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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-F.BF.3b

Using f(x) + k, k f(x), f(kx), and f(x + k): i) identify the effect on the graph when replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); ii) find the value of k given the graphs; iii) write a new function using the value of k; and iv) use technology to experiment with cases and explore the effects on the graph. *Include recognizing even and odd functions from their graphs.

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