Transformations of Power Functions
Recall that functions with a form like $y=x^2$y=x2, $y=2x^2$y=2x2 and $y=3x^3$y=3x3, etc. are known as power functions.
Power functions have the general form $y=ax^n$y=axn where $n$n is a positive integer, and we now need to extend our understanding of these functions when they undergo certain transformations.
A transformation of a curve might involve a distortion in the shape of the curve. It might involve a reflection of the curve in the $x$x - axis or $y$y - axis. It might simply involve shifting the curve either horizontally or vertically. The important point however is that the essential character of the curve doesn't change.
We can stretch, compress or reflect the curve $y=ax^n$y=axn by changing the value of the coefficient $a$a.
A large value of $a$a causes the curve to become steeper faster. A small value of $a$a causes the curve to be become shallower. A negative value of $a$a causes the curve to reflect in the $x$x - axis.
We can also change a curve's position relative to the origin. We can shift it horizontally or vertically (or both). This type of transformation is known as a translation.
We can perform a translation on $y=ax^n$y=axn either vertically or horizontally:
- A vertical translation of $k$k units so that the power function looks like $y=ax^n+k$y=axn+k.
- A horizontal translation of $h$h units so that the function looks like $y=a\left(x-h\right)^2$y=a(x−h)2.
Watch how the graph of the function $y=x^3$y=x3 progressively changes to the graph of $y=\frac{1}{2}\left(x-3\right)^3+5$y=12(x−3)3+5.
From the graph of $y=x^3$y=x3 we:
- Halve each $y$y value to compress the curve.
- Shift the graph to the right by $3$3 units.
- Lift the curve up by $5$5 units .
Now its your turn. The following applet shows $y=a\left(x-h\right)^n+k$y=a(x−h)n+k and you can change all four constants $a$a, $h$h, $n$n and $k$k to see the effects. It's important that you experiment with different combinations of constants to really understand the way transformations work.
Worked Examples
QUESTION 1
Consider the equation $y=-x^2$y=−x2
Complete the following table of values.
$x$x $-3$−3 $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $3$3 $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Plot the points in the table of values.
Loading Graph...Hence plot the curve.
Loading Graph...Are the $y$y values ever positive?
No
AYes
BWhat is the maximum $y$y value?
Write down the equation of the axis of symmetry.
QUESTION 2
Consider the quadratic function $y=\left(x+3\right)^2-5$y=(x+3)2−5.
Calculate the $y$y-intercept.
Is the graph concave up or concave down?
Concave up
AConcave down
BWhat is the minimum $y$y value?
What $x$x value corresponds to the minimum $y$y value?
What are the coordinates of the vertex?
Vertex $=$=$\left(\editable{},\editable{}\right)$(,)
Graph the parabola.
Loading Graph...What is the axis of symmetry of the parabola?
QUESTION 3
Consider the function $y=\left(x-2\right)^3$y=(x−2)3.
Complete the following table of values.
$x$x $0$0 $1$1 $2$2 $3$3 $4$4 $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Sketch a graph of the function.
Loading Graph...What transformation of the graph $y=x^3$y=x3 results in the graph of $y=\left(x-2\right)^3$y=(x−2)3?
horizontal translation $2$2 units to the left
Avertical translation $2$2 units down
Bhorizontal translation $2$2 units to the right
Cvertical translation $2$2 units up
D