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 |
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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 |
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Each of these parent functions can be generalized further by its 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.
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.
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.
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 $0
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.
When we combine translations, dilations, and reflections we can reposition and stretch/compress the parent function.
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
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.
Translations
$g(x)=f(x)+k$g(x)=f(x)+k is the graph of $f(x)$f(x) translated vertically |
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$g(x)=f(x-h)$g(x)=f(x−h) is the graph of $f(x)$f(x) translated horizontally |
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Dilations
$g(x)=af(x)$g(x)=af(x) is the graph of $f(x)$f(x) dilated vertically | |
$g(x)=f(ax)$g(x)=f(ax) is the graph of $f(x)$f(x) dilated horizontally |
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 |
Consider the function $y=-\frac{1}{2}x^2$y=−12x2
Complete the following table of values.
$x$x | $-2$−2 | $-1$−1 | $0$0 | $1$1 | $2$2 |
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$y$y | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |
Plot the graph.
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)?
Move the graph to the left by $6$6 units.
Move the graph to the right by $6$6 units.
Consider the function $y=\frac{2}{x}$y=2x
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 | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |
Plot the graph.
In which quadrants does the graph lie?
$3$3
$2$2
$1$1
$4$4
Consider the cubic function $y=4x^3-3$y=4x3−3
Is the cubic increasing or decreasing from left to right?
Increasing
Decreasing
Is the cubic more or less steep than the function $y=x^3$y=x3 ?
More steep
Less steep
What are the coordinates of the point of inflection of the function?
Inflection ($\editable{}$, $\editable{}$)
Plot the graph $y=4x^3-3$y=4x3−3