[6.1-5] Function Transformations

[6] Function Transformations



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Heating schedule for a building

To save money, an office building is kept warmest during business hours. During the night, the building's temperature is cool, 50 F. From 6 am to 8 am the building is warmed linearly up to 70 F, and from 4 pm to 8 pm the building is allowed to cool linearly back down to 50 F. Let $f(t)$ be the temperature $t$ hours after midnight.


$t$ 0 2 4 6 8 10 12 14 16 18 20 22 24
$f(t)$ 50 50 50 50 70 70 70 70 70 60 50 50 50
$g(t)$
$h(t)$



Let $g(t)=f(t)+5$. Fill in as much of the table as you need to, graph $g(t)$ and interpret $g$. Describe *visually* what you would do to the graph of $f(t)$ to get the graph of $g(t)$.

Let $h(t)=f(t+4)$. Fill in as much of the table as you need to, graph $h(t)$, and describe *visually* what you would do to the graph of $f(t)$ to get the graph of $g(t)$.

First summary

Where $k$ is some positive constant.

Function
definition		 How does the graph compare to the graph of $f(x)$?
$f(x)+k$
$f(x)-k$

$f(x+k)$
$f(x-k)$
 shifted up by $k$ units
shifted down by $k$ units

shifted left by $k$ units
shifted right by $k$ units

Shifts problem

Express each of the four functions shown below in terms of $m(r)$.

What about lines?

What is the slope of $f(x)=x$? $f(x)+3$? $f(x-2)$?

"Reflections"

Let $f(0)=1$, $f(1)=4$, $f(2)=3$, and let $f$ be changing linearly between each pair of successive points.

  • graph $f$.
  • let $g(x)=-f(x)$. Graph $g$.
  • let $h(x)=f(-x)$. Graph $h$.

2nd summary

Where $k$ is a positive constant.

Function definition How does the graph compare to the graph of $f(x)$?
$-f(x)$
$f(-x)$

Reflects (or flips) the graph over the $x$ axis
Reflects the graph over the $y$ axis

Reflections problem

Match up the functions with the graphs

Stretches / compressions


$t$ 0 2 4 6 8 10 12 14 16 18 20 22 24
$f(t)$ 50 50 50 50 70 70 70 70 70 60 50 50 50
$a(t)$
$b(t)$



Let $a(t)=1.1f(t)$. Fill in as much of the table as you need to, graph $a(t)$ and interpret $a$.

Let $b(t)=f(2t)$. Fill in as much of the table as you need to, graph $b(t)$, and interpret $b$.

Again, Let $f(0)=1$, $f(1)=4$, $f(2)=3$, and let $f$ be changing linearly between each pair of successive points.

  • graph $f$.
  • let $a(x)=\frac 12f(x)$. Graph $a$.
  • let $b(x)=f(\frac 12x)$. Graph $b$.

3rd summary

Where $k$ is a positive constant.

Function definition How does the graph compare to the graph of $f(x)$?
[$k > 1$]
$kf(x)$
$f(kx)$

[$0 < k < 1$]
$kf(x)$
$f(kx)$
stretched vertically by a factor of $k$, away from $x$-axis
compressed horizontally by a factor $1/k$, towards the $y$-axis

Compressed vertically by a factor of $k$, towards the $x$-axis
Stretched horizontally by a factor of $1/k$ away from the $y$-axis

What about lines?

What is the slope of $f(x)=2x+1?$? of $1.7*f(x)$? of $f(2*x)$? of $f(\frac 13 x)$?

Working with functions and formulas

  1. From the graph, define $g(x)$ in terms of visual transformations that you make to $f(x)$.
  2. Figure out a formula for $f(x)$.
  3. Use your answer from 1) and your knowledge of function transformations to find a formula for $g(x)$.
  4. Verify by calculating a few points.

    Even

    If $h(x)= f(-x)$, describe what the graph of $h(x)$ looks like, in relation to the graph of $f(x)$.

    An even function has the property that $f(-x)=f(x)$.

    So, what must a function that is even look like on a graph?


    Is either one / both /neither of these functions even?

    Odd

    Now let $f(-1)=1$, $f(0)=1$, $f(1)=4$, $f(2)=3$, and let $f$ be changing linearly between each pair of successive points.

    • graph $f$.
    • let $b(x)=-f(-x)$. Graph $b$.

    Describe $b$ in terms of transformations of $f$.

    A function is odd if $$-f(-x)=f(x).$$

    Which of these polynomials are odd? (Try out x=1, x=2,)

    • $f(x)=x$,
    • $x^2$,
    • $x^3$,
    • $(x-3)(x)(x+3)=x^3-6x^2-9x$.

    If $f(x)$ is an odd function, and its domain is [-10,10], what one point *must* the graph of $f(x)$ pass through?

    Combining transformations

    See this practice sheet

    Exponentials


    These are the graphs of $f(x)=2^{-x}+1$ and $g(x)=2^{-(x+1)}+1$. How are these graphs related?

    Visually, is there an alternative way that you see to think about this transformation?