#### Definitions

**Tangent Line** - A straight line that passes through a point on a function and has the same slope as that function at that point.

**Normal Line** - A straight line that passes through a point on a function is perpendicular to the function at that point.

A common question is to find an equation for the normal or tangent line at a given point on a function. Since the equation is linear the answer will take the form of:

(1)If you are looking for a **tangent line** then *m* will simply be the value of the derivative at the given point on the function.

If you are looking for a **normal line** then *m* will be the **negative reciprocal** of the value of the derivative at the given point on the function

The *y-intercept* can then be found using the coordinates of the given point on the function.

##### Example

Find equations for both the normal and the tangent lines on the function $f(x)=2x^2+x$ at the point where $x=2$.

**Let's work on the tangent line first:**

The derivative of $f(x)$ is $f'(x)=4x+1$ so $f'(x=2)=9$. This gives us an equation for the tangent line:

(2)We want the line to be tangent when $x=2$ which occurs at the point $(2,10)$. Use those values in the equation above.

(3)Solving for *b* we get a final equation for the tangent line of $y=9x-8$

**Now on to the normal line.**

The slope of the normal line will be the negative reciprocal of the slope of the tangent line. So we can write an equation for the tangent line as:

(4)Again this will go through the point $(2,10)$. Using these point to solve for *b* we can write an equation for the normal line as:

A good explanation; however, I don't follow how you ended up with 46/45 on the last stage. Please could you explain. Thank you

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