Share ThisHow to Graph Parent Functions
There are a handful of common families of functions that you see over and over again in many areas of mathematics. Some of the most common are quadratic and square root functions, cubic and cube root functions, and absolute value functions. The simplest version of each of these function groups are sometimes referred to as parent functions — because they can be altered to produce their slightly more difficult offspring functions. You learn about how to transform a parent function into an offspring function in the article, “How to Transform Functions.” You should know the parent functions by heart because of the central role they play in countless problems.
Quadratic functions (or parabolas)
Quadratic functions are equations in which the second power, or square, is the highest power to which the variable is raised.
The equation
is a quadratic function and is the parent function for all other quadratic functions like f(x) = x2 – 5x +12 or f(x) = 5x2 – 1.
The graph of any quadratic function is called a parabola. All parabolas have the same basic cup shape. The simple parent parabola you see above has its vertex at (0, 0), the origin. Note the nice pattern of how this parabola goes up from the vertex: going left or right, each time you go across 1, you go up 1, then 3, then 5, then 7 etc. For example, you go from (0, 0) to (1, 1) — which is over 1 up 1 — then from (1, 1) to (2, 4) — which is over 1 up 3, etc.
The offspring of this simple parabola will all have the same basic cup shape, but they could have their vertex at any point of the coordinate plane, they could be narrower or wider than the parabola above, and they might open down instead of up.
Note that the above function is an even function which means that it’s symmetric with respect to the y-axis, or, in other words, that the left and right halves of the parabola are mirror images of each other.
Square root functions
A square root graph is related to a quadratic graph; the quadratic graph is
while the square root graph is
You can also write the square root function as
The graph of the above square-root function looks like the right half of the parent parabola that has been flipped over the imaginary line y = x. This fact makes these two functions inverses of each other.
This parent square root graph starts at the origin (0, 0) and then goes over 1 up 1 to (1, 1), then over 3 up 1 to (4, 1), then over 5 up 1 to (9, 3), etc.
Cubic functions
In a cubic function, the highest power on any variable is three.
is the parent function. It’s the parent of all cubic function such as . All cubic functions have the same basic “S” shape as the above parent graph, but the offspring might be narrower or wider, the centers of the S might be somewhere other than the origin, and the S shape might go like a regular S (as opposed to the above graph which curves like a backward elongated S).
The above cubic function is an odd function which means that it’s symmetric with respect to the origin; or, in other words, if you rotate the graph 180 degrees, you’ll get the exact same graph.
Cube root functions
Cube root functions are related to cubic functions in the same way that square root functions are related to quadratic functions: Namely, that they’re inverses of each other. In other words, the cube root function graph above is the cubic function graph flipped over the line y = x. You write cubic functions as f(x) = x1/3 or f(x) = 3√x.
Like the parent cubic function, f(x) = x3, the above cube root function is an odd function.
Absolute value functions
The parent of all absolute value functions is shown above. Its equation is . Its symmetry with respect to the y-axis makes it an even function. All offspring of this function have this same basic V shape, but they may be narrower or wider, they might have their vertex someplace other than the origin, and they might open downward rather than upward.
