TERMINOLOGY:
If we say “y” is the value of the output for function “f” at x,
then the formula f(x) = y also means f(x) = x and so y or f(x) are equal to x: y = x = f(x) or f(x) = y = x or y = f(x) = x
For the linear function f(x) = mx + b, where b = 0 and m = 1, f(x) = 1x + 0 = x = y.
Writing f(x) = x is necessary because we need to solve the formula. Therefore, if x = 2, f(2) = 2 = y because f(x) = x = y.
Another way to say this:
x is in the domain of input values for function f, f(x) = x.
y is in the range of output values for function f, f(x) = y
and so f(x) = x = y or f(x) is y when we solve f(x) = x.
A Parent or identity “linear” function: f(x) = x = y
TRANSFORMATIONS are
translations, reflections, and stretches or shrinks
and move the parent or identity linear function from
it’s position at f(x) = x = y.
IF we subtract a value of “h” or “-h” from x (inside parenthesis), then f(x - h) or f(x – [-h]) = f(x + h)
and the result is a
Horizontal translation along x-axis which means f(x) = x = y moves to right (+h units) along x-axis or
to left (-h units) along x-axis.
See Core Concept p. 146.
Example:
Parent linear function, f(x) = y = x,
will move h units right when f(x - h)
Parent linear, f(x) = y = x,
will move h units left when f(x – [-h]) = f(x + h)
VERTICAL translation means f(x) = x = y moves
upward (k units from x) or
downward (-k units) along y-axis. See Core Concept p. 146.
Example:
Parent linear function,f(x) = y = x, will move k units upward when f(x) + k (but we think either x + k or y + k, because we know that
f(x) = y = x).
Parent linear function,f(x) = y = x, will move k units downward when f(x) + (-k) = f(x) - k (but we think either x - k or y - k, because
we know that f(x) = y = x).
SUMMARY of Horizontal/Vertical Translations:
f(x) = y = x moves left or right along x-axis when f(x h);
f(x) = y = x moves up or down along y-axis when f(x) k.
A Reflection transformation along x-axis means f(x) = x = y moves across the x-axis when f(x) = -f(+x) or -y is the output for
positive x and positive y is output for negative x.
See Core Concept p. 147.
Example: Parent linear function,f(x) = y = x, will reflect across the x-axis when f(x) is -f(x); so think if I use +2 for x, then f(+2) = -2
= -y and not +y; and if I use -3 for x, then
-f(-3) = -(-3) = +3 = +y and not negative y.
A Reflection transformation along y-axis means f(x) = x = y moves across the y-axis when f(x) is f(-x) or -y is the output for
negative x and positive y is output for positive x.
See Core Concept p. 147.
Example: Parent linear function, f(x) = y = x, will reflect across the y-axis when f(x) is f(-x); so think if I use +2 for x, then f(- (+2)) =
-2 = -y and not +y like in parent; and if I use -3 for x, then f(- (-3)) = f(3) = +3 = +y and not -y like in parent.
SUMMARY of Reflections:
f(x) = y = x moves across the x-axis when -f(x) = -x = -y (or positive x inserted into -f(x) produces negative y and visa versa);
f(x) = y = x moves across the y-axis when f(-x) = -x = -y (or negative x in f(- [-x]) produces positive y and visa versa.
STRETCHes or SHRINKs (find the “a”)
When “a or 1/a” is a multiplier inside the parenthesis (see p 148 Core Concept on left):
Horizontal stretch = (vertical shrink): when a is a fraction (0<a<1 or
ay, and the graph f(x) stretches toward the x-axis;
1/a) the parent f(x) = y = x translates to f( a x) = ax =
Horizontal shrink = (vertical stretch): when a>1 (positive and greater than 1) or a<-1 (negative and less than 1) the parent f(x) = y =
x translates to f( a•x) = ax = ay, and the graph f(x) shrinks away from x-axis and toward the y-axis;
When “a or 1/a” is a multiplier outside the parenthesis (see p. 148 Core Concept on right):
Vertical shrink = (horizontal stretch): when a is a fraction (0<a<1 or 1/a) the parent f(x) = y = x translates to a•f(x) = ax =
ay, and the graph f(x) shrinks away from the y-axis and toward the x-axis.
Vertical stretch = (horizontal shrink): when a>1 (positive and greater than 1) or a<-1 (negative and less than 1) the parent f(x) = y =
x translates to to a•f(x) = ax = ay, and the graph f(x) stretches toward the y-axis.
SUMMARY of Horizontal and Vertical Stretches or Shrinks (depends on location of the “a” multiplier):
When “a” is a fraction ( 1/a), the graph of f(x) = x = y moves towards the x-axis;
When “a” <1 (negative) or “a” > 1 (positive), the graph of f(x) = x = y moves towards the y-axis.
When “a” is inside parenthesis:
f(x) = x = y will stretch toward the horizontal x-axis when a is inside the parenthesis and shrink toward the y-axis;
When “a” is outside parenthesis:
f(x) = x = y will stretch toward the vertical y-axis when a is outside the parenthesis and shrink toward the x-axis;