Graphing Polynomial Functions: Intervals, End Behavior, and Domain/Range

Polynomial Functions and Their Graphs

Intervals of Unknown Behavior

When analyzing polynomial functions, it is important to understand how the graph behaves between known points. This involves identifying end behavior, x-intercepts, y-intercepts, and turning points. By breaking the graph into intervals and plotting a point in each interval, we can determine the overall shape and behavior of the function.

• End Behavior: Describes how the function behaves as $x \to \infty$ or $x \to -\infty$.

• x-intercepts: Points where the graph crosses the x-axis ($f(x) = 0$).

• y-intercept: The point where the graph crosses the y-axis ($x = 0$).

• Turning Points: Points where the graph changes direction from increasing to decreasing or vice versa.

• Intervals: Sections of the graph between known points, used to analyze behavior.

Example Table: Known points and intervals for a polynomial function:

x	-3	-1	0	2
f(x)	f3	f1	f2	f4

By plotting these points and analyzing the intervals between them, we can sketch the graph and determine unknown behavior.

Steps to Graph a Polynomial Function

To accurately graph a polynomial function, follow these systematic steps:

1. Determine End Behavior: Analyze the leading term to see if the graph rises or falls on each side.

   • For $f(x) = ax^n$, if $n$ is even, both ends go in the same direction; if $n$ is odd, ends go in opposite directions.

   • If $a > 0$, right side rises; if $a < 0$, right side falls.

2. Find x-intercepts and Their Behavior: Solve $f(x) = 0$ to find x-intercepts. Check multiplicity:

   • If multiplicity is odd, the graph crosses the x-axis.

   • If multiplicity is even, the graph touches but does not cross the x-axis.

3. Find the y-intercept: Compute $f(0)$.

4. Determine Intervals and Plot Points: Break the x-axis into intervals between intercepts and plot a point in each interval to check the graph's behavior.

5. Connect Points: Draw a smooth, continuous curve through the plotted points.

6. Check for Turning Points: Use calculus or graphing to find local maxima and minima (turning points).

Example: Graphing a Polynomial Function

Consider the function $f(x) = 2x^4 - 6x^3 + 6x - 2$.

• End Behavior: Leading term $2x^4$ (even degree, positive coefficient) means both ends rise.

• x-intercepts: Solve $f(x) = 0$ for x-intercepts. Check multiplicity to determine if the graph touches or crosses the axis.

• y-intercept: $f(0) = -2$.

• Intervals: Choose values between intercepts and calculate $f(x)$ to plot additional points.

• Connect: Draw a smooth curve through all points.

Domain and Range of Polynomial Functions

The domain of any polynomial function is all real numbers ($(-\infty, \infty)$). The range depends on the degree and leading coefficient:

• Even degree, positive leading coefficient: Range is $[\text{minimum}, \infty)$.

• Even degree, negative leading coefficient: Range is $(-\infty, \text{maximum}]$.

• Odd degree: Range is $(-\infty, \infty)$.

Worked Example

Graph $f(x) = 3x^2 + 12x + 12$ and determine domain and range.

• End Behavior: $3x^2$ (even degree, positive coefficient) → both ends rise.

• x-intercepts: Solve $3x^2 + 12x + 12 = 0$.

• y-intercept: $f(0) = 12$.

• Domain: $(-\infty, \infty)$

• Range: $[\text{minimum}, \infty)$

Practice Problem

Graph $f(x) = (3x^2 + 1)(x - 1)^2$ and determine domain and range.

• End Behavior: Leading term $3x^4$ (even degree, positive coefficient) → both ends rise.

• x-intercepts: $x = 0$ and $x = 1$ (check multiplicity).

• y-intercept: $f(0) = 1$.

• Domain: $(-\infty, \infty)$

• Range: $[\text{minimum}, \infty)$

Summary Table: Polynomial Graphing Features

Feature	Description
End Behavior	Determined by degree and leading coefficient
x-intercepts	Where $f(x) = 0$; check multiplicity
y-intercept	Where $x = 0$
Turning Points	Maximum number is $n-1$ for degree $n$
Domain	All real numbers
Range	Depends on degree and leading coefficient

Additional info: These notes expand on the graphical analysis of polynomial functions, including step-by-step procedures and examples for determining intervals, intercepts, and domain/range, as well as the importance of multiplicity and end behavior in sketching accurate graphs.