**The quadratic approximation is one way to approximate a curve.** Quadratic polynomial approximations are specific examples of a useful class of quadratic approximations called **Taylor polynomials**.

The basic idea is that you want to approximate a function with a parabola.

## General Form of Quadratic Approximation

The general form of a quadratic approximation is:

If it looks complicated, don’t worry: you don’t have to *solve *the equation; all you have to do is plug in some terms.

Let’s say you were trying to approximate a function at x = 1. The formula is basically saying to find three values at point x = 1 and add them up:

- f: The function at x = 1
- f′: The first derivative at x = 1
- f′′: The second derivative at x = 1.

## Quadratic Approximation: Example

**Example problem**: Find the quadratic approximation for f(x) = xe^{-2x} near x = 1

Step 1: **Find the first derivative of the function**. Use the product rule for this function (with x and e^{-2x}) and then the chain rule (for e^{-2x}):

f′(x) = e^{-2x} – 2xe^{-2x} = e^{-2x}(1 – 2x).

Step 2: **Find the second derivative of the function**. In other words, find the derivative of the derivative you calculated in Step 1.

f′′(x) = -2e^{-2x} – 2(e^{-2x} – 2xe^{-2x})

= -4e^{-2x} + 4xe^{-2x}

= -4e^{-2x}(1 – x).

Step 3: **Find values at x = 1 for the function, and the first and second derivatives** you calculated in Steps 1 and 2:

- f(1) = (1)e
^{-2(1)}= e^{-2} - f′(1) = e
^{-2(1)}(1 – 2(1)) = -e^{-2} - f′′(1) = -4(1)
^{-2(1)}(1 – (1)) = 0.

Step 4: **Take the three values you calculated** in Step 3 and **insert them into the general formula **. Note that the second derivative (from Step 3) was zero, so we can ignore the third part of the formula:

f(x) ≈ f(1) + f′(1)(x – 1)

≈ e^{-2} – e^{ – 2}(x – 1).

*That’s it!*

## Use of Quadratic Approximation

In addition to modeling functions, approximations are used to:

- Study asymptotic behavior,
- Evaluate definite integrals (i.e. integrals that have a defined starting and stopping point),
- Understanding the growth of functions,
- Solve differential equations.

**Next**: How to use Taylor polynomials to approximate a function.