Polynomial Functions and Their Graphs Flashcards
What’s the definition of a polynomial function?
Simple Definition: A function that involves only non-negative integer powers or only positive integer exponents of a variable in an equation.
Complex Definition: Let n be a nonnegative integer and let aₙ, aₙ₋₁, …, a₂, a₁, a₀ be real numbers, with aₙ ≠ 0. The function defined by
f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … a₂x² + a₁x¹ + a₀
is called a polynomial function of degree n.
What’s the name of the coefficient of the variable to the highest power called, how is it denoted?
The leading coefficient is the coefficient of the term with the highest power.
It is denoted by aₙ.
Is this a valid polynomial function:
f(x) = -3x^1/2.
Explain your answer.
No, it’s not vlaid because the exponent 1/2 is not a nonnegative integer, which is required for polynomials.
What numbers can only be used for raising variables to powers in a polynomial function?
Nonnegative integers.
What does the term “end behavior” mean?
The behavior of the graph of a function to the far left or far right, more specifically the point or points where a graph falls or rises without bound.
What determines the end behaviour?
End behavior is determined by the leading coefficient (aₙ) and the degree (n). Specifically, whether aₙ is positive or negative and whether n is even or odd.
What are the 4 types of end behaviors?
Odd n, positive aₙ: Falls on the left, rises on the right.
Odd n, negative aₙ: Rises on the left, falls on the right.
Even n, positive aₙ: Rises on both ends.
Even n, negative aₙ: Falls on both ends.
What’s a model breakdown?
An impossibility that the model predicts. For example, a function that models the speed ,more specifically the deceleration of a car, per elapsed minutes. Without bounds, at some point the function will predict the car is driving at negative speed, which is impossible.
What are the zeros of a graph?
The zeros of a graph are the points where the graph crosses or touches the x-axis, also called the x-intercepts.
What’s a multiplicity of zero? Give an example.
The multiplicity of a zero is the number of times a root appears. For example, in
𝑓(𝑥)=(𝑥−2)(𝑥−2) the root 2 appears twice. Therefore, the zero 2 has multiplicity 2.
What’s the definition for zeros with multiplicity when factoring?
In a factored polynomial f(x), if the factor (x - r) appears exactly k times, r is called a zero with multiplicity k..
r is a zero of a graph, k is the multiplicity of r, describe what happens if k is even or odd. Provide 1 explanation in 2 different ways, first use k as a guideline for multiplicity, then use r.
If k (the multiplicity of r) is even, the graph touches the x-axis at r and turns around without crossing. If k is odd, the graph crosses the x-axis at r. Alternatively, if r is a zero of even multiplicity, the graph touches and turns at r. If r is of odd multiplicity, the graph crosses the x-axis at r.
What does the intermediate value theorem prove?
That there is at least one zero between two points with opposite signs.
What’s the definition for the intermediate value theorem?
Let f be a polynomial function with real coefficients. If f(a) and f(b) have opposite signs, then there’s at least one value for c between a and b for which f(c) = 0.
What’s a turning point on a graph?
A point where the graph smoothly goes from decreasing to increasing or vice-versa. These points are always relative maximums or minimums.
