How to calculate the equations of the normal and tangent lines to a function at a given point.
Consider the function: y = f(x), with point (x1,y1) lying on the function graph. The tangent line to the function atÂ (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',Â 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):Â Â Â
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.
Solution:Â Part 1, tangent line.
Solution:Â Part 2, normal line.
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.
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 1Â 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 2Â Graph the function y = f(x)
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,Â you need to adjust the
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'.Â
Â Step 3Â 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:
Â At x1 = 1, y1 = 1
Â and the gradient, m = dy/dx = 2
Â The equation of the tangent at (x1,y1) is given by:
y - y1 = m(x - x1)
Â so y = m(x - x1) + y1
Â Â Â Â Â Â = mx - mx1 + y1
Â Â Â Â Â Â = mx + (y1 - mx1)
Â Â Â Â Â Â = 2x + (1 - 2 × 1)
Â Â Â Â Â 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:
Â At x1 = 1, y1 = 1
Â The gradient of the normal line is:
Â 'M' = -(1/m) = -(1/2) = -0.5
The equation of the normal at (x1,y1) is given by:
y - y1 = M(x - x1)
Â so y = M(x - x1) + y1
Â Â Â Â Â Â = Mx - Mx1 + y1
Â Â Â Â Â Â = Mx + (y1 - Mx1)
Â Â Â Â Â Â = -0.5x + (1 - -0.5 × 1)
Â Â Â Â Â so y = -0.5x + 1.5