Lesson

Functions with a form like $y=x^2$`y`=`x`2, $y=2x^2$`y`=2`x`2 and $y=3x^3$`y`=3`x`3, etc. are known as power functions.

The following applet lets you see the effect of increasing the powers of $x$`x` for the function $y=x^n$`y`=`x``n`.

- 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.

Power functions have the general form $y=ax^n$`y`=`a``x``n` 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 $-1`x`$<$<$1$1, squaring it or cubing it (or indeed raising it to any integer power) reduces the absolute value of its size. So $\left(-0.5\right)^3=-0.125$(−0.5)3=−0.125 and $\left(0.9\right)^2=0.81$(0.9)2=0.81 .

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`=`x`2, $y=x^3$`y`=`x`3 and $y=x^4$`y`=`x`4 between $x=-2$`x`=−2 and $x=2$`x`=2.

Note the following four properties:

- Each graph passes through the origin.
- The function values for $y=x^2$
`y`=`x`2, $y=x^3$`y`=`x`3 and $y=x^4$`y`=`x`4 within the interval $\left[-1,1\right]$[−1,1] lie closer to the $x$`x`-axis than the function values for $y=x$`y`=`x`. As $x$`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 $y=x$`y`=`x`. - The
*odd*powered graphs of $y=x$`y`=`x`and $y=x^3$`y`=`x`3 pass through $\left(-1,-1\right)$(−1,−1) and $\left(1,1\right)$(1,1) and have rotational symmetry about the origin. Specifically the function value $f\left(-x\right)$`f`(−`x`) is the negative of the function value $f\left(x\right)$`f`(`x`). That is $f\left(-x\right)=-f\left(x\right)$`f`(−`x`)=−`f`(`x`). - The
*even*powered graphs of $y=x^2$`y`=`x`2 and $y=x^4$`y`=`x`4 pass through $\left(-1,1\right)$(−1,1) and $\left(1,1\right)$(1,1) and possess reflective symmetry about the $y$`y`- axis. specifically the function value $f\left(-x\right)$`f`(−`x`) is the same as the function value $f\left(x\right)$`f`(`x`). That is $f\left(-x\right)=f\left(x\right)$`f`(−`x`)=`f`(`x`).

These properties hold for all integral powers.

The coefficient $a$`a` in the power function form $y=ax^n$`y`=`a``x``n` scales the function values of $y=x^n$`y`=`x``n` 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`=`x``n` 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`=`x`2,`y`=2`x`2,`y`=−12`x`2 and $y=-3x^2$`y`=−3`x`2 as shown here:

How does the graph of $y=\frac{1}{2}x^3$`y`=12`x`3 differ to the graph of $y=x^3$`y`=`x`3?

One is a reflection of the other about the $y$

`y`-axisA$y$

`y`increases more rapidly on $y=\frac{1}{2}x^3$`y`=12`x`3 than on $y=x^3$`y`=`x`3B$y=\frac{1}{2}x^3$

`y`=12`x`3 is a horizontal shift of $y=x^3$`y`=`x`3C$y$

`y`increases more slowly on $y=\frac{1}{2}x^3$`y`=12`x`3 than on $y=x^3$`y`=`x`3DOne is a reflection of the other about the $y$

`y`-axisA$y$

`y`increases more rapidly on $y=\frac{1}{2}x^3$`y`=12`x`3 than on $y=x^3$`y`=`x`3B$y=\frac{1}{2}x^3$

`y`=12`x`3 is a horizontal shift of $y=x^3$`y`=`x`3C$y$

`y`increases more slowly on $y=\frac{1}{2}x^3$`y`=12`x`3 than on $y=x^3$`y`=`x`3D

Consider the parabola $y=x^2-3$`y`=`x`2−3.

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`=`x`2 to sketch a graph of $y=x^2-3$`y`=`x`2−3.Loading Graph...What is the $y$

`y`value of the $y$`y`-intercept of the graph $y=x^2-3$`y`=`x`2−3?Adding a constant to the equation $y=x^2$

`y`=`x`2 corresponds to which transformation of its graph?Vertical shift

ASteepening of the graph

BHorizontal shift

CReflection about an axis

DVertical shift

ASteepening of the graph

BHorizontal shift

CReflection about an axis

D

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

BConcave 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?

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

Apply graphical methods in solving problems