# 3.5 Transformation of functions  (Page 4/21)

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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.

## Graphing combined vertical and horizontal shifts

Given $\text{\hspace{0.17em}}f\left(x\right)=|x|,\text{\hspace{0.17em}}$ sketch a graph of $\text{\hspace{0.17em}}h\left(x\right)=f\left(x+1\right)-3.$

The function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is our toolkit absolute value function. We know that this graph has a V shape, with the point at the origin. The graph of $\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ has transformed $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ in two ways: $\text{\hspace{0.17em}}f\left(x+1\right)\text{\hspace{0.17em}}$ is a change on the inside of the function, giving a horizontal shift left by 1, and the subtraction by 3 in $\text{\hspace{0.17em}}f\left(x+1\right)-3\text{\hspace{0.17em}}$ is a change to the outside of the function, giving a vertical shift down by 3. The transformation of the graph is illustrated in [link] .

Let us follow one point of the graph of $\text{\hspace{0.17em}}f\left(x\right)=|x|.$

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

[link] shows the graph of $\text{\hspace{0.17em}}h.$

Given $\text{\hspace{0.17em}}f\left(x\right)=|x|,\text{\hspace{0.17em}}$ sketch a graph of $\text{\hspace{0.17em}}h\left(x\right)=f\left(x-2\right)+4.$

## Identifying combined vertical and horizontal shifts

Write a formula for the graph shown in [link] , which is a transformation of the toolkit square root function.

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

$h\left(x\right)=f\left(x-1\right)+2$

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

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

Write a formula for a transformation of the toolkit reciprocal function $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}\text{\hspace{0.17em}}$ that shifts the function’s graph one unit to the right and one unit up.

$g\left(x\right)=\frac{1}{x-1}+1$

## Graphing functions using reflections about the axes

Another transformation that can be applied to a function is a reflection over the x - or y -axis. A vertical reflection reflects a graph vertically across the x -axis, while a horizontal reflection reflects a graph horizontally across the y -axis. The reflections are shown in [link] .

Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the x -axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the y -axis.

## Reflections

Given a function $\text{\hspace{0.17em}}f\left(x\right),$ a new function $\text{\hspace{0.17em}}g\left(x\right)=-f\left(x\right)\text{\hspace{0.17em}}$ is a vertical reflection    of the function $\text{\hspace{0.17em}}f\left(x\right),\text{\hspace{0.17em}}$ sometimes called a reflection about (or over, or through) the x -axis.

Given a function $\text{\hspace{0.17em}}f\left(x\right),\text{\hspace{0.17em}}$ a new function $\text{\hspace{0.17em}}g\left(x\right)=f\left(-x\right)\text{\hspace{0.17em}}$ is a horizontal reflection    of the function $\text{\hspace{0.17em}}f\left(x\right),\text{\hspace{0.17em}}$ sometimes called a reflection about the y -axis.

Given a function, reflect the graph both vertically and horizontally.

1. Multiply all outputs by –1 for a vertical reflection. The new graph is a reflection of the original graph about the x -axis.
2. Multiply all inputs by –1 for a horizontal reflection. The new graph is a reflection of the original graph about the y -axis.

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