Example: Graphing Combined Vertical and Horizontal Shifts

Given [latex]f\left(x\right)=|x|[/latex], sketch a graph of [latex]h\left(x\right)=f\left(x+1\right)-3[/latex]. The function [latex]f[/latex] is our toolkit absolute value function. We know that this graph has a V shape, with the point at the origin. The graph of [latex]h[/latex] has transformed [latex]f[/latex] in two ways: [latex]f\left(x+1\right)[/latex] is a change on the inside of the function, giving a horizontal shift left by 1, and the subtraction by 3 in [latex]f\left(x+1\right)-3[/latex] is a change to the outside of the function, giving a vertical shift down by 3. The transformation of the graph is illustrated below. Let us follow one point of the graph of [latex]f\left(x\right)=|x|[/latex].

• The point [latex]\left(0,0\right)[/latex] is transformed first by shifting left 1 unit: [latex]\left(0,0\right)\to \left(-1,0\right)[/latex]
• The point [latex]\left(-1,0\right)[/latex] is transformed next by shifting down 3 units: [latex]\left(-1,0\right)\to \left(-1,-3\right)[/latex]

Below is the graph of [latex]h[/latex].

Example: Identifying Combined Vertical and Horizontal Shifts

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

Answer: 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

[latex]h\left(x\right)=f\left(x - 1\right)+2[/latex]

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

[latex]h\left(x\right)=\sqrt{x - 1}+2[/latex]

Analysis of the Solution

Note that this transformation has changed the domain and range of the function. This new graph has domain [latex]\left[1,\infty \right)[/latex] and range [latex]\left[2,\infty \right)[/latex].

Example: Applying a Learning Model Equation

A common model for learning has an equation similar to [latex]k\left(t\right)=-{2}^{-t}+1[/latex], where [latex]k[/latex] is the percentage of mastery that can be achieved after [latex]t[/latex] practice sessions. This is a transformation of the function [latex]f\left(t\right)={2}^{t}[/latex] shown below. Sketch a graph of [latex]k\left(t\right)[/latex].

Answer: This equation combines three transformations into one equation.

• A horizontal reflection: [latex]f\left(-t\right)={2}^{-t}[/latex]
• A vertical reflection: [latex]-f\left(-t\right)=-{2}^{-t}[/latex]
• A vertical shift: [latex]-f\left(-t\right)+1=-{2}^{-t}+1[/latex]

We can sketch a graph by applying these transformations one at a time to the original function. Let us follow two points through each of the three transformations. We will choose the points [latex](0, 1)[/latex] and [latex](1, 2)[/latex].

1. First, we apply a horizontal reflection: [latex](0, 1) (–1, 2)[/latex].
2. Then, we apply a vertical reflection: [latex](0, −1) (1, –2)[/latex].
3. Finally, we apply a vertical shift: [latex](0, 0) (1, 1)[/latex].

This means that the original points, [latex](0,1)[/latex] and [latex](1,2)[/latex] become [latex](0,0)[/latex] and [latex](1,1)[/latex] after we apply the transformations. In the graphs below, the first graph results from a horizontal reflection. The second results from a vertical reflection. The third results from a vertical shift up 1 unit.

Analysis of the Solution

As a model for learning, this function would be limited to a domain of [latex]t\ge 0[/latex], with corresponding range [latex]\left[0,1\right)[/latex].

Example: Finding a Triple Transformation of a Tabular Function

Given the table below for the function [latex]f\left(x\right)[/latex], create a table of values for the function [latex]g\left(x\right)=2f\left(3x\right)+1[/latex].

[latex]x[/latex]	6	12	18	24
[latex]f\left(x\right)[/latex]	10	14	15	17

Answer: There are three steps to this transformation, and we will work from the inside out. Starting with the horizontal transformations, [latex]f\left(3x\right)[/latex] is a horizontal compression by [latex]\frac{1}{3}[/latex], which means we multiply each [latex]x\text{-}[/latex] value by [latex]\frac{1}{3}[/latex].

[latex]x[/latex]	2	4	6	8
[latex]f\left(3x\right)[/latex]	10	14	15	17

Looking now to the vertical transformations, we start with the vertical stretch, which will multiply the output values by 2. We apply this to the previous transformation.

[latex]x[/latex]	2	4	6	8
[latex]2f\left(3x\right)[/latex]	20	28	30	34

Finally, we can apply the vertical shift, which will add 1 to all the output values.

[latex]x[/latex]	2	4	6	8
[latex]g\left(x\right)=2f\left(3x\right)+1[/latex]	21	29	31	35

Example: Finding a Triple Transformation of a Graph

Use the graph of [latex]f\left(x\right)[/latex] to sketch a graph of [latex]k\left(x\right)=f\left(\frac{1}{2}x+1\right)-3[/latex].

Answer: To simplify, let’s start by factoring out the inside of the function.

[latex]f\left(\frac{1}{2}x+1\right)-3=f\left(\frac{1}{2}\left(x+2\right)\right)-3[/latex]

By factoring the inside, we can first horizontally stretch by 2, as indicated by the [latex]\frac{1}{2}[/latex] on the inside of the function. Remember that twice the size of 0 is still 0, so the point [latex](0,2)[/latex] remains at [latex](0,2)[/latex] while the point [latex](2,0)[/latex] will stretch to [latex](4,0)[/latex]. Next, we horizontally shift left by 2 units, as indicated by [latex]x+2[/latex]. Last, we vertically shift down by 3 to complete our sketch, as indicated by the [latex]-3[/latex] on the outside of the function.