Common Core Algebra 2 - 2020 Edition
1.02 Function families and transformations
Lesson

## Parent functions

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

$f(x)=x^2$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)=3x

Absolute value

$f(x)=\left|x\right|$f(x)=|x|

Rational - Linear

$f(x)=\frac{1}{x}$f(x)=1x

$f(x)=\frac{1}{x^2}$f(x)=1x2

Exponential

$f(x)=b^x$f(x)=bx , $b>1$b>1

Logarithmic

$f(x)=\log_bx$f(x)=logbx, $b>1$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(xh))+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 $\left(x-h\right)$(xh) to get the transformed function $g(x)=f(x-h)$g(x)=f(xh)$f(x)$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=1x35. 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=(x1)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=(x2)3+1

Did you know?

Transformations inside the function $f(x)$f(x), such as $f(x+3)$f(x+3), are horizontal.

Transformations outside the function $f(x)$f(x), such as $f(x)+3$f(x)+3, are vertical.

### 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=12log2(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

Describe the transformation of $f(x)=\frac{1}{x}$f(x)=1x to  $g(x)=\frac{5}{x-2}+3$g(x)=5x2+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)=5x2+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