# Tangent Line to a Graph

How to find the tangent line to a given graph: formula, examples, and their solutions.

## Formula

The tangent line to *y* = *f*(*x*) at *x* = *a* is*y* = *f*'(*a*)(*x* - *a*) + *f*(*a*).

The slope of the tangent line is *f*'(*a*).

And the tangent line is on (*a*, *f*(*a*)).

Then, the tangent line in point-slope form is*y* = *f*'(*a*)(*x* - *a*) + *f*(*a*).

## Example 1

Put 1 into *f*(*x*).

Then *f*(1) = 4.

So the tangent line is on (1, 4).

Find the slope of the tangent line.*f*'(*x*) = 3*x*^{2} - 8*x*

Derivatives of polynomials

So *f*'(1) = -5.

*f*'(1) = -5

(1, 4)

Then the tangent line is *y* = -5(*x* - 1) + 4.

Point-slope form

## Example 2

Put 0 into *f*(*x*).

Then *f*(0) = 1.

So the tangent line is on (0, 1).

Find the slope of the tangent line.*f*'(*x*) = *e*^{x}

Derivative of *e*^{x}

So *f*'(0) = 1.

*f*'(0) = 1

(0, 1)

Then the tangent line is *y* = 1(*x* - 0) + 1.

Point-slope form