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Graphing Square Root & Cube Root Functions

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

This lesson will briefly review the most basic square root and cube root functions. Then we will look at transformations of these basic functions and their corresponding algebraic operations that we can use to graph more involved square root and cube root functions.

Basic Square Root and Cube Root Functions

Have you ever noticed how graphing basic functions within a family is fairly simple? We plot a few points and connect the dots accordingly. For instance, consider the families of square root functions and cube root functions.

A square root function is a function with the variable under the square root. Similarly, a cube root function is a function with the variable under the cube root. The most basic of these functions are √(x) and 3√(x), respectively. We can graph these basic functions by finding some points that satisfy each function, plotting them and then connecting the dots.


grphsqcu1


As we expected, that's not so hard, but what if we want to graph square root and cube root functions that are more involved than the basic ones? For instance, what if we wanted to graph y = 2√(x + 3) or y = -3√(x) - 4? We can still do this! The answer lies in transformations of functions. Let's explore!

Transformations of Functions

Notice that the two non-basic functions we mentioned are algebraic variations of the basic functions. These algebraic variations correspond to moving the graph of the function around in different ways, and they are called transformations.

There are four types of transformations:

  • Horizontal - This shifts the graph left or right. It corresponds to adding or subtracting a number, c, from x in the function. If we are adding c, we shift the graph c units to the left, and if we are subtracting c, then we shift the graph c units to the right. Example: The 3 added to x in y = 2√(x + 3) corresponds to shifting the graph of y = √(x) 3 units to the left.
  • Stretching/Shrinking - This stretches or compresses the graph vertically or horizontally. It corresponds to multiplying the whole function by a number c or just the x-variable by the number c. If we multiply the whole function by c, then we stretch the graph vertically by a factor of c if c > 1, and we compress the graph vertically by a factor of c if 0 < c < 1. If we multiply just the x variable by c, then we stretch the graph horizontally by a factor of c if 0 < c < 1, and we compress the graph horizontally by a factor of c if c > 1. Example: The 2 multiplied by √(x) in y = 2√(x + 3) corresponds with stretching the graph of y = √(x) vertically by a factor of 2.
  • Reflection - This reflects the graph over the x- or y-axes. It corresponds to multiplying by a negative. If we multiply the whole function by a negative, then we reflect the graph over the x-axis, and if we multiply just the x-variable by a negative, then we reflect the graph over the y-axis. Example: The negative in y = -3√(x) - 4 corresponds to reflecting the graph of y = 3√(x) over the x-axis.
  • Vertical - This shifts the graph up or down. It corresponds to adding or subtracting a number, c, from the function. If we add c to the function, then we shift the graph up c units. If we subtract c from the function, then we shift the graph down c units. Example: Subtracting 4 in y = -3√(x) - 4 corresponds with shifting the graph of y = 3√(x) down 4 units.


graphsqrcub2


Graphing Square Root and Cube Root Functions With Transformations

To graph non-basic square root and cube root functions, we can use the following steps:

  1. Identify the algebraic operations with their corresponding transformation
  2. Take the graph of the basic function through these transformations in the order horizontal, reflections, stretching/shrinking, vertical.

Let's use these steps and transformations to graph the non-basic functions we mentioned earlier. First, consider y = 2√(x + 3). The first step is to identify the algebraic operations with their corresponding transformations. We did this in the examples of horizontal transformations and stretching/shrinking transformations.


graphsqurcub3


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