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Equation Families

by Tim Isbell, October 2015

Several elements of this chart are valuable for Algebra 1, more so geometry students, and is absolutely essential to pass the first semester of Algebra 2. It shows a wide variety of parent equations in both their graphing and standard forms. 

If y = f(x) is the Parent, the Graphing Form for its family of functions is: y = a · f(x − h) + k. The point (h, k) is the locator point that helps you place the graph like this:
  • Horizontally translated by the value of h
  • Vertically translated by the value of k
  • Vertically stretched if the absolute value of a is greater than 1
  • Vertically compressed if the absolute value of a is less than 1
  • Reflected across the x‑axis if a is negative



 Graphing Form

 Standard Form

 y = x


 y = a(x − h) + k

point slope form:        (y-y1) = m(x-x1)

 Ax + By = C

 y = |x|

 Absolute Value

 y = a|x - h| + k


 y = x2


 y = a(x − h)2 + k

 y = ax2+ bx + c

To convert Standard to Graphing Form, complete the square. See below.

 y = x3


 y = a(x − h)3 + k

 y = ax3 + bx2 + cx + d

 y = xn

 nth degree polynomial

 y = a(x - h)n + k

 y = a0xn + a1xn-1 +... + an

 y = 1/x


 y = a/(x - h)  + k


 y = x)

 Square Root

 y = ax-h) + k


 y = bx


 y = ab(x − h) + k


 y = logbx

 Logarithmic (inverse of Exponential)

 y = a logb (x-h) + v


 x2  + y2 = r2


 (x − h)2 + (y − k)2 =  r2

 x2 + y2 + Cx + Dy + E= 0

It is especially useful to “play” with these equations using an online graphing calculator such as Desmos ( because you can parameterize the constants and vary them to get an intuitive grasp of how they impact the graph. This is valuable in engineering solutions when you need to create an equation that models your data.

Converting a quadratic equation from Graphing Form to Standard Form, and back again:

I stared at the quadratic's graphing form and standard form for a while, knowing that the "a" value in one was the same as the "a" value in the other. So there had to be a way to convert an equation from one form to another. So I started with a quadratic in the Graphing Form, graphed it, then converted it to the Standard Form. Now the question was, "can I get it back to the Graphing Form." I knew the answer had to be yes, but it was worth actually plugging through it. Here's the work.
For a larger view of the hand analysis below, just click on it.

You can click on Equation Families for a Google Doc version of this whole web page.

Tim Isbell, 10/21/2015

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