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In a quadratic equation, the $y$ represents the coordinate for how high up each point on the curve is. Take a look at the $6$ graphs below. They all have the same slope, or they all are the same steepness, but the only difference is that their vertexes are at different points on the $y$ - axis!

Okay. There are $6$ graphs on the coordinate plane. The red one is $y = x^2$.

The blue graph has the equation $y = x^2 + 3$. Therefore, the purple graph has equation $y = x^2 + 6$, the green graph has equation $y = x^2 - 3$, and the black and orange ones lie even further below $y = x^2 - 3.$

Vertical Translation

Translation is moving a graph around on a plane. Vertical translation is moving a graph exclusively up or down on the plane. Let's look at an example:

Here we have $y=|x|$. To translate the function up and down, you simply add or subtract numbers from the whole function. If you add a positive number (or subtract a negative number), you translate the function up. If you subtract a positive number (or add a negative number), you translate the function down. For example, the function in red below is $y=|x| + 2$.

The answer above is $-3$ because the graph is shifted $3$ units down from the graph of $y = |x|$. The equation for the blue graph is:

$$y = |x| - 3$$

If you want to learn about horizontal translation, click right through here!

(Video) Translating Absolute Value Functions

by Keith Mann

This video by Keith Mann explains how to translate functions vertically (in other words, move them up or down on a graph).

$\large{\text{Summary}}$

When you translate functions vertically, you add to or subtract from the whole function. This means that you add or subtract outside of the absolute value.

Adding a positive shifts the graph upwards. Subtracting a positive shifts the graph downward. Let's look at an example in a graph:

The green function is the base function. You have to subtract $7$ from the whole equation to get the blue function. That way, you're just shifting the graph along the $y$-axis.

We add $5$ to the whole equation to move the graph up $5$ points.

Now, what would the graph of $y = |3x - 3|$ and $y = |3x - 3| + 5$ look like?

Vertical Translation

The first lesson in our series on function transformations is vertical translations!

A vertical shift allows you to move a function up or down along the $y$-axis. We can add or subtract a constant. Let's write a formula for a quadratic parent function.

$$y=(x)^2-1$$

Adding a constant will shift the function up, and subtracting a constant will shift the function down.

Example

Starting with the parent quadratic function, $y=x^2$,

Say we want to shift this function one unit down. We can subtract $1$ from $x$ in the equation. Our new equation will look like: $y=(x)^2-1$.

Let's sketch a graph.

As you can see, we have shifted the function down by $1$ unit. This is a vertical translation!

In the next lesson, you will learn about the horizontal shift.