Lesson 5.2 : Power Functions and Polynomial Functions

New Concept

This lesson introduces power functions, in the form $f(x) = kx^p$, and polynomial functions, which are sums of power functions. We will learn to identify each, determine their degree and leading coefficients, and analyze their end behavior.

What’s next

Next, you'll work through examples and practice problems to master identifying these functions and their key features, starting with power functions.

Property

A power function is a function that can be represented in the form

where $k$ and $p$ are real numbers, and $k$ is known as the coefficient.

Property

The end behavior of a function describes the behavior of the graph of the function as the input values get very small ($x \to -\infty$) and get very large ($x \to \infty$).

Given a power function $f(x) = kx^n$ where $n$ is a non-negative integer, identify the end behavior.

1. Determine whether the power $n$ is even or odd.
2. Determine whether the constant $k$ is positive or negative.
3. Use the following rules to identify the end behavior:
4. Even Power ($n$): If $k>0$, $f(x) \to \infty$ as $x \to \pm\infty$. If $k<0$, $f(x) \to -\infty$ as $x \to \pm\infty$.
5. Odd Power ($n$): If $k>0$, $f(x) \to -\infty$ as $x \to -\infty$ and $f(x) \to \infty$ as $x \to \infty$. If $k<0$, $f(x) \to \infty$ as $x \to -\infty$ and $f(x) \to -\infty$ as $x \to \infty$.

Examples

• For $f(x) = 3x^6$, the power is even (6) and the coefficient is positive (3). So, as $x \to \infty, f(x) \to \infty$ and as $x \to -\infty, f(x) \to \infty$. Both arms point up.
• For $g(x) = -2x^5$, the power is odd (5) and the coefficient is negative (-2). So, as $x \to \infty, g(x) \to -\infty$ and as $x \to -\infty, g(x) \to \infty$. The left arm points up, and the right arm points down.
• For $h(x) = -x^{10}$, the power is even (10) and the coefficient is negative (-1). So, as $x \to \infty, h(x) \to -\infty$ and as $x \to -\infty, h(x) \to -\infty$. Both arms point down.

Property

Let $n$ be a non-negative integer. A polynomial function is a function that can be written in the form

This is called the general form of a polynomial function. Each $a_i$ is a coefficient and can be any real number, but $a_n \neq 0$. Each expression $a_i x^i$ is a term of a polynomial function.

Property

We often rearrange polynomials so that the powers are descending. When a polynomial is written this way, we say that it is in general form. The degree of the polynomial is the highest power of the variable. The leading term is the term containing the highest power of the variable. The leading coefficient is the coefficient of the leading term. For any polynomial, the end behavior of the polynomial will match the end behavior of the power function consisting of the leading term.

Examples

• For $f(x) = -7x^4 + 2x^3 - 9$, the leading term is $-7x^4$. The degree is 4 and the leading coefficient is -7. The end behavior is: as $x \to \pm\infty, f(x) \to -\infty$.
• For $g(x) = 5x^2 - 8x^6 + 1$, reorder it to $g(x) = -8x^6 + 5x^2 + 1$. The leading term is $-8x^6$, the degree is 6, and the leading coefficient is -8.
• For $h(x) = -2x^3 + 5x - 1$, the leading term is $-2x^3$. The degree is odd (3) and the coefficient is negative (-2), so as $x \to \infty, h(x) \to -\infty$ and as $x \to -\infty, h(x) \to \infty$.

Explanation

The leading term is the most powerful term in a polynomial. As $x$ gets very large, this single term overpowers all others and dictates the graph's end behavior. Its degree and sign tell the whole story for the graph's arms.

Property

A turning point of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The y-intercept is the point at which the function has an input value of zero. The x-intercepts are the points at which the output value is zero.

A polynomial of degree $n$ will have, at most, $n$ $x$-intercepts and $n-1$ turning points.

Examples

• For $f(x) = (x-5)(x+2)(x-1)$, the x-intercepts are $(5,0)$, $(-2,0)$, and $(1,0)$. The y-intercept is $f(0) = (-5)(2)(-1) = 10$, which is the point $(0,10)$.
• For $g(x) = x^3 - 9x$, we factor it as $g(x) = x(x-3)(x+3)$. The x-intercepts are $(0,0)$, $(3,0)$, and $(-3,0)$. The y-intercept is also $(0,0)$.
• The polynomial $h(x) = 2x^8 - 5x^3 + 1$ has a degree of 8. Therefore, it can have at most 8 x-intercepts and at most $8-1=7$ turning points.