3'4 II Graphs and Transformations

1
3.4 IIGraphs and Transformations
2
Vertical Stretching/Compressing -stretched/compre
ssed with respect to the x-axis -affects the
y-value
Let c be a positive number. 1) If (x,y) is a
point on f(x) then (x, cy) is a point on g(x)
c f(x) 2) If cgt1, the graph is stretched
vertically, away from the x-axis by a factor of
c. 3) If clt1, the graph is compressed
vertically, toward the x-axis.
3
Ex Graph g(x) 3x2 and h(x) (1/8)x2. Describe
how the graph differs from the parent function
f(x) x2.
g(x) 3x2
Every y-coordinate of f(x) is multiplied by
3. Stretches the graph vertically making it
move away from the x-axis.
4
h(x) (1/8)x2
Every y-coordinate of f(x) is multiplied by
(1/8). Compresses the graph vertically making
is move toward the x-axis.
5
Horizontal Stretching/Compressing -stretched/comp
ressed with respect to the y-axis -affects the
x-value
Let c be a positive number. 1) If (x,y) is a
point on f(x) then ((1/c)x, y) is a point on
g(x) f(c x) 2) If cgt1, the graph is
compressed horizontally, toward the y-axis by a
factor of 1/c. 3) If clt1, the graph is
stretched horizontally, away from the y-axis by
1/c.
6
Ex Graph g(x) (3x)2. Describe how the graph
differs from the parent function f(x) x2.
g(x) (3x)2
Because c 3 gt 1, every x-coordinate of f(x)
is multiplied by (1/3). Compresses the graph
horizontally making it move toward from the
y-axis.
7
I get that my calculator can do all the work for
me, but I want to know WHY the graph of a
function looks like it does!!! Please enlighten
me
8
g(x) f( (x ))
c
a
b
d
Uhh.thats a lot of letters. You expect me to
understand this?
c ? reflection across x-axis ? vertical
stretch/compression by a factor of c
a ? reflection across y-axis ? horizontal
stretch/compression by a factor of 1/c
b ? horizontal shifts
d ? vertical shifts
9
Ex Graph the function. h(x) -2x - 1 4
Step 1 Rewrite the function in the form g(x)
c f(a(x b)) d
h(x) -1 2(x ½) 4
So, a 2 b ½ c -1 d 4
Tells us 1) there is no reflection about the
y-axis 2) there is a horizontal compression
because 2gt1 so, compresses toward y-axis by ½
Tells us our graph shifts right by 1/2
Tells us 1) there is a reflection about the
x-axis 2) there is no vertical
stretch/compression
Tells us our graph shifts up by 4
10
h(x) -2(x 1)2 3
a 1 b -1 c -2 d 3