Recall that functions like $y=x^2$y=x2, $y=2x^2$y=2x2 and $y=3x^3$y=3x3, etc. are known as power functions.
Exploration
The following applet lets us see the effect of increasing the powers of $x$x for the function $y=x^n$y=xn.
- Note how for odd powers the graph moves in opposite directions at the extremities, and for even powers the graph moves off in the same direction at the extremities.
- Also look carefully at the shape of the curves as they come close to the origin.
- What else do you notice about the curves?
For odd powers greater than $1$1, the curves rise toward the origin with a decreasing positive gradient until, when the origin is reached, they become momentarily horizontal and then continue upward with an ever increasing gradient. That "terracing" shape around the origin is known as a horizontal inflection.
For even powers, the curve changes direction at the origin. This is called a minimum turning point.
Concepts of power functions
Power functions have the general form $y=ax^n$y=axn where $n$n is any number. When $n$n is a positive integer, we can learn to sketch these functions by considering a few simple principles of powers.
Whenever a non-zero number is raised to an even integer power, the result is always positive. So for example $\left(3\right)^4=+81$(3)4=+81 and also $\left(-3\right)^4=+81$(−3)4=+81. This results in the graphs of all power functions that have even integral powers to have some similar properties and general shape.
Whenever a non-zero number is raised to an odd integer power, the result is positive when that number is also positive, and negative when that number is also negative. So $\left(3\right)^3=+27$(3)3=+27 and $\left(-3\right)^3=-27$(−3)3=−27. This results in the graphs of all power functions that have odd integer powers to have some similar properties and general shape.
Whenever a number lies in the interval $-1
To demonstrate the effect of these principles on power graphs, we have graphed the functions given by $y=x$y=x, $y=x^2$y=x2, $y=x^3$y=x3 and $y=x^4$y=x4 between $x=-2$x=−2 and $x=2$x=2.
Effect of the coefficient $a$a
The coefficient $a$a in the power function form $y=ax^n$y=axn scales the function values of $y=x^n$y=xn by a factor of $a$a. So for example, if $a=\frac{1}{2}$a=12, then each value of the function $y=x^n$y=xn is halved, so the graph looks compressed. If $a=2$a=2, the function values are doubled and the graph looks vertically stretched instead.
If $a$a happens to be negative, the effect on the graph is a reflection across the $x$x-axis. That is, function values that are negative become positive and function values that are positive become negative.
As an example compare the graphs of $y=x^2,y=2x^2,y=-\frac{1}{2}x^2$y=x2,y=2x2,y=−12x2 and $y=-3x^2$y=−3x2 as shown here:
Practice questions
Question 1
How does the graph of $y=\frac{1}{2}x^3$y=12x3 differ to the graph of $y=x^3$y=x3?
One is a reflection of the other across the $y$y-axis
A$y$y increases more rapidly on $y=\frac{1}{2}x^3$y=12x3 than on $y=x^3$y=x3
B$y=\frac{1}{2}x^3$y=12x3 is a horizontal shift of $y=x^3$y=x3
C$y$y increases more slowly on $y=\frac{1}{2}x^3$y=12x3 than on $y=x^3$y=x3
D
Question 2
Consider the parabola $y=x^2-3$y=x2−3.
Complete the table of values.
$x$x $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Use the graph of $y=x^2$y=x2 to sketch a graph of $y=x^2-3$y=x2−3.
Loading Graph...What is the $y$y value of the $y$y-intercept of the graph $y=x^2-3$y=x2−3?
Adding a constant to the equation $y=x^2$y=x2 corresponds to which transformation of its graph?
Vertical shift
ASteepening of the graph
BHorizontal shift
CReflection about an axis
D
Question 3
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?
Transformations of power functions
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 translating the curve either horizontally or vertically. The important point, however, is that the essential character of the curve doesn't change.
The essential character of the curve doesn't change.
We can dilate 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 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 translate it horizontally or vertically (or both). 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 $y=a\left(x-h\right)^n$y=a(x−h)n.
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 dilate the curve away from the y-axis.
- Translate the graph to the right by $3$3 units.
- Translate the curve up by $5$5 units.
Explore
The following applet shows $y=a\left(x-h\right)^n+k$y=a(x−h)n+k. Change the 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.
Practice questions
QUESTION 4
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 5
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