Transform Upside-Down: Reflect Polynomials About x-Axis

(Video) Graph Transformations About the $x$- and $y$-Axis

by PatrickJMT

The above video by PatrickJMT describes how to reflect about the x-axis and y-axis.

Summary

To reflect a polynomial over the $x$-axis, we multiply the entire polynomial by $-1$.

$$y=-f(x)$$

The negative sign in front of the entire polynomial flips the graph about the $x$-axis.

(Video) Reflections

by MrMaisonet

In the above video by MrMaisonet it is shown an intuitive example of what is happening with reflections. Triangles are used, but this can be generalized to functions, including polynomials.

In this next lesson in the series of polynomial transformations, we will talk about reflecting polynomials about $x$-axis. We've already learn about shifting from right to left, shifting up and down, and reflections.

Reflecting Polynomials About the $x$-Axis

When we reflect a polynomial function about the $x$-axis, we are essentially flipping the function upside down! That means that the $y$-values get negated. So the points on top of the $x$-axis flip to the bottom of the $x$-axis, and vice versa.

$$f(x)=-f(x)$$

Example

Say we want to reflect the parent square root function, $y=\sqrt{x}$, over the $x$-axis.

If we want to flip this function over the $x$-axis, we negate all $y$ values.

$$\begin{align} (0,0) \rightarrow (0,0)\\ (1,1) \rightarrow (1,-1)\\ (4,2) \rightarrow (4,-2)\\ (9,3) \rightarrow (9,-3)\\ \end{align}$$

As we can see from the second graph, the function shape did not change, and it was not translated horizontally or vertically. The only feature that changed are the $y$-values of the function.

Reflect Polynomials About the $x$-Axis

When we talk about reflecting a polynomial about the x-axis, we mean flipping it upside-down. This happens when we take each point on the graph of the polynomial and make its y-coordinate the opposite of what it was. For example, if we have a point at $(3,4)$, it will move to $(3,-4)$. To do this to the whole graph, we multiply the polynomial by $-1$.

This changes the shape of the graph. If the polynomial was going upwards, it will now go downwards, and if it was going down, it will go upwards. Remember that this reflection doesn't change the $x$-coordinates, only the $y$-coordinates. So, when you reflect a polynomial about the $x$-axis, you're flipping the graph up to down, but it still passes through the same $x$-values. This is useful when you're analyzing how a function behaves or comparing different graphs.