Power functions
Functions with a form like $y=x^2$y=x2, $y=5x^3$y=5x3 and $y=ax^n$y=axn are known as power functions.
The following applet lets you see the general shape of power functions where $a=1$a=1 and the powers are positive integers.
- What happens at the extremities of the graph? Is it different for odd and even powers?
- Also look carefully at the shape of the curves as they come close to the origin.
- Do you notice anything else about the curves?
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Odd degree power functions all have the following properties:
- The graph has $180$180$^\circ$° rotational symmetry about the origin, that is $f(-x)=-f(x)$f(−x)=−f(x) for all $x$x,
- The graph moves in opposite directions at the extremities
- They all pass through $\left(0,0\right)$(0,0), $\left(-1,-1\right)$(−1,−1) and $\left(1,1\right)$(1,1)
- Domain and range are both $\left(-\infty,\infty\right)$(−∞,∞)
- As $n$n increases, within the interval $\left[-1,1\right]$[−1,1] the graphs lie closer to the $x$x-axis, 'flatter' about the point of inflection
- As $n$n increases and $x$x extends beyond the interval $\left[-1,1\right]$[−1,1], the functions values grow further away from the $x$x-axis than those of lower powers
Even degree power functions all have the following properties:
- The graph is symmetric about the y-axis, that is $f(-x)=f(x)$f(−x)=f(x) for all $x$x
- The graph moves in same directions at the extremities
- They all pass through $\left(0,0\right)$(0,0), $\left(-1,1\right)$(−1,1) and $\left(1,1\right)$(1,1)
- Domain is and $\left(-\infty,\infty\right)$(−∞,∞)and the range is$\left[0,\infty\right)$[0,∞)
- As $n$n increases, within the interval $\left[-1,1\right]$[−1,1] the graphs lie closer to the $x$x-axis, 'flatter' about the turning point
- As $n$n increases and $x$x extends beyond the interval $\left[-1,1\right]$[−1,1], the functions values grow further away from the $x$x-axis than those of lower powers
Beyond linear, quadratic, cubic functions: Quartic functions
We have already seen quadratics and cubics, but we can go one step further to examine quartics. Just like quadratics and cubics, quartics are made up of a sum of power functions, except that the largest power is $4$4.
Let's look at quartics more generally. That is functions of the form :
$P(x)=ax^4+bx^3+cx^2+dx+e$P(x)=ax4+bx3+cx2+dx+e
Quartics are also known as 'polynomials of degree $4$4'. In fact all these types of functions are polynomials of different degree (even constant functions!) but for this course quartics are the largest degree we look at.
Since the degree of a quartic is even ($4$4), both ends of the graph will have the same sign, either both positive or both negative. Determining which depends on the leading coefficient, $a$a. We can see from the diagram that for large positive values (or large negative values) of $x$x the sign of $y$y is determined by the leading coeffiecient.
| Quartics | $a>0$a>0 | $a<0$a<0 |
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Just as quadratics or cubics can have up to $2$2 or $3$3 unique $x$x-intercepts respectively, a quartic can have up to $4$4.
In the same way, quadratics or cubics can have up to $1$1 or $2$2 turning points respectively. A quartic can have up to $3$3. Keep in mind that these are upper limits and it is not uncommon to see a quartic with less than $4$4 $x$x-intercepts and less than $3$3 turning points.
Details of key features can be found using calculus and technology or from factored forms of the function. Even though there is the more general form mention above, we are going to concentrate on a specific type of quartic.
Transformations of the power function $x^4$x4.
We can manipulate the function $x^4$x4 in ways we have seen before, by translation, dilation or reflection.
| Transformation | Formula | Effect |
|---|---|---|
| Vertical Dilation | $y=ax^4$y=ax4 |
The $y$y-values are now $a$a times their original value. This stretches the graph away from the $x$x-axis by a factor of $a$a. |
| Horizontal Translation | $y=(x-b)^4$y=(x−b)4 |
The $y$y-value of the original function is now associated with the $x$x-value $b$b units to the right. This shifts the graph to the right by $b$b units. |
| Vertical Translation | $y=x^4+c$y=x4+c |
The $y$y-value is now $c$c units more than it was originally. This means the graph is shifted upwards by $c$c units. |
| Vertical Reflection | $y=-x^4$y=−x4 |
Each $y$y-value is swapped for its negative. This means that the graph is flipped over the $x$x-axis. |
We can see these transformation take place with the following applet:
Practice questions
Question 1
Consider the function $y=-x^4+4$y=−x4+4
Determine the leading coefficient of the polynomial function.
Is the degree of the polynomial odd or even?
Odd
AEven
BWhich of the following is true of the graph of the function?
It rises to the left and falls to the right
AIt rises to the left and rises to the right
BIt falls to the left and rises to the right
CIt falls to the left and falls to the right
DWhich of the following is the graph of $y=-x^4+4$y=−x4+4?
Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...D
Question 2
We want to plot the graph of the function $y=-\left(x-1\right)^4+3$y=−(x−1)4+3.
Which of the following best describes the behaviour of this function?
As $x$x$\to$→$-\infty$−∞, $y$y$\to$→$\infty$∞.
As $x$x$\to$→$\infty$∞, $y$y$\to$→$-\infty$−∞.
AAs $x$x$\to$→$-\infty$−∞, $y$y$\to$→$-\infty$−∞.
As $x$x$\to$→$\infty$∞, $y$y$\to$→$-\infty$−∞.
BAs $x$x$\to$→$-\infty$−∞, $y$y$\to$→$\infty$∞.
As $x$x$\to$→$\infty$∞, $y$y$\to$→$\infty$∞.
CAs $x$x$\to$→$-\infty$−∞, $y$y$\to$→$-\infty$−∞.
As $x$x$\to$→$\infty$∞, $y$y$\to$→$\infty$∞.
DWhat is the value of $y$y at the $y$y-intercept of the function?
Complete the following table of values.
$x$x $-1$−1 $0$0 $1$1 $2$2 $3$3 $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Using the table of values above, plot the graph of the function.
Loading Graph...