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Given a function and both a vertical and a horizontal shift, sketch the graph.
Given $\text{\hspace{0.17em}}f(x)=\left|x\right|,\text{\hspace{0.17em}}$ sketch a graph of $\text{\hspace{0.17em}}h(x)=f(x+1)-3.$
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: $\text{\hspace{0.17em}}f(x+1)\text{\hspace{0.17em}}$ is a change on the inside of the function, giving a horizontal shift left by 1, and the subtraction by 3 in $\text{\hspace{0.17em}}f(x+1)-3\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}f(x)=\left|x\right|.$
[link] shows the graph of $\text{\hspace{0.17em}}h.$
Given $\text{\hspace{0.17em}}f(x)=\left|x\right|,\text{\hspace{0.17em}}$ sketch a graph of $\text{\hspace{0.17em}}h(x)=f(x-2)+4.$
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
Using the formula for the square root function, we can write
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.
$g\left(x\right)=\frac{1}{x-1}+1$
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.
Given a function $\text{\hspace{0.17em}}f(x),$ a new function $\text{\hspace{0.17em}}g(x)=-f(x)\text{\hspace{0.17em}}$ is a vertical reflection of the function $\text{\hspace{0.17em}}f(x),\text{\hspace{0.17em}}$ sometimes called a reflection about (or over, or through) the x -axis.
Given a function $\text{\hspace{0.17em}}f(x),\text{\hspace{0.17em}}$ a new function $\text{\hspace{0.17em}}g(x)=f(-x)\text{\hspace{0.17em}}$ is a horizontal reflection of the function $\text{\hspace{0.17em}}f(x),\text{\hspace{0.17em}}$ sometimes called a reflection about the y -axis.
Given a function, reflect the graph both vertically and horizontally.
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