1329.sagews

This chapter is about:

1) Translations
2) Reflections
3) Scale Changes
4) Function Composition
5) Inverse Functions
6) $z$-Scores

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1)  Translations

You can think of a translation $T$ as a kind of function that operates on a point:  $T(x, y) = (x+h, y+k)$

If $f(x)$ is some algebraic function, and if all of the points in its graph are translated by $T$,

then the new function $g(x)$ that describes those points will be $g(x) = f(x-h) + k$.

def T(x, y): return (x+2, y-3)

L = [(0, 0), (3, 4), (-1, 5)]
L

[(0, 0), (3, 4), (-1, 5)]

[T(x, y) for (x, y) in L]

[(2, -3), (5, 1), (1, 2)]

@interact
def _(h = (-5..5), k = (-5..5)):
    def T(L): return [(x+h, y+k) for (x, y) in L]
    def f(x): return x^2
    F = [(x, f(x)) for x in [-5..5]]
    def g(x): return f(x-h) + k
    G = T(F)
    print 'T(x, y) = (x +',h,', y +',k,')'
    print '\nf(x) = x^2'
    print F
    print '\ng(x) = (x-',h,')^2 + ', k
    print G
    show(points(G, color = 'red') + points(F) + plot(g, -5+h, 5+h, color = 'red') + plot(f, -5, 5))

h  k  [removed] [removed] [removed] [removed]

If $f(x) = x^2$ and $g(x) = (x-3)^2$ , then the graph of $g$ will look just like the graph of $f$ shifted horizontally 3 units.

But in which direction?  To the left or to the right of $f$?

You might expect $g$ to appear to the left, because of the $(x-3)$.

But no, it actually appears to the right.  This fact is often confusing to students.

Here is another interactive cell to experiment with just horizontal shifts:

@interact

def horizontal_translation(h = (-10..10)):
    f(x) = x^2
    g(x) = f(x-h)
    print 'g(x) = f(x - ',h,')'
    show(plot(f, -5, 5) + plot(g, -5+h, 5+h, color = 'red'))

h  [removed] [removed] [removed] [removed]

We say that $g$ is the image and that $f$ is the pre-image (original). 

Now suppose that $g(x) = x^2 + 3$.

 The graph of $g$ will look just like the graph of $f$ shifted vertically 3 units.

But in which direction?  Above or below $f$?

You might expect $g$ to appear above, because of the $+3$.

And yes, that's what happens.  This fact is usually not confusing to students:

 

@interact
def translate(k = (-25..25)):
    f(x) = x^2
    g(x) = f(x) + k
    show(plot(f, -5, 5)+plot(g, -5, 5, color = 'red'))

k  [removed] [removed] [removed] [removed]

Here are both kinds of translations put together:

@interact

def translate(h = (-5..5), k = (-5..5)):
    f(x) = x^2
    g(x) = f(x - h) + k
    print 'g(x) = f(x -',h,') +',k
    show(plot(f, -5, 5)+plot(g, -5+h, 5+h, color = 'red')+points([(0,0), (h, k)], color = 'black', pointsize = 25), ymax = 25)

h  k  [removed] [removed] [removed] [removed]

Summary: 

If $T(x, y) = (x + h, y + k)$, then $g(x) = f(x - h) + k$ where

$f$ is some function and $g$ is the translated function.

2)  Reflections

Reflection over the $y$ axis:      $r_y(x, y) = (-x, y)$

Reflection over the $x$ axis:      $r_x(x, y) = (x, -y)$

f(x) = 2^x

def r_y(L): return [(-x, y) for (x, y) in L]

F = [(x, f(x)) for x in [-5..5, step = 1/5]]

show(points(F)+points(r_y(F), color = 'red'))

def r_x(L): return [(x, -y) for (x, y) in L]

show(points(F)+points(r_x(F), color = 'red'))

@interact

def reflections(r_x = [False, True], r_y = [False, True]):
    f(x) = 2^x
    g(x) = f(x)
    if r_x: 
        g(x) = -g(x)
    if r_y: 
        g(x) = g(-x)
    show(plot(g, -5, 5, color = 'red')+plot(f, -5, 5), ymin = -25, ymax = 25)

r_x  r_y  [removed] [removed] [removed] [removed]

Note $\rightarrow$ A reflection over both the $x$ and $y$ axes is equivalent to a reflection over the origin.

Reflection over $(0, 0)$:      $r_{(0, 0)}(x, y) = (-x, -y)$

A reflection over the line $y = x$ swaps the $x$ and $y$ coordinates:  $r_{y = x}(x, y) = (y, x)$

f(x) = 2^x

F = [(x, f(x)) for x in [-5..5, step = 1/5]]
G = [(y, x) for (x, y) in F]
I = [(F[0][0], F[0][0]), (G[-1][0], G[-1][0])]

show(points(F) + points(G, color = 'red') + line(I, color = 'black'), aspect_ratio = 1)

3)  Scale Changes

$S(x, y) = (ax, by)$

@interact
def _(a = (-5..5), b = (-5..5)):
    def S(L): return [(a*x, b*y) for (x, y) in L]
    def f(x): return x^2
    F = [(x, f(x)) for x in [-5..5]]
    def g(x): return f(x/a)*b
    G = S(F)
    print 'S(x, y) = (',a,'x, ',b,'y)'
    print '\nf(x) = x^2'
    print F
    print '\ng(x) = (x/',a,')^2 * ', b
    print G
    show(points(G, color = 'red') + points(F) + plot(g, -5*abs(a), 5*abs(a), color = 'red') + plot(f, -5, 5))

a  b  [removed] [removed] [removed] [removed]

Summary:

If $S(x, y) = (ax, by)$, then $g(x) = f(x/a)\cdot b$ where

$f$ is some function and $g$ is the scaled function.

4)  Function Composition

Function composition means creating new functions from combinations of other functions.

f(x) = x^2
g(x) = 4*x

f(g(x))

16*x^2

g(f(x))

4*x^2

Note - function composition is not generally commutative!

5)  Inverse Functions

If $f(x)$ and $g(x)$ are inverse functions, then $f(g(x)) = g(f(x)) = x$.

f(x) = pow(2, x)
g(x) = log(x, 2)

show(plot(f, -5, 5)+plot(g, 1/32, 32, color = 'red')+plot(x, -5, 32, color = 'black'), aspect_ratio = 1)

6)  $z$-scores

$z(x) = \frac{x - \mu}{\sigma}$

A $z$-score is a translation followed by a scale change:

$T(x) = x - \mu$

$S(x) = \frac{1}{\sigma}x$

$S(T(x)) = \frac{x - \mu}{\sigma}$

@interact
def normal_curve(m=75, s=10): 
    z(x) = (x-m)/s
    P(x) = 1/(s*sqrt(2*pi*exp(z(x)^2)))
    show(plot(P, m-3*s, m+3*s)+\
         line([(m,0),(m,P(m))], color = 'red', linestyle = '--')+\
         line([(m-s,0),(m-s,P(m-s))], color = 'red', linestyle = '--')+\
         line([(m+s,0),(m+s,P(m+s))], color = 'red', linestyle = '--')+\
         line([(m-2*s,0),(m-2*s,P(m-2*s))], color = 'red', linestyle = '--')+\
         line([(m+2*s,0),(m+2*s,P(m+2*s))], color = 'red', linestyle = '--'))

m  s  [removed] [removed] [removed] [removed]