3.5 Transformation of functions  (Page 10/21)

Finding a triple transformation of a graph

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

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

$$f\left(\frac{1}{2}x+1\right)-3=f\left(\frac{1}{2}(x+2)\right)-3$$

By factoring the inside, we can first horizontally stretch by 2, as indicated by the $\frac{1}{2}$ on the inside of the function. Remember that twice the size of 0 is still 0, so the point (0,2) remains at (0,2) while the point (2,0) will stretch to (4,0). See [link] .

Next, we horizontally shift left by 2 units, as indicated by $x+2.$ See [link] .

Last, we vertically shift down by 3 to complete our sketch, as indicated by the $-3$ on the outside of the function. See [link] .

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Key equations

Key concepts

• A function can be shifted vertically by adding a constant to the output. See [link] and [link] .
• A function can be shifted horizontally by adding a constant to the input. See [link] , [link] , and [link] .
• Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts. See [link] .
• Vertical and horizontal shifts are often combined. See [link] and [link] .
• A vertical reflection reflects a graph about the $x\text{-}$ axis. A graph can be reflected vertically by multiplying the output by –1.
• A horizontal reflection reflects a graph about the $y\text{-}$ axis. A graph can be reflected horizontally by multiplying the input by –1.
• A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph. See [link] .
• A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly. See [link] .
• A function presented as an equation can be reflected by applying transformations one at a time. See [link] .
• Even functions are symmetric about the $y\text{-}$ axis, whereas odd functions are symmetric about the origin.
• Even functions satisfy the condition $f(x)=f(-x).$
• Odd functions satisfy the condition $f(x)=-f(-x).$
• A function can be odd, even, or neither. See [link] .
• A function can be compressed or stretched vertically by multiplying the output by a constant. See [link] , [link] , and [link] .
• A function can be compressed or stretched horizontally by multiplying the input by a constant. See [link] , [link] , and [link] .
• The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in the order given. However, a vertical transformation may be combined with a horizontal transformation in any order. See [link] and [link] .

Section exercises

Verbal