𝑔 𝑥 =𝑎·𝑓 𝑏 𝑥−𝑐
TRANSFORMATIONS OF FUNCTIONS
+𝑑
Shifts and stretches
TRANSFORMATIONS OF FUNCTIONS
Vertical Translation
Horizontal Translation
Vertical Stretches/Compressions
Horizontal Stretches/Compressions
VERTICAL TRANSLATION
Vertical translation (d)
Consider the parent function of the
parabolas, f(x)=x2, the black curve on
the right. It's not too difficult to imagine
that if we simply add a constant number
(it's 2 in the figure) to every value of the
function, we just raise (translate) the
curve upward along the y-axis by 2
units.
If we add a constant number, d, to a
function, we translate it upward (d > 0)
or downward (d < 0) along the y-axis
by that amount.
HORIZONTAL TRANSLATION
Horizontal translation (c)
When a number, usually denoted by c, is subtracted from
the independent variable insideof a function, the function
is translated by c units to the right if c > 0 and to the left
if c < 0.
This can be tricky. Remember that the transformation is
written f(x)→f(x - c); the c is subtracted. When c is
positive, the translation is in the positive x direction.
When c is negative, (x-(-c)) = (x+c), and the translation is
to the left. For example,
f(x) = (x - 2)2 is a parabola translated to the right by
two units.
f(x) = (x + 2)2 is a parabola translated to the left by
two units.
VERTICAL STRETCHES/COMPRESSIONS
Vertical Stretches/Compressions (a)
When a function is multiplied by a
constant, usually denoted by a, the result
is vertical scaling of the graph. In this
case, f(x) becomes a·f(x)
When a > 1, the graph is stretched
vertically.
When 0 < a < 1, the graph is
compressed vertically, and
when a < 0, the graph is flipped or
reflected across the x-axis
HORIZONTAL STRETCHES/COMPRESSIONS
Horizontal Stretches/Compressions (b)
𝑓 𝑥 = 4𝑥 2
When the independent variable is multiplied by
a constant, usually denoted by b, the result is
scaling of the graph along the x-axis.
When 0 < b < 1, the graph is stretched
horizontally (made wider).
When b > 1, the graph is compressed (made
smaller) horizontally, and
when b < 0, the graph is reflected across the yaxis (and stretched or compressed depending on
the absolute value of b).
TRANSFORMATIONS OF FUNCTIONS
𝑔 𝑥 =𝑎·𝑓 𝑏 𝑥−𝑐
Vertical Stretches/
Compressions
Stretch (a > 1)
Compress (0 < a < 1)
Horizontal Stretches/
Compressions
Stretch (0 < b < 1)
Compress (b > 1)
Vertical translation
Upward (d > 0)
Downward (d < 0)
+𝑑
Horizontal translation
Left (+c)
Right (-c)