Combining transformations
Below is a summary of the transformations we have encountered for a general function $f\left(x\right)$f(x):
| Transformation | Effect on graph | Effect on coordinates |
|---|---|---|
| $-f\left(x\right)$−f(x) | reflects $f\left(x\right)$f(x) across the $x$x-axis | multiplies $y$y-coordinate by $-1$−1 |
| $kf\left(x\right)$kf(x) | dilates $f\left(x\right)$f(x) by a factor of $k$k from the $x$x-axis | multiplies $y$y-coordinate by $k$k |
| $f\left(x\right)+c$f(x)+c | translates $f\left(x\right)$f(x) by $c$c units vertically | adds $c$c to each $y$y-coordinate |
| $f\left(x-b\right)$f(x−b) | translates $f\left(x\right)$f(x) by $b$b units horizontally | adds $b$b to each $x$x-coordinate |
| $f\left(-x\right)$f(−x) | reflects $f\left(x\right)$f(x) across the $y$y-axis | multiplies $x$x-coordinate by $-1$−1 |
| $f\left(ax\right)$f(ax) | dilates $f\left(x\right)$f(x) by a factor of $\frac{1}{a}$1a from the $y$y-axis | divides $x$x-coordinates by $a$a |
When we combine translations, dilations and reflections we can reposition and stretch/compress the parent function. We can combine any number of these transformations to arrive at the general form $y=kf(ax+b)+c$y=kf(ax+b)+c. The geogebra applet below allows us to see what happens when we change the values of $k$k, $a$a, $b$b and $c$c using the sliders provided. Be sure to look at both the graph and the equation when you move the sliders. Transformations affect the sign, the symmetry and often, the asymptotes of a 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, and hence sketch a graph of $g(x)$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 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.
- Next we have a vertical translation up $3$3 to give us $y=\frac{5}{x}+3$y=5x+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.
The importance of the order of transformations
Although it is possible to look at a function like $g(x)=\frac{5}{x-2}+3$g(x)=5x−2+3 above and consider the transformations independently, the order in which we apply a series of transformations to a basic function can mean we end up with a different final equation and sketch.
Consider the example of the parabola $y=x^2$y=x2. Let's say we decide to translate it $3$3 units to the right, then compress it towards the $y$y-axis by a factor of $2$2. The equation would have progressed through these transformations, from $y=x^2$y=x2 $\rightarrow$→ $y=(x-3)^2$y=(x−3)2 $\rightarrow$→ $y=(2x-3)^2$y=(2x−3)2. In this case we can see that the vertex would move from the origin to $x=3$x=3 and then finally to $x=\frac{3}{2}$x=32.
But, if we were to compress towards the $y$y-axis first then translate $3$3 units, we will end up with a different vertex. We would have passed from $y=x^2$y=x2 $\rightarrow$→ $y=(2x)^2$y=(2x)2 $\rightarrow$→ $y=(2(x-3))^2$y=(2(x−3))2. The vertex would have stayed at the origin through the first transformation, then shifted $3$3 units to $x=3$x=3. Hence, order is important.
The above example also shows us one very important step in finding horizontal translations of functions - if a horizontal dilation has occurred, we must factorise to find the horizontal shift. For our parabola $y=(2x-3)^2$y=(2x−3)2 we must rewrite the equation as $y=(2(x-\frac{3}{2}))^2$y=(2(x−32))2 to find the vertex at $x=\frac{3}{2}$x=32 .
In general terms, the horizontal shift of $y=kf(ax+b)+c$y=kf(ax+b)+c is $\frac{b}{a}$ba units to the left, since the form can also be written as $y=kf(a(x+\frac{b}{a}))+c$y=kf(a(x+ba))+c.
Practice questions
Question 1
Use the graph of $y=\left|x\right|$y=|x| to draw the graph of $y=\left|x-4\right|+4$y=|x−4|+4.
- Loading Graph...
Question 2
Write a fully simplified equation for when the graph of $y=\log_4x$y=log4x is translated seven units downward, six units to the left, and then reflected across the $x$x-axis.
Question 3
Consider the exponential equation $y=5^{3x+3}-1$y=53x+3−1.
Identify the:
horizontal translation
vertical translation
horizontal dilation factor
Question 4
For the function $y=-10\left(x+8\right)^2-9$y=−10(x+8)2−9, which of the following describes the transformations that have occurred?
Starting with $y=x^2$y=x2, the function is translated $8$8 units left, undergoes a change in dilation of $10$10 and is translated $9$9 units down.
AStarting with $y=x^2$y=x2, the function is translated $9$9 units left, undergoes a change in dilation of $10$10, is reflected across the horizontal axis and is translated $8$8 units down.
BStarting with $y=x^2$y=x2, the function is translated $8$8 units right, undergoes a change in dilation of $10$10, is reflected across the horizontal axis and is translated $9$9 units up.
CStarting with $y=x^2$y=x2, the function is translated $8$8 units left, undergoes a change in dilation of $10$10, is reflected across the horizontal axis and is translated $9$9 units down.
D
Transforming unknown functions
We don't need an equation, or a familiar graph of one of our key families of functions, to perform these transformations. Our understanding of function notation will guide us to make a sketch of a new graph. To create a transformed graph we can look at the key features of the original function, such as any intercepts or turning points, and transform these accordingly as a first step.
Practice questions
Question 5
Use the graph of $y=f\left(x\right)$y=f(x) below to sketch the graph of $y=f\left(x\right)+4$y=f(x)+4 on the same axes:
- Loading Graph...
Question 6
Use the graph of $y=f\left(x\right)$y=f(x) to sketch a graph of $y=\frac{1}{2}f\left(x+4\right)$y=12f(x+4)$-$−$2$2.
- Loading Graph...
Question 7
If the maximum value of $y=f\left(x\right)$y=f(x) is $6$6, what is the maximum value of $y=\frac{f\left(x+2\right)}{2}-3$y=f(x+2)2−3?