The Central Idea
A basic power function has the form
\[ \boxed{f(x)=ax^n}, \]where \(a\) and \(n\) are constants and \(a\neq0\).
In the polynomial setting developed here, \(n\) will usually be a nonnegative integer:
\[ 0,\ 1,\ 2,\ 3,\ldots \]Examples include
\[ f(x)=3x^2, \] \[ g(x)=-2x^5, \]and
\[ h(x)=\frac12x^4. \]Each contains only one power term.
Power Functions as Polynomial Building Blocks
Every polynomial is built by adding power terms.
For example,
\[ P(x)=4x^5-3x^2+7x-6 \]combines
\[ 4x^5,\qquad -3x^2,\qquad 7x,\qquad -6. \]Studying individual power functions helps us understand the larger polynomials formed from them.
The Parent Power Functions
The simplest member of each power family is
\[ f(x)=x^n. \]The first several examples are
\[ x^0=1,\qquad x^1=x,\qquad x^2,\qquad x^3,\qquad x^4,\qquad x^5. \]These basic functions reveal a strong distinction between even and odd exponents.
The Constant Power
When \(n=0\), a polynomial power term becomes a constant.
For nonzero \(x\),
\[ x^0=1. \]Thus
\[ ax^0=a. \]In standard polynomial notation, we simply write
\[ \boxed{f(x)=a}. \]The Linear Power
When \(n=1\),
\[ f(x)=ax. \]This is a linear function passing through the origin, with
\[ \boxed{\text{slope}=a}. \]Linear relationships are therefore the first nonconstant power family.
The Quadratic Power
When \(n=2\),
\[ f(x)=ax^2. \]The parent function \(y=x^2\) has a U-shaped graph called a parabola. If \(a>0\), it opens upward. If \(a<0\), it opens downward.
Quadratic functions receive a full treatment in the next lesson.
The Cubic Power
When \(n=3\),
\[ f(x)=ax^3. \]The parent function \(y=x^3\) passes through the origin and has origin symmetry. Its shape is fundamentally different from \(x^2\) because the exponent is odd.
Even Powers
Examples include
\[ x^2,\qquad x^4,\qquad x^6,\ldots \]For an even integer power,
\[ (-x)^{2k}=x^{2k}. \]Therefore,
\[ \boxed{f(-x)=f(x)}. \]Such functions are called even functions.
Even Symmetry
The equation
\[ f(-x)=f(x) \]means the graph is symmetric about the \(y\)-axis.
For example,
\[ (-3)^4=3^4=81. \]Thus the points \((-3,81)\) and \((3,81)\) occur at equal heights on the graph of \(y=x^4\).
Odd Powers
Examples include
\[ x,\qquad x^3,\qquad x^5,\ldots \]For an odd integer power,
\[ (-x)^{2k+1}=-x^{2k+1}. \]Therefore,
\[ \boxed{f(-x)=-f(x)}. \]Such functions are called odd functions.
Odd Symmetry
The condition
\[ f(-x)=-f(x) \]gives symmetry about the origin.
For example,
\[ (-2)^3=-8 \]while
\[ 2^3=8. \]Thus \((-2,-8)\) and \((2,8)\) are symmetric through the origin.
Even and Odd Power Families
| Exponent | Example | Symmetry | Basic Sign Pattern |
|---|---|---|---|
| Even | \(x^4\) | \(y\)-axis | Nonnegative for all real \(x\) |
| Odd | \(x^5\) | Origin | Same sign as \(x\) |
The Role of the Coefficient
In
\[ f(x)=ax^n, \]the exponent determines the basic family, while \(a\) controls vertical scale and orientation.
Compare
\[ x^2,\qquad 3x^2,\qquad \frac12x^2. \]All belong to the same power family, but their outputs are scaled by different amounts.
Vertical Stretch and Compression
If
\[ |a|>1, \]output magnitudes are enlarged. This is commonly called a vertical stretch.
If
\[ 0<|a|<1, \]output magnitudes are reduced. This is commonly called a vertical compression.
Reflection Across the \(x\)-Axis
If \(a<0\), every output changes sign.
Therefore,
\[ y=-x^n \]is the reflection of
\[ y=x^n \]across the \(x\)-axis.
Example: Read the Structure
Consider
\[ f(x)=-3x^4. \]- The exponent is even.
- The graph has \(y\)-axis symmetry.
- The magnitude \(|a|=3\) produces a vertical stretch.
- The negative coefficient reflects the graph across the \(x\)-axis.
- Both ends fall.
Domain for Positive Integer Powers
If \(n\) is a positive integer, then \(ax^n\) is defined for every real input.
\[ \boxed{\text{Domain}=(-\infty,\infty)}. \]Range for Even Positive Powers
If \(n\) is positive and even, then \(x^n\ge0\).
Therefore, if \(a>0\),
\[ \boxed{\text{Range}=[0,\infty)}. \]If \(a<0\),
\[ \boxed{\text{Range}=(-\infty,0]}. \]Range for Odd Positive Powers
If \(n\) is positive and odd, the function can take every real value.
\[ \boxed{\text{Range}=(-\infty,\infty)}. \]Intercepts
For \(n>0\),
\[ f(0)=a(0)^n=0. \]Therefore, the graph passes through
\[ \boxed{(0,0)}. \]The origin is both the \(x\)-intercept and the \(y\)-intercept.
Evaluating a Power Function
Let
\[ f(x)=2x^5. \]Then
\[ f(3)=2(3)^5=2(243)=\boxed{486}. \]Negative Inputs
Compare the even-power function
\[ f(x)=3x^4 \]with the odd-power function
\[ g(x)=3x^5. \]At \(x=-2\),
\[ f(-2)=3(16)=\boxed{48}, \]while
\[ g(-2)=3(-32)=\boxed{-96}. \]Even powers remove the sign of the input; odd powers preserve it.
End Behavior
End behavior describes what happens as
\[ x\to\infty \]and
\[ x\to-\infty. \]For positive integer powers, end behavior depends on the parity of the exponent and the sign of the coefficient.
Even Power with Positive Coefficient
If \(n\) is even and \(a>0\),
\[ f(x)\to\infty \quad\text{as}\quad x\to\infty \]and
\[ f(x)\to\infty \quad\text{as}\quad x\to-\infty. \]Both ends rise.
Even Power with Negative Coefficient
If \(n\) is even and \(a<0\),
\[ f(x)\to-\infty \]at both ends.
Both ends fall.
Odd Power with Positive Coefficient
If \(n\) is odd and \(a>0\),
\[ f(x)\to-\infty \quad\text{as}\quad x\to-\infty \]and
\[ f(x)\to\infty \quad\text{as}\quad x\to\infty. \]The graph falls to the left and rises to the right.
Odd Power with Negative Coefficient
If \(n\) is odd and \(a<0\),
\[ f(x)\to\infty \quad\text{as}\quad x\to-\infty \]and
\[ f(x)\to-\infty \quad\text{as}\quad x\to\infty. \]The graph rises to the left and falls to the right.
End-Behavior Summary
| Exponent | Coefficient | Left End | Right End |
|---|---|---|---|
| Even | Positive | Up | Up |
| Even | Negative | Down | Down |
| Odd | Positive | Down | Up |
| Odd | Negative | Up | Down |
Higher Powers and Growth
For \(|x|>1\), higher positive powers grow faster in magnitude.
At \(x=3\),
\[ 3^2=9,\qquad 3^3=27,\qquad 3^4=81,\qquad 3^5=243. \]This is why high-degree terms eventually dominate lower-degree terms in a polynomial.
Behavior Near the Origin
For
\[ 0<|x|<1, \]higher positive powers become smaller in magnitude.
For example,
\[ \left(\frac12\right)^2=\frac14, \qquad \left(\frac12\right)^3=\frac18, \qquad \left(\frac12\right)^4=\frac1{16}. \]Higher powers therefore appear flatter near the origin and steeper farther away.
The Scaling Law
If
\[ f(x)=ax^n, \]then
\[ f(kx)=a(kx)^n. \]Therefore,
\[ \boxed{f(kx)=k^n f(x)}. \]Multiplying the input by \(k\) multiplies the output by \(k^n\).
Example: Area Scaling
The area of a square with side length \(s\) is
\[ A(s)=s^2. \]Doubling the side gives
\[ A(2s)=(2s)^2=4s^2. \]Therefore, doubling the side multiplies the area by
\[ \boxed{4}. \]Example: Volume Scaling
The volume of a cube is
\[ V(s)=s^3. \]Doubling the side gives
\[ V(2s)=(2s)^3=8s^3. \]Therefore, doubling all linear dimensions multiplies the volume by
\[ \boxed{8}. \]A Ratio Property
For nonzero \(x_1\) and \(x_2\),
\[ \frac{f(x_2)}{f(x_1)} = \frac{ax_2^n}{ax_1^n}. \]Therefore,
\[ \boxed{ \frac{f(x_2)}{f(x_1)} = \left(\frac{x_2}{x_1}\right)^n }. \]The exponent determines how input ratios become output ratios.
Power Functions Are Not Exponential Functions
A power function places the variable in the base:
\[ f(x)=x^3. \]An exponential function places the variable in the exponent:
\[ g(x)=3^x. \]These are different function families.
Power Functions Beyond Polynomials
In broader mathematics, power functions can also include
\[ x^{-2}, \qquad x^{1/2}, \qquad x^{3/2}. \]These are power functions because the variable is raised to a fixed power.
They are not polynomial functions because polynomial exponents must be nonnegative integers.
Negative-Power Example
Consider
\[ f(x)=x^{-2}. \]Since
\[ x^{-2}=\frac1{x^2}, \]the input \(x=0\) is excluded from the domain.
This differs fundamentally from positive-integer polynomial power functions.
Fractional-Power Example
Consider
\[ f(x)=x^{1/2}=\sqrt{x}. \]Over the real numbers,
\[ \boxed{\text{Domain}=[0,\infty)}. \]This is a power function but not a polynomial function.
Why Integer Powers Matter Here
Positive integer powers are the direct building blocks of polynomials.
Negative and fractional powers are important, but they introduce domain restrictions and other behavior that belongs to broader function study.
This lesson therefore emphasizes polynomial power functions while distinguishing the broader terminology.
Power Functions and Leading Terms
Consider
\[ P(x)=4x^6-3x^2+2x-1. \]For large \(|x|\), the term
\[ 4x^6 \]dominates the lower-degree terms.
Therefore, the polynomial eventually behaves much like
\[ y=4x^6. \]This is why power-function behavior becomes the foundation for analyzing polynomial end behavior.
Example: Predict the End Behavior
Consider
\[ f(x)=-5x^7. \]The exponent is odd and the coefficient is negative.
Therefore,
\[ f(x)\to\infty \quad\text{as}\quad x\to-\infty \]and
\[ f(x)\to-\infty \quad\text{as}\quad x\to\infty. \]Example: Determine Symmetry and Range
Consider
\[ f(x)=-2x^6. \]The exponent is even, so the graph has \(y\)-axis symmetry.
Since \(x^6\ge0\) and the coefficient is negative,
\[ f(x)\le0. \]Therefore,
\[ \boxed{\text{Range}=(-\infty,0]}. \]Common Error: Confusing Coefficient and Exponent
In
\[ f(x)=4x^3, \]the coefficient is \(4\), while the exponent is \(3\).
The coefficient scales the function. The exponent determines the power family.
Common Error: Assuming Every Power Function Is Polynomial
The functions
\[ x^{-3} \]and
\[ x^{1/2} \]are power functions, but they are not polynomial functions.
Common Error: Assuming Every Polynomial Is a Power Function
The polynomial
\[ x^3+2x+1 \]is not a single power function because it contains several different powers of \(x\).
It is a sum of power terms.
A Reliable Procedure for Analyzing \(ax^n\)
- Identify the exponent \(n\).
- Determine whether \(n\) is even or odd.
- Identify the sign of \(a\).
- Use \(|a|\) to describe vertical scaling.
- Use the sign of \(a\) to identify reflection.
- Determine symmetry.
- Determine domain and range.
- Determine end behavior.
- Identify the origin as an intercept when \(n>0\).
Practice 1: Identify the Structure
For
\[ f(x)=-4x^5, \]identify the coefficient, exponent, symmetry, and end behavior.
Practice 2: Evaluate
Let
\[ f(x)=3x^4. \]Find \(f(2)\) and \(f(-2)\).
Practice 3: Range
Determine the range of
\[ f(x)=-7x^8. \]Practice 4: End Behavior
Describe the end behavior of
\[ f(x)=2x^9. \]Practice 5: Scaling
Suppose
\[ f(x)=x^3. \]If the input is multiplied by \(5\), by what factor is the output multiplied?
Practice 6: Power or Exponential?
Classify each expression:
- \(x^6\)
- \(6^x\)
- \(3x^4\)
- \(2^x+1\)
Practice 7: Polynomial or Non-Polynomial Power Function?
- \(x^7\)
- \(x^{-2}\)
- \(x^{1/3}\)
- \(-5x^4\)
Practice 8: Modeling
The volume of a cube is
\[ V(s)=s^3. \]If the side length is multiplied by \(3\), by what factor does the volume change?
Practice Answers
Practice 1
- coefficient: \(\boxed{-4}\);
- exponent: \(\boxed{5}\);
- the exponent is odd;
- the function has origin symmetry;
- the left end rises and the right end falls.
Practice 2
\[ f(2)=3(2)^4=\boxed{48}, \] \[ f(-2)=3(-2)^4=\boxed{48}. \]Practice 3
Since \(x^8\ge0\) and the coefficient is negative,
\[ \boxed{\text{Range}=(-\infty,0]}. \]Practice 4
The exponent is odd and the coefficient is positive:
\[ f(x)\to-\infty \quad\text{as}\quad x\to-\infty, \] \[ f(x)\to\infty \quad\text{as}\quad x\to\infty. \]Practice 5
\[ 5^3=\boxed{125}. \]Practice 6
- \(x^6\): power function.
- \(6^x\): exponential function.
- \(3x^4\): power function.
- \(2^x+1\): exponential-type function, not a power function.
Practice 7
- \(x^7\): polynomial power function.
- \(x^{-2}\): power function, not polynomial.
- \(x^{1/3}\): power function, not polynomial.
- \(-5x^4\): polynomial power function.
Practice 8
Scaling the input by \(3\) scales the output by
\[ 3^3=\boxed{27}. \]What to Remember
A basic power function has the form
\[ \boxed{f(x)=ax^n}. \]For polynomial power functions, \(n\) is a nonnegative integer.
- Even positive powers have \(y\)-axis symmetry.
- Odd positive powers have origin symmetry.
- The coefficient controls vertical scale and reflection.
- End behavior depends on exponent parity and coefficient sign.
Power functions obey the scaling law
\[ \boxed{f(kx)=k^n f(x)}. \]They provide the basic shapes and growth patterns that later explain the behavior of general polynomials.
Where We Go Next
The most important degree-\(2\) power relationship is the quadratic family. Adding linear and constant terms turns the basic function \(ax^2\) into the general quadratic function.
The next lesson, Quadratic Functions and Equations, develops quadratic graphs, vertices, intercepts, standard and factored forms, and the relationship between quadratic functions and quadratic equations.