Moving Polynomials Up or Down

In the earlier lesson, we talked about how we can transform polynomials. This lesson will cover how to move polynomials up or down.

Moving Polynomials Up or Down

We can move polynomials vertically (up or down) to translate the function. We'll use the quadratic function as an example.

Translating a Quadratic Function

Before we dive in, let's go over some of the characteristics of parent quadratic functions.

• Vertex at $(0,0)$

• Function goes through $(1,1)$ and $(-1,1)$

• Parabola opens upwards

• Standard form: $ax^2+bx+c$

When we are considering polynomial transformations for quadratic functions, we use vertex form of quadratics.

$$y=a(x-h)^2 +k$$

This equation tells us that the vertex is $(h, k)$.

This lesson will look at the $k$ value, which controls vertical translations. The $h$ value controls horizontal translations.

Let's look at the parent function of the quadratic function:

Translating functions is an example of a polynomial transformation. Say we want to translate the (above) parent quadratic function up by 2 units. That means that $k=2$. We're not dilating the function, and we're not doing a horizontal translation, so $a=1$ and $h=0$, respectively.

Let's graph $y=x^2+2$.

The parent function of a quadratic has the vertex at $(0,0)$, but the above, new, function has a vertex of $(0,2)$ because it was translated up by two units.

Translating a Cubic Function

Translating a cubic function up or down is very similar to translating a quadratic function up or down. While there is no vertex form of a cubic function, like we saw with quadratic functions, we can still use a similar cubic equation.

$$y=a(x-h)^2 +k$$

$h$ affects horizontal translations, and $k$ affects vertical translations.

In this example, $k=-4$. That means that the function has been shifted down by $4$ units.

Moving Polynomials Up or Down

In the previous lesson, we learned about polynomial transformations, which involve shifting, stretching, or reflecting polynomials. Today, we’ll focus on how to move polynomials up or down. This is known as a vertical translation.

To move a polynomial up or down, we add or subtract a number from the whole polynomial. In the above example, we see the polynomial in purple, $y=x^2+x+1$, shifted up by five units. The new polynomial is $y=x^2+x+6$. This means the entire graph of the polynomial will shift $5$ units upward.

On the other hand, if we subtract a number, the entire graph moves down.

These vertical shifts don't change the general shape of the graph, they just move the whole graph up or down. It's an important concept when analyzing how a polynomial function behaves, especially when combined with other transformations like stretching or reflecting.

In the next lessons, we will learn to move a polynomial right and left, to do an upside-down reflection, and to do a mirror reflection.

(Video) Moving Polynomials Up or Down

by Anusha Rahman

In this lesson by Anusha Rahman, you will learn how to move polynomials up or down (a vertical transformation).

Summary

Vertical transformations involve shifting the graph of a polynomial up or down. These transformations occur when we add or subtract a constant from the polynomial function.