How to Find Tangent Line of Parabola - Tutorial

Definition:

In a parabola, two tangent lines in a graph meets at a point which is horizontally equidistant from the tangent points.

Example:

Draw the tangent line for the equation, y = x² + 3x + 1 at x=2

Given:

Equation = x² + 3x + 1 x = 2

Solution:

Step 1:

To find the y value, substitute the x value in given equation. y = 2² + 3(2) + 1 y = 4 + 6 + 1 y = 11

Step 2:

Differentiate the given equation, y = x² + 3x + 1 dy/dx = d(x² + 3x + 1)/dx dy/dx = 2x+3

Step 3:

Now, substitute x value in the above result. dy/dx = 2x+3 dy/dx = 2(2)+3 dy/dx = 7 Consider the above value as m, i.e. m = 7.

Step 4:

Substitute m value in the tangent line formula . tangent line => y-y0=m(x-x0), (x0,y0)=(2,11) y-11=7(x-2) y=7x-7(2)+11 y=7x-14+11 y=7x-3 (which is the equation of tangent line)

Step 5:

Now, substitute x value from -5 to +5 in the given equation to draw the parabola. For, x=-5 => y=-5²+3(-5)+1 = 25-15+1 = 16 For, x=-4 => y=-4²+3(-4)+1 = 16-12+1 = 5 Likewise, calculate the balance points. (-5,16),(-4,5),(-3,1),(-2,-1),(-1,-1),(0,1),(1,5),(2,11),(3,19),(4,29),(5,41)

Step 6:

Substitute x value from 0 to given x*2 in the tangent line equation. tangent line => y-y0=m(x-x0), (x0,y0) For, x=0 => y=7(0)-4 = -4 For, x=1 => y=7(1)-4 = 3 Now, calculate the balance points. (0,-4),(1,3),(2,10),(3,17),(4,24)

Step 7:

Now, draw the parabola and tangent line with these points