entering functions, see the 'Easy Start' tutorial in the Maths Helper Plus 'help'. To access the tutorial, click 'Help' on the Maths Helper Plus menu bar, then select: 'Tutorial...'

If the required part of the function graph does not display,  you need to adjust the graph scale. 
First reduce the graph scale. Press the F10 key enough times until the main parts of the graph are visible.
To enlarge the graph, hold down 'Ctrl' while you press F10.

For more help on setting graph scales, click 'Help' on the Maths Helper Plus menu bar, then select 'Index'. Click the button at the top left corner of the help screen. Click 'Training', then 'Essential skills', then 'Graph scale magic'. 

 

 Step 3  Calculate and plot the tangent and normal lines

Carefully point to the function curve with the mouse pointer. Double click to display the options dialog box for that function. (This may work better on parts of the graph that are not so steep.)

Click the 'Tangents & Normals' tab at the top of the dialog box. See below:

To find a tangent to your function, click on the 'Tangent line...' edit box and type the 'x' value at the tangent.

TIP: To type several 'x' values at once, simply separate them with commas.

Select the 'expanded working' option to display working steps for calculating the tangent line.

Click the 'plot tangents' button to plot your tangent line(s) on the graph.

The settings shown on the dialog box above display this working for calculating the tangent line:

TANGENT line at x = 1

 At x1 = 1, y1 = 1

 and the gradient, m = dy/dx = 2

 The equation of the tangent at (x1,y1) is given by:

y - y1 = m(x - x1)

  so y = m(x - x1) + y1

       = mx - mx1 + y1

       = mx + (y1 - mx1)

       = 2x + (1 - 2 × 1)

      so y = 2x - 1

To find a normal to your function, click on the 'Normal line...' edit box and type an 'x' value. The procedure is exactly the same as for calculating and plotting the tangent line.

The settings shown on the dialog box above display this working:

NORMAL line at x = 1

 At x1 = 1, y1 = 1

 The gradient of the normal line is:

 'M' = -(1/m) = -(1/2) = -0.5

The equation of the normal at (x1,y1) is given by:

y - y1 = M(x - x1)

  so y = M(x - x1) + y1

       = Mx - Mx1 + y1

       = Mx + (y1 - Mx1)

       = -0.5x + (1 - -0.5 × 1)

      so y = -0.5x + 1.5

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