Calculus Definitions > Standard Form

Contents:

## What is standard form?

The **standard form **of any line in a Cartesian plane is **Ax + By = C** where:

- A, B, and C are placeholders for constants (at least one of A and B must be nonzero),
- x and y are variables.

This form differs from the “usual” *slope-intercept form *you come across in calculus: y = mx + b or the *point-slope form* (which is really just a rearrangement of slope-intercept): y – y_{1} = m (x – x_{1}).

As a quick check of your understanding, which two of the following equations are in standard form?

See the solution here.

## Example of Rewriting in Standard Form

**Example Question**: Rewrite y = 3x – 4 in standard form.

Step 1: Flip the equation (this gets the Ax term in the right place):

3x – 4 = y

Step 2: Subtract “y” from both sides:

3x – y – 4 = 0

Step 3: Add 4 to both sides:

3x – y = 4

That’s it!

**More examples**:

## Why do we need it?

Sometimes, it’s easier to work with equations that are standardized. For example, it’s easier to identify the center of a circle (h, k) and the radius (r) if the equation is in standard form: (x – h)^{2} + (y – k)^{2} = r^{2}. Compare that to the general form x^{2} + y^{2} + Dx + Ey + F = 0, where D, E, F are constants.

On the other hand, the slope and y-intercept of a line are easier to see in slope-intercept form, so that’s it’s handy to be able to switch between the different forms.

In calculus, you’ll mostly be working with the standard generalized form of the linear equation (Ax + Bx + C). However, you’ll occasionally need some others. They include:

For a **parabola**, the standard form is either:

- (x – h)
^{2}= a (y – k), or - (y – k)
^{2}= a(x – h).

**Ellipsoid**:

Two of the equations in the example above are in standard form: 1 and 4:

## References

Lazari, A. (2020). Equation of a Circle. Retrieved October 20, 2020 from: https://mypages.valdosta.edu/alazari/math1111/Circle.html

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