Polynomial Function

A polynomial function is a type of function where the output value, denoted by $f (x)$ , is a polynomial expression in the input variable x. The degree of a polynomial function is the highest degree of any of its terms after the function has been simplified. For example, consider the function $f (x) = x^{7} + 2 x^{3} + 6 x - x^{7}$ . The highest exponent is 7, but the $x^{7}$ and ${- x}^{7}$ will cancel each other out when you combine like terms. The greatest remaining exponent is the 3 in $2 x^{3}$ , making this a polynomial function in the third degree.

It's important to note that a constant polynomial function, such as $f (x) = 7$ , is still considered a polynomial function, despite having no variables. In this case, the degree of the function is zero. In the same way, a polynomial function like $f (x) = 3 x + 7$ that doesn't have any terms with exponents is still a polynomial function, and its degree is determined by the highest power of $x$ . In this example, the highest power of $x$ is technically 1, making the degree of the function equal to one.

All polynomial functions are defined and continuous for the entire set of real numbers. The rest of this article will explore what you can do with different types of polynomial functions. Are you ready?

Exploring polynomial functions: Constant functions

A constant function is a linear function for which the range does not change no matter which member of the domain is used. In other words, regardless of the input, you always get the same output. Expressed mathematically, we could say that $f (x_{1}) = f (x_{2})$ for any $x_{1}$ and $x_{2}$ in the domain.

You know you're working with a constant function if there isn't a variable in it. For instance, $f (x) = 3$ is a classic example of a constant function. Since a constant function has no variables, it has a degree of zero. Graphing a constant function yields a straight, horizontal line such as the one below for $f (x) = 3$ :

Exploring polynomial functions: linear functions

A linear function may be defined as any function that can be written in the form $f (x) = m x + b$ where m 0. For instance, $f (x) = 2 x - 1$ is an example of a linear function. Linear functions have a degree of 1 since the variable has a power of 1, though you generally won't see the subscript in the expression.

The graph of a linear function is always a line with either a positive or negative slope. If the line rises from left to right, the slope is positive. If it falls from left to right, the slope is negative. Below, you'll find a graph for $f (x) = 2 x - 1$ . What kind of slope does it have?

If you said positive, you're correct! The line rises from left to right. If you cannot graph the function, linear functions have a positive slope when a is positive and a negative slope when a is negative.

Exploring polynomial functions: quadratic functions

A quadratic function can be defined as any function with the general form $f (x) = a x^{2} + b x + c$ when a does not equal 0. The degree of a quadratic function is always 2. The graph of a quadratic function is a parabola, a type of two-dimensional curve. Here is an example of what the parabola created by the parent quadratic function, $f (x) = x^{2}$ , looks like:

As shown below, the value of the coefficient a changes how wide or thin the parabola is. If a is negative, the entire parabola turns upside down.

The vertex of the parabola is the point at the bottom of the U shape (or the top if the parabola opens downward). The equation for a parabola may be written in "vertex form," or

$y = {(a - h)}^{2} + k$

where the vertex of the parabola is located at point $(h, k)$ . Here is an illustration:

If your quadratic equation is in standard form ${a x}^{2} + b x + c = 0$ instead of vertex form, you can use the following formula to determine the x-coordinate of its vertex:

$x = - \frac{b}{2 a}$

Once you have a value for x, you can sub it into the quadratic function to get a value for y. For example, the quadratic function $f (x) = 3 x^{2} + 12 x - 12$ has an a value of 3 and a b value of 12. Using the formula above gives us:

$- \frac{12}{2 (3)}$ or $- \frac{12}{6}$ or $-2$

Now, we plug in -2 for x and solve for y:

$3 {(-2)}^{2} + 12 (-2) - 12 = -24$

Therefore, the vertex of our parabola is $(-2, -24)$ . You can also use this method to find the axis of symmetry for a parabola. The axis of symmetry is the vertical line that cuts straight through the vertex. It has the same formula as that for the x-intercept above, $- \frac{b}{2 a}$ . You would give your answer as $x = - \frac{b}{2 a}$ .

The last thing you might be asked to do with a parabola is to identify the x and y-intercepts. The y-intercept is easier as all you have to do is substitute 0 for x and work out the equation. If your quadratic function is in standard form, the value of c is automatically the y-intercept since every other term has an x in it.

X-intercepts are a bit more challenging. You can try factoring, completing the square, or the quadratic formula depending on the numbers you are given, but remember that some parabolas don't have x-intercepts at all.

Exploring polynomial functions: cubic functions

A cubic function may be defined as any function in the format $f (x) = a x^{3} + b x^{2} + c x + d$ where a does not equal 0. The degree of a cubic function is always 3. The parent cubic function is $f (x) = x^{3}$ which is graphed below:

The function of coefficient a in the general equation is to make the graph wider or skinnier. A negative value will reflect the image as shown below:

The constant in the equation, $d$ , represents its y-intercept. The effects of $b$ and $c$ are more complicated though. If you can factor the right side of the equation, you can find one or more $x$ -intercepts and use those to sketch the graph. Some cubic functions cannot be factored, however, so this method won't always work. A cubic function may have 1, 2, or 3 $x$ -intercepts depending on the real roots of the related cubic equation.

Exploring polynomial functions: quartic functions

A quartic function takes the form $f (x) = a x^{4} + b x^{3} + c x^{2} + d x + e$ where a does not equal 0. The degree of a quartic function is always 4. It has been theorized that quartic functions are the highest degree to which finding solutions is possible, and the graph of a quartic function often resembles the letter $W$ . However, you will not be asked to work with quartic functions at this level of mathematics.

Practice problems on polynomial functions

a. How would you describe the graph of a constant function such as $f (x) = 4$ ?

Horizontal line

b. Is the slope of the linear function $f (x) = 8 x - 16$ positive or negative?

Positive

c. Would the parabola created by graphing the quadratic function $f (x) = -5 x^{2} + 17 x + 3$ open upward or downward? Why?

Downward because a is negative.

d. What is the axis of symmetry for the parabola represented by the function $f (x) = x^{2} - 6 x + 5$ ?

$x = 3$

e. What is the vertex of the parabola given by the following equation: $f (x) = -2 x^{2} - 12 x - 23$ ?

$(-3, -5)$

f. What is the y-intercept of the cubic function $f (x) = 3 x^{3} + 18 x^{2} - 9 x + 10$ ?

Topics related to the Polynomial Function

Factor Theorem

Adding and Subtracting Polynomials

Multiplying Polynomials

Flashcards covering the Polynomial Function

Algebra II Flashcards

College Algebra Flashcards

Practice tests covering the Polynomial Function

Algebra II Diagnostic Tests

College Algebra Diagnostic Tests

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