TRANSFORMATIONS OF FUNCTIONS

Shifts and stretches

𝑔 (𝑥 )=𝑎 · 𝑓 (𝑏 (𝑥−𝑐 ) )+𝑑

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)

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 y-axis (and stretched or compressed depending on the absolute value of b).

𝑓 (𝑥 )=4 𝑥2

TRANSFORMATIONS OF FUNCTIONS

𝑔 (𝑥 )=𝑎 · 𝑓 (𝑏 (𝑥−𝑐 ) )+𝑑

Vertical translation

Upward (d > 0) Downward (d

< 0)

Horizontal translationLeft (+c) Right (-c)

Horizontal Stretches/

CompressionsStretch (0 < b <

1) Compress (b >

1)

Vertical Stretches/

CompressionsStretch (a > 1)

Compress (0 < a < 1)