4.3 Transformations of Functions Reflecting Graphs; Symmetry
-vertical shifts -horizontal shifts -reflections over x and y axis
Slide 2 Vertical Shifts Adding a constant to a function shifts the
graph vertically. UP if the constant is positive DOWN if the
constant is negative. Slide 3 we will want to start with the parent
function. Parent functions are functions like. These are functions
that we are familiar with, nothing has happened to them yet Slide 4
y=x 2 Slide 5 y = x 2 y = x 2 + 2 Slide 6 y = x 2 y = x 2 - 3 Slide
7 Slide 8 Horizontal Shift To move the graph either left or right
LEFT you add a constant c to x RIGHT you subtract a constant c to x
Slide 9 y = x 2 y = (x+3) 2 Slide 10 y = x 2 y = (x-2) 2 y = (x+4)
2 Slide 11 Consider any graph y = f(x) We can now shift it up or
down, and left and right one movement at a time, we cannot get off
of the x or y axis. But what if we want to move that graph y = f(x)
up and left at the same time? (diagonally) So we can move the graph
anywhere on the Cartesian Plane. Slide 12 We can combine a vertical
shift and a horizontal shift to move the graph of y = f(x) to any
spot on the Cartesian Plane. Transformation Combinations Slide 13
Slide 14 GraphSight Examples. Slide 15 Vertical
Stretching/Compressing y = f(x) Vertical stretching or compressing
refers to how the y value changes between y = f(x) and y = af(x) at
the same value x. Looking at y = f(x) and y = af(x), here the y
value changes by a multiple of a. Slide 16 Complete the following
tables xy = x 2 0 1 2 3 4 5 6 xy = 3x 2 0 1 2 3 4 5 6 Slide 17 y =
x 2 y = 3x 2 Notice that the value that we multiplied f(x) by was
3. So each y value is now three times greater in y=af(x) than it
was in y=f(x) Slide 18 a > 1 When a > 1. Then the graph is
stretched vertically 0 < a < 1 When 0 < a < 1. Then the
graph is compressed vertically Slide 19 Now a is between 0 and 1 so
you can see that the pink graph has shrunk vertically. Slide 20
Slide 21 Now when we stretch or shrink something horizontally, then
we are multiplying each x value by a multiple a. Not the whole
function, just the individual x values. Stretches when 0 < a
< 1 Shrinks when a > 1 Slide 22 Horizontal
Shrinking/Stretching Slide 23 Slide 24 stretch shrink Slide 25
Reflections Across the x axis (multiply the entire function by -1)
Across the y axis (multiply each x by a -1) Slide 26 Slide 27
Reflection in the y-axis The graph of y=f(-x) is obtained by
reflecting the graph of y=f(x) in the y-axis. Notice that each
point (x,y) on the original graph (red) becomes the point (-x,y) on
the reflected blue graph. Slide 28 Reflecting along the line y=x
This refection requires the interchanging of x and y in the
equation. Slide 29 Weve discussed how a quadratic has a vertex at
b/2a, it also has a line of symmetry through that point. A cubic
has a point of symmetry at the point b/3a. Slide 30 Types of
Symmetry X-AXIS if (x,y) then (x,-y) Y-AXIS if (x,y) then (-x,y)
ORIGIN if (x,y) then (-x,-y) Y=X if (x,y) then (y,x) Slide 31
X-AXIS if (x,y) then (x,-y) (2,4) (2,-4) Slide 32 Y-AXIS if (x,y)
then (-x,y) (2,4) (-2,4) Slide 33 Some x-axis symmetry graphs Slide
34 Y-axis symmetry graphs Slide 35 Symmetry with respect to the
origin Best way I can explain how to visually see if a graph is
symmetric with respect to the origin is to fold the x-axis, and
then fold the y- axis does the graph end up overlapping itself?
Slide 36 Symmetry about the line y=x. Notice that x and y values
are interchanged. Slide 37 To check algebraically if an equation is
symmetric about any of these 3 lines and 1 point, then you simply
apply the rule to each equation and see if the result is an
equivalent equation to the original. Slide 38 Find the symmetry for
the equation x 2 + y 2 = 9. X-AXIS if (x,y) then (x,-y) Y-AXIS if
(x,y) then (-x,y) ORIGIN if (x,y) then (-x,-y) Y=X if (x,y) then
(y,x) Slide 39 Find the symmetry for the equation x 4 = y - 3.
X-AXIS if (x,y) then (x,-y) Y-AXIS if (x,y) then (-x,y) ORIGIN if
(x,y) then (-x,-y) Y=X if (x,y) then (y,x) Slide 40 Hwk. Pg. 136
1-4, 7-12, 15, 16
