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 – y1 = m (x – x1).
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
Why Do We Need Standard Form?
Sometimes, it’s easier to work with equations that are in a standard form. 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 = r2. Compare that to the general form x2 + y2 + Dx + Ey + F = 0, where D, E, F are constants.
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).
Lazari, A. (2020). Equation of a Circle. Retrieved October 20, 2020 from: https://mypages.valdosta.edu/alazari/math1111/Circle.html
Stephanie Glen. "Standard Form: Simple Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/standard-form/
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