Use the graph of
$\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ in
[link] to sketch a graph of
$\text{\hspace{0.17em}}k(x)=f\left(\frac{1}{2}x+1\right)-3.$
To simplify, let’s start by factoring out the inside of the function.
By factoring the inside, we can first horizontally stretch by 2, as indicated by the
$\text{\hspace{0.17em}}\frac{1}{2}\text{\hspace{0.17em}}$ 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
$\text{\hspace{0.17em}}x+2.\text{\hspace{0.17em}}$ See
[link] .
Last, we vertically shift down by 3 to complete our sketch, as indicated by the
$\text{\hspace{0.17em}}-3\text{\hspace{0.17em}}$ on the outside of the function. See
[link] .
$g(x)=f(x)+k\text{\hspace{0.17em}}$ (up for
$\text{\hspace{0.17em}}k>0$ )
Horizontal shift
$g(x)=f(x-h)$ (right for
$\text{\hspace{0.17em}}h>0$ )
Vertical reflection
$g(x)=-f(x)$
Horizontal reflection
$g(x)=f(-x)$
Vertical stretch
$g(x)=af(x)\text{\hspace{0.17em}}$ (
$a>0$ )
Vertical compression
$g(x)=af(x)\text{\hspace{0.17em}}$$(0<a<1)$
Horizontal stretch
$g(x)=f(bx)$$(0<b<1)$
Horizontal compression.
$g(x)=f(bx)\text{\hspace{0.17em}}$ (
$b>1$ )
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
$\text{\hspace{0.17em}}x\text{-}$ 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
$\text{\hspace{0.17em}}f(x)=f(-x).$
Odd functions satisfy the condition
$\text{\hspace{0.17em}}f(x)=-f(-x).$
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
When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal shift from a vertical shift?
A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output.