Віддзеркалення графіку многочлена (відбиття відносно осі Oy)

(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.

The previous lesson talked about reflecting polynomial functions over the $x$-axis. This lesson will talk about the mirror transformation, or reflecting polynomials about the $y$-axis.

Reflecting Polynomials About the $y$-Axis

When we are reflecting polynomials about the $y$-axis, we are keeping the $y$-values the same, but we're negating the $x$-values. So the points on the right side of the $y$-axis flip to the left side of the $y$-axis.

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

Example

Let's take a look at the square root parent function, $y=\sqrt{x}$. Here's the graph of the parent function:

If we want to reflect this function over the $y$-axis, we want to negate the $x$ 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 shape of the function has stayed the same. Even the $y$ values of the function have stayed the same. The only things that changed are the $x$ values!

Reflecting Polynomials Over the $y$-Axis

When we reflect a polynomial over the y-axis, we flip it across the vertical line where $x=0$. This means that every point on the graph of the polynomial moves to the opposite side of the y-axis. For example, if a point on the graph is at $(2,3)$, after the reflection, it would move to $(-2,3)$.

They key idea is that reflection over the $y$-axis changes the signs of the odd powers of $x$, but it keeps the signs of the even powers the same. So, in the equation $y=x^2 + 3x-4$, the $x^2$ stays positive because $2$ is an even number, but the $3x$ changes to $-3x$ because $1$ is an odd number.

If the polynomial has only even powers of $x$ like $y=x^2+4x^4$, reflecting it over the $y$-axis will not change the graph at all. This is because the even powers of $x$ stay the same when we replace $x$ with $-x$.

Reflecting polynomials over the y-axis helps us understand how the graph of a polynomial behaves when flipped. By practicing with different polynomials, you’ll become more comfortable with recognizing these changes!

In the next lesson, you will learn how to combine polynomial transformations.