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.
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5) and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.