Virginia SOL Algebra 2 - 2020 Edition
topic badge
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

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)=3x

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)=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(xh) 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

 

Practice questions

Question 1

Consider the function $y=-\frac{1}{2}x^2$y=12x2

  1. Complete the following table of values.

    $x$x $-2$2 $-1$1 $0$0 $1$1 $2$2
    $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  2. Plot the graph.

    Loading Graph...

Question 2

How do we shift the graph of $y=g\left(x\right)$y=g(x) to get the graph of $y=g\left(x+6\right)$y=g(x+6)?

  1. Move the graph to the left by $6$6 units.

    A

    Move the graph to the right by $6$6 units.

    B

    Move the graph to the left by $6$6 units.

    A

    Move the graph to the right by $6$6 units.

    B

Question 3

Consider the function $y=\frac{2}{x}$y=2x

  1. Complete the following table of values.

    $x$x $-2$2 $-1$1 $\frac{-1}{2}$12 $\frac{1}{2}$12 $1$1 $2$2
    $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  2. Plot the graph.

    Loading Graph...

  3. In which quadrants does the graph lie?

    $3$3

    A

    $2$2

    B

    $1$1

    C

    $4$4

    D

    $3$3

    A

    $2$2

    B

    $1$1

    C

    $4$4

    D

Question 4

Consider the cubic function $y=4x^3-3$y=4x33

  1. Is the cubic increasing or decreasing from left to right?

    Increasing

    A

    Decreasing

    B

    Increasing

    A

    Decreasing

    B
  2. Is the cubic more or less steep than the function $y=x^3$y=x3 ?

    More steep

    A

    Less steep

    B

    More steep

    A

    Less steep

    B
  3. What are the coordinates of the point of inflection of the function?

    Inflection ($\editable{}$, $\editable{}$)

  4. Plot the graph $y=4x^3-3$y=4x33

    Loading Graph...

 

Outcomes

AII.6a

For absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic functions recognize the general shape of function families

AII.6b

For absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic functions use knowledge of transformations to convert between equations and the corresponding graphs of functions

What is Mathspace

About Mathspace