5.06 Transformations of functions

Parent functions

A family of functions is a set of functions whose equations have a similar form. The parent function of the family is the equation in the family with the simplest form. Let's first take a quick look at the graphs of parent functions as shown here in the diagrams below. The function's description and its equation are given above each graph. 

Linear $f(x)=x$f(x)=x	Quadratic $f(x)=x^2$f(x)=x2	Square root $f(x)=\sqrt{x}$f(x)=√x	Cubic $f(x)=x^3$f(x)=x3	Cube root $f(x)=\sqrt[3]{x}$f(x)=3√x

Absolute value $f(x)=\left|x\right|$f(x)=|x|	Rational - Linear $f(x)=\frac{1}{x}$f(x)=1x​	Rational - Quadratic $f(x)=\frac{1}{x^2}$f(x)=1x2​	Exponential $f(x)=b^x$f(x)=bx , $b>1$b>1	Logarithmic $f(x)=\log_bx$f(x)=logb​x, $b>1$b>1

Each of these parent functions can be generalized further by its transformations.

 

Transformations

Parent functions can be transformed to create other members in a family of graphs. We will look at translations, reflections, and dilations.

For any function $f(x)$f(x), the family of functions can be represented by the transformed function $f(x)=a\left(n\left(x-h\right)\right)+k$f(x)=a(n(x−h))+k. The GeoGebra applet below allows you to see what happens when you change the values of $a$a, $h$h and $k$k for the family of quadratics. Be sure to look at both the graph and the equation when you move the sliders.

 

Translations

One common transformation is known as a translation.  This is a horizontal and/or vertical shift in the position of the curve relative to the $xy$xy-plane. 

Horizontal translations: For $h>0$h>0, if we replace the $x$x in $f(x)$f(x) with $\left(x-h\right)$(x−h) to get the transformed function $g(x)=f(x-h)$g(x)=f(x−h), $f(x)$f(x) is translated $h$h units to the right. If we replace $x$x with $\left(x+h\right)$(x+h) to get the transformed function $g(x)=f(x+h)$g(x)=f(x+h), $f(x)$f(x) is translated $h$h units to the left.

Vertical translations: For $k>0$k>0 and $f(x)$f(x), the transformed function $g(x)=f(x)+k$g(x)=f(x)+k is $f(x)$f(x) translated up $k$k units. The transformed function $g(x)=f(x)-k$g(x)=f(x)−k is $f(x)$f(x) translated down $k$k units.

So suppose we take the rational function $y=\frac{1}{x}$y=1x​ and translate it $3$3 units to the right and $5$5 units down. The new function becomes $y=\frac{1}{x-3}-5$y=1x−3​−5. The vertical and horizontal asymptotes are respectively $x=3$x=3 and $y=-5$y=−5.

If we take the parabola $y=x^2$y=x2 and shift it $1$1 unit to the right and $5$5 units up, the new function becomes $y=\left(x-1\right)^2+5$y=(x−1)2+5.

If we take the function $y=x^3$y=x3 and translate it $2$2 units to the right and $1$1 unit up, the new function is given by $y=\left(x-2\right)^3+1$y=(x−2)3+1. 

 

Dilations

Another type of transformation is commonly referred to as a dilation. This is when a curve is stretched or compressed by some factor other than $1$1 in the function's equation.

For example, the difference between $y=x^2$y=x2 to $y=3x^2$y=3x2 is the vertical dilation factor $3$3. For the same $x$x-value, every $y$y-value in $y=3x^2$y=3x2 is $3$3 times the $y$y-value in $y=x^2$y=x2. This means that the curve becomes steeper. Similarly, every function value of the function $y=\frac{1}{2}\log_2(x)$y=12​log2​(x) is half the associated function value of $y=\log_2(x)$y=log2​(x), so the curve will be compressed.

Vertical dilations: For $a>1$a>1 and $f(x)$f(x), the transformed function $g(x)=af(x)$g(x)=af(x) is $f(x)$f(x) stretched away from the $x$x-axis by a factor of $a$a. For $00<a<1 and $f(x)$f(x), the transformed function $g(x)=af(x)$g(x)=af(x) is $f(x)$f(x) compressed toward the $x$x-axis by a factor of $a$a.

Horizontal dilations: For $n>1$n>1 and $f(x)$f(x), the transformed function $g(x)=f(nx)$g(x)=f(nx) is $f(x)$f(x) compresses toward the $y$y-axis by a factor of $n$n. For $00<n<1 and $f(x)$f(x), the transformed function $g(x)=f(nx)$g(x)=f(nx) is $f(x)$f(x) stretched away from the $y$y-axis by a factor of $n$n.

 

Reflections

The last type of transformation is a reflection. Similar to what we have seen so far, we can have a vertical or horizontal reflection.

Vertical reflection: For $f(x)$f(x), the transformed function $g(x)=-f(x)$g(x)=−f(x) is $f(x)$f(x) reflected over the $x$x-axis. 

Horizontal reflection: For $f(x)$f(x), the transformed function $g(x)=f(-x)$g(x)=f(−x) is $f(x)$f(x) reflected over the $y$y-axis.

 

Combining transformations

When we combine translations, dilations, and reflections we can reposition and stretch/compress the parent function. 

Worked example

example 1

Describe the transformation of $f(x)=\frac{1}{x}$f(x)=1x​ to  $g(x)=\frac{5}{x-2}+3$g(x)=5x−2​+3. Hence, graph $g(x)$g(x).

Think: Transformation "inside" $f(x)$f(x) are horizontal and "outside" are vertical.

Do: Our parent function is $y=\frac{1}{x}$y=1x​ 

1. The first transformation gives us $y=\frac{5}{x}$y=5x​, so is a vertical dilation by factor of $5$5 away from the $x$x-axis.
2. Next we have a vertical translation up $3$3 to give us $y=\frac{5}{x}+3$y=5x​+3
3. Finally, we have a horizontal translation right $2$2 to give us $g(x)=\frac{5}{x-2}+3$g(x)=5x−2​+3

The graph of $g(x)$g(x) will have asymptotes of  $x=2$x=2 and $y=3$y=3 and be stretched away from the $x$x-axis to give us the graph in red below.

Summary

Translations

$g(x)=f(x)+k$g(x)=f(x)+k is the graph of $f(x)$f(x) translated vertically	If $k>0$k>0, then translated $k$k units up If $k<0$k<0, then translated $k$k units down
$g(x)=f(x-h)$g(x)=f(x−h) is the graph of $f(x)$f(x) translated horizontally	If $h>0$h>0, then translated $h$h units right If $h<0$h<0, then translated $h$h units left

 

Dilations

$g(x)=af(x)$g(x)=af(x) is the graph of $f(x)$f(x) dilated vertically	If $a>1$a>1, then vertical dilation (stretch) away from the $x$x-axis If$00<a<1, then vertical dilation (compression) toward the $x$x-axis
$g(x)=f(ax)$g(x)=f(ax) is the graph of $f(x)$f(x) dilated horizontally	If $a>1$a>1, then horizontal dilation (compression) toward the $y$y-axis If $00<a<1, then horizontal dilation (stretch) away from the $y$y-axis

 

Reflections

$g(x)=-f(x)$g(x)=−f(x) is the graph of $f(x)$f(x) reflected vertically	Reflected over the $x$x-axis
$g(x)=f(-x)$g(x)=f(−x) is the graph of $f(x)$f(x) reflected horizontally	Reflected over the $y$y-axis

Worked example

example 2

Point $A$A is  a point on the graph of the function $y=f\left(x\right)$y=f(x) with coordinates ($2$2, $5$5). 

(a) Determine the coordinates of point $A$A' after the transformation $y=2f\left(x+3\right)-3$y=2f(x+3)−3

Think: Start inside the bracket and work outwards and for each transformation consider if it is affecting the $x$xvalues or $y$y values. First $x+3$x+3 means $3$3 is subtracted from every $x$x value. Then the $2$2 means multiply every $y$y value by $2$2 and finally $-3$−3 means subtract $3$3 from every $y$y value. Apply each transformation to the coordinates in order.

Do: ($2$2, $5$5) → ($2-3$2−3, $5$5) → ($-1$−1, $5$5) → ( $-1$−1, $2\times5$2×5) → ($-1$−1,$10$10) → ($-1$−1, $10-3$10−3) → ($-1$−1,$7$7)

Therefore $A$A' is the point ($-1$−1, $7$7)

(b) Describe the transformations on $y=f\left(x\right)$y=f(x) by $y=2f\left(x+3\right)-3$y=2f(x+3)−3 

Do:  Write: There is a horizontal translation of $3$3 units left, followed by a vertical dilation of factor $2$2 from the $x$x axis, followed by a vertical translation of $3$3 units down. 

 

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5