EQUATIONS OF TANGENT LINES AND NORMAL LINES
Suppose we have a point  x1 , y1  and a slope m of a line. Then the equation of the line through
the point with that slope is
 y  y1   m  x  x1  .
You should remember this formula from algebra. It’s the point-slope form of the equation of a
line. If you don’t remember this formula, memorize it!
Next, suppose that we have an equation y  f  x  , where  x1 , y1  satisfies that equation. Then,
f   x   m , and we can plug all our values into the equation for a line and get the equation of the
tangent line.
Example 1
Find the equation of the tangent line to the curve y  5 x 2 at the point  3, 45 .
Example 2
Find the equation of the tangent line to the curve y  x3  x 2 at  3,36
Naturally, there are a couple of things that can be done to make the problems harder. First, you
can be given only the x-coordinate. Second, the equation can be more difficult to differentiate.
In order to find the y-coordinate, all you have to do is plug in the x value into the equation for the
curve and solve for y.
Example 3
Find the equation of the tangent line to y 
2x  5
at x  1 .
x2  3
Sometimes, instead of finding the equation of a tangent line, you will be asked to find the
equation of the normal line. A normal line is simply the line perpendicular to the tangent line at
the same point. You follow the same steps as with the tangent line, but you use the slope that
will give you a perpendicular line. Remember what that is? It’s the negative reciprocal of the
slope of the tangent line.
Example 4
Find the equation of the normal to y  x5  x 4  1 at x  2 .
Example 5
Find the equations of the tangent and normal lines to the graph of y 
10 x
at the point  2, 4  .
x2  1
Example 6
The curve y  ax 2  bx  c passes through the point  2, 4  and is tangent to the line y  x  1 at
the point  0,1 . Find the values of a, b, and c.
Example 7
Find the points on the curve y  2 x3  3x 2  12 x  20 where the tangent is parallel to the x-axis.
PRACTICE
1.
Find the equation of the tangent to the curve y  3x 2  x at x  1 .
2.
Find the equation of the tangent to the curve y  x3  3x at x  3 .
3.
Find the equation of the normal to graph of y  8 x at x  2 .
4.
Find the equation of the tangent to the graph of y 
5.
Find the equation of the normal to the graph of y 
6.
Find the equation of the tangent to the graph of y  4  3x  x 2 at the point  0, 4 .
7.
Find the equation of the tangent to the graph of y  2 x3  3x 2  12 x  20 at x  2 .
8.
Find the equation of the tangent to the graph of y 
9.
Find the equation of the tangent to the graph of y  x3  15 at  4, 7  .
10.
Find the equation of the tangent to the graph of y   x 2  4 x  4  at x  2 .
11.
Find the values of x where the tangent to the graph of y  2 x3  8 x has a slope equal to
1
x 7
2
at x  3 .
x3
at x  4 .
x3
x2  4
at x  5 .
x6
2
the slope of y  x .
3x  5
at x  3 .
x 1
12.
Find the equation of the normal to the graph of y 
13.
Find the values of x where the normal to the graph of y   x  9  is parallel to the y-axis.
14.
Find the coordinates where the tangent to the graph of y  8  3x  x 2 is parallel to the
2
x-axis.
15.
Find the values of a, b, and c. where the curve y  x 2  ax  b and y  x 2  cx have a
common tangent line at  1, 0 .
ANSWERS
1.
y  2  5  x 1
or
y  5x  3
2.
y 18  24  x  3
or
y  24 x  54
3.
y  4    x  2
or
y  x  6
4.
y
5.
y7 
6.
y  4  3  x  0
or
y  3x  4
7.
y0
8.
y  29  39  x  5
9.
y7 
10.
y0
11.
x
3
6

2
2
12.
y7 
1
 x  3
2
13.
x 9
14.
 3 41 
 ,

 2 4 
15.
a  1 , b  0 , and c  1
1
3
   x  3
4
64
1
 x  4
6
24
 x  4
7