Problem Solving > *Vertical shift of a function*

## What is a Vertical Shift of a Function?

**Vertical shift of a function** is where the graph of a function moves up or down the y-axis:

The graph above shows the graph of f(x) = x (in red) shifted up three units from y = 0 to y = 3 (in blue) on the vertical (y) axis. The new graph (blue) is now f(x) = x + 3— the original graph plus the 3 unit shift. This leads to a simple rule for any function:

**1. Compared to the graph of f(x), a graph f(x) + k is shifted up k units.**

We can also work this in reverse:

**2. Compared to the graph of f(x), a graph f(x) – k is shifted down k units.**

Generally speaking, a vertical shift adds (or subtracts) a constant to (from) each y-value while leaving the x-value unchanged. In other words, the only changes going on here are y-values moving up or down. The x-values stay exactly where they are.

## Vertical Shift of a Function: Examples

**Example question #1:** How are the graphs of f(x) = |x + 2| and f(x) = |x| related?

**Solution**:

Step 1: **Compare the right hand side of both equations**:

- |x + 2|
- |x|.

The difference between these two statements is the “+ 2”.

Step 2: **Choose one of the above statements** based on the result from Step 1. We have a positive 2, so choose statement 1:

*Compared to the graph of f(x), a graph f(x) + k is shifted up k units.*

Step 3: **Replace the f(x) and f(x) + k the statement with your functions.**

Compared to the graph of

*f(x) = |x|*, the vertical shift of a function

*f(x) = |x + 3|*is a shift

*up*2 units.

**Example question #2:** How are the graphs of f(x) = 3x^{2} and f(x) = 3x^{2} – 8 related?

**Solution**:

Step 1: **Compare the right hand side of both equations:**

- 3x
^{2} - 3x
^{2}– 8

The difference between these two statements is the “- 8”.

Step 2: **Choose one of the above statements** based on the result from Step 1. We have a negative 8, so choose statement 2:

*Compared to the graph of f(x), a graph f(x) + k is shifted down k units.*

Step 3: **Replace the f(x) and f(x) + k the statement with your functions.**

Compared to the graph of

*f(x) = 3x*, the vertical shift of a function

^{2}*f(x) = 3x*is a shift

^{2}– 8*down*k units.

## References

Larson, R. & Edwards, B. (2016). Calculus (10th Edition). Cengage Learning.