Graphing Functions Using Vertical and Horizontal Shifts

HOW TO

Given a function and both a vertical and a horizontal shift, sketch the graph.

1. Identify the vertical and horizontal shifts from the formula.

2. The vertical shift results from a constant added to the output. Move the graph up for a positive constant and down for a negative constant.

3. The horizontal shift results from a constant added to the input. Move the graph left for a positive constant and right for a negative constant.

4. Apply the shifts to the graph in either order.

Example 7

Graphing Combined Vertical and Horizontal Shifts

Given $f(x)=|x|$, sketch a graph of $h(x)=f(x+1)-3$.

Solution

The function $f$ is our toolkit absolute value function. We know that this graph has a $V$ shape, with the point at the origin. The graph of $h$ has transformed $f$ in two ways: $f(x+1)$ is a change on the inside of the function, giving a horizontal shift left by $1$, and the subtraction by 3 in $f(x+1)-3$ is a change to the outside of the function, giving a vertical shift down by 3 . The transformation of the graph is illustrated in Figure 9.

Let us follow one point of the graph of $f(x)=|x|$.

• The point $(0,0)$ is transformed first by shifting left 1 unit: $(0,0) \rightarrow(-1,0)$
• The point $(-1,0)$ is transformed next by shifting down 3 units: $(-1,0) \rightarrow(-1,-3)$

Figure 9

Figure 10 shows the graph of $h$.

Figure 10

Try It #3

Given $f(x)=|x|$, sketch a graph of $h(x)=f(x-2)+4$.

Example 8

Identifying Combined Vertical and Horizontal Shifts

Write a formula for the graph shown in Figure 11, which is a transformation of the toolkit square root function.

Figure 11

Solution

The graph of the toolkit function starts at the origin, so this graph has been shifted 1 to the right and up 2. In function notation, we could write that as

$h(x)=f(x-1)+2$

Using the formula for the square root function, we can write

$h(x)=\sqrt{x-1}+2$

Analysis

Note that this transformation has changed the domain and range of the function. This new graph has domain $[1, \infty)$ and range $[2, \infty)$.

Try It #4

Write a formula for a transformation of the toolkit reciprocal function $f(x)=\frac{1}{x}$ that shifts the function's graph one unit to the right and one unit up.