What Is a Polynomial Transformation?

We know that the standard form of a quadratic function is $ax^2+bx+c=0$, where $a$, $b$, and $c$ are coefficients. We know that the $c$ value dictates the vertex of the parabola, and $a$ tells you the opening direction and the stretching/compression of the parabola. We can manipulate these values to change how we want our parabola to look.

We can use this idea when considering polynomial transformations!

Polynomial Transformation

A polynomial transformation is when we alter a basic function to change the way it looks. Essentially, we are translating the function to a different position on a coordinate plane.

Let's look at an example.

The following is a graph of $y=x^2$.

We can move this function to the left! The equation is $y=(x+2)^2$.

Let's think about what would happen if we wanted to move this function up! It would look like the next graph! The equation is

So, as you can tell, you can move around your function on a graph using different transformations!

We can do the following transformations:

• Horizontal shift

• Vertical shift

• Reflection on the $x$-axis

• Reflection on the $y$-axis

The following lessons will consider these transformations in-depth!

(Video) Transformations on Polynomials

by Kyle Hall

In this video by Kyle Hall, you will learn about a few different polynomial transformations together.

Use this video as an introduction to polynomial transformations, but don't spend too much time on the details! We will break down the individual concepts in the following lessons.

Polynomials Transformations

Polynomials are expressions made up of terms with variables raised to powers, like $x^2$ or $3x^3$. A polynomial function is a graph that shows how these expressions change as you plug in different values for $x$. But sometimes, we don't just want the regular graph of a polynomial — we want to shift, stretch, or flip it. This is where polynomial transformations come in!

When we change a polynomial, we can move its graph up, down, left, or right. These changes are called translations. You can also dilate a graph vertically (stretch) or horizontally (compress). Lastly, if you multiply the entire function by a negative number, it flips or reflects the graph upside down. Similarly, changing the power of $x$ can change the shape of the graph, like when you move from a quadratic function to a cubic function.

Understanding how these transformations work helps you predict how the graph of a polynomial will look just by looking at its equation. In the next lesson, we’ll explore more specific examples of these transformations with different polynomials!