Revise later
SpaceTo flip
If confident
All Flashcards
How does the leading term determine end behavior?
For large absolute values of x, the leading term dominates the polynomial's value, dictating its end behavior.
Explain the end behavior of a polynomial with a positive leading coefficient and even degree.
As $x$ approaches $\pm \infty$, $f(x)$ approaches $\infty$.
Explain the end behavior of a polynomial with a negative leading coefficient and even degree.
As $x$ approaches $\pm \infty$, $f(x)$ approaches $-\infty$.
Explain the end behavior of a polynomial with a positive leading coefficient and odd degree.
As $x$ approaches $\infty$, $f(x)$ approaches $\infty$, and as $x$ approaches $-\infty$, $f(x)$ approaches $-\infty$.
Explain the end behavior of a polynomial with a negative leading coefficient and odd degree.
As $x$ approaches $\infty$, $f(x)$ approaches $-\infty$, and as $x$ approaches $-\infty$, $f(x)$ approaches $\infty$.
Why focus on the leading term when determining end behavior?
As x approaches infinity, the leading term's contribution to the function's value becomes overwhelmingly larger than all other terms.
How does an even degree affect the end behavior?
Even degree polynomials have the same end behavior as x approaches both positive and negative infinity.
How does an odd degree affect the end behavior?
Odd degree polynomials have opposite end behaviors as x approaches positive and negative infinity.
What is the relationship between end behavior and limits at infinity?
End behavior describes the limits of the function as x approaches positive or negative infinity.
Why is understanding end behavior important?
It provides a general understanding of how the function behaves for very large or very small values of x and is foundational for further analysis.
How does the graph of an even-degree polynomial with a positive leading coefficient look as $x$ approaches $\pm \infty$?
The graph rises on both the left and right sides.
How does the graph of an odd-degree polynomial with a positive leading coefficient look as $x$ approaches $\pm \infty$?
The graph rises to the right and falls to the left.
How does the graph of an even-degree polynomial with a negative leading coefficient look as $x$ approaches $\pm \infty$?
The graph falls on both the left and right sides.
How does the graph of an odd-degree polynomial with a negative leading coefficient look as $x$ approaches $\pm \infty$?
The graph falls to the right and rises to the left.
What does a graph with both ends approaching infinity indicate about the polynomial's degree and leading coefficient?
It suggests an even degree and a positive leading coefficient.
What does a graph with both ends approaching negative infinity indicate about the polynomial's degree and leading coefficient?
It suggests an even degree and a negative leading coefficient.
What does a graph rising to the right and falling to the left indicate about the polynomial's degree and leading coefficient?
It suggests an odd degree and a positive leading coefficient.
What does a graph falling to the right and rising to the left indicate about the polynomial's degree and leading coefficient?
It suggests an odd degree and a negative leading coefficient.
How can you identify the leading term from a graph's end behavior?
By observing the direction of the graph as x approaches positive and negative infinity.
What does the end behavior of a polynomial graph tell you about possible horizontal asymptotes?
Polynomials do not have horizontal asymptotes; they either increase or decrease without bound.
Define 'end behavior' of a function.
Describes the function's output values as input values approach positive or negative infinity.
What is a 'leading term'?
The term with the highest degree in a polynomial function.
Define 'leading coefficient'.
The coefficient of the term with the highest degree.
What does $lim_{x \to \infty} f(x) = \infty$ mean?
As x approaches infinity, the function f(x) also approaches infinity.
What does $lim_{x \to -\infty} f(x) = -\infty$ mean?
As x approaches negative infinity, the function f(x) also approaches negative infinity.
What is the significance of the degree of a polynomial?
The highest power of the variable in the polynomial; influences the shape and end behavior.
What is the significance of the sign of the leading coefficient?
Determines whether the function increases or decreases without bound as x approaches infinity or negative infinity.
What is meant by 'increases without bound'?
The function's values become infinitely large (positive infinity).
What is meant by 'decreases without bound'?
The function's values become infinitely small (negative infinity).
Define polynomial function.
A function that can be expressed in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$, where n is a non-negative integer.