Algebra 2 Honors Week 16 Flashcards

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Questions and Answers

What is the standard form of a polynomial?

All coefficients are positive

Any like terms are combined and the terms descend in numerical order by degree (correct)

The leading coefficient is 0

The terms are arranged in ascending order by degree

What is a leading coefficient?

The non-zero factor that is multiplied by the greatest power of x.

What does the degree of a polynomial represent?

The greatest degree of any of the terms.

What is the name of a polynomial with 0 as the degree?

Constant Signup and view all the answers

What is the name of a polynomial with 1 as the degree?

Linear Signup and view all the answers

What is the name of a polynomial with 2 as the degree?

Quadratic Signup and view all the answers

What is the name of a polynomial with 3 as the degree?

Cubic Signup and view all the answers

What is the name of a polynomial with 4 as the degree?

Quartic Signup and view all the answers

What is the name of a polynomial with 5 as the degree?

Quintic Signup and view all the answers

What do we usually call a polynomial after the quartic (4th degree) function?

Polynomial of higher degree Signup and view all the answers

What does end behavior describe?

What happens to the function values as x approaches positive or negative infinity. Signup and view all the answers

When there is an odd degree, what is the way of the end behavior?

One up-one down Signup and view all the answers

When there is an odd degree and a positive leading coefficient, what does the end behavior look like?

One up in quadrant 1, and one down in quadrant 3. Signup and view all the answers

When there is an odd degree and a negative leading coefficient, what does the end behavior look like?

One down in quadrant 4, and one up in quadrant 2. Signup and view all the answers

When there is an even degree, what is the way of the end behavior?

Both up or both down. Signup and view all the answers

When there is an even degree and a positive leading coefficient, what does the end behavior look like?

Both up in quadrants 1 and 2. Signup and view all the answers

When there is an even degree and a negative leading coefficient, what does the end behavior look like?

Both down in quadrants 3 and 4. Signup and view all the answers

What does a picture example of all end behaviors with the even and odd degrees show?

It shows how various polynomial degrees behave. Signup and view all the answers

How do you write the notation for end behavior?

X and an arrow to - or + infinity, f(x) and an arrow to - or + infinity. Signup and view all the answers

What does a picture example of finding and writing the end behavior notation show?

It shows how to analyze the function's limits. Signup and view all the answers

What are turning points?

Where the function changes from increasing to decreasing. Signup and view all the answers

What is a relative minimum?

The point where the function has the least value over an interval. Signup and view all the answers

What is a relative maximum?

The point where the function has the greatest value over an interval. Signup and view all the answers

What does a picture example of finding the turning points and maximum and minimums show?

It illustrates the behavior of a polynomial function. Signup and view all the answers

How can you ensure your intervals of increasing and decreasing are correct?

Make sure all the numbers cover the whole number line. Signup and view all the answers

What does the term 'intervals of increasing and decreasing' mean?

Using interval notation to indicate where the graph is increasing or decreasing. Signup and view all the answers

What do the two problems related to finding turning points and maximum and minimums represent?

They are examples of analyzing polynomial functions. Signup and view all the answers

What does an E in the decimal on the graphing calculator indicate?

It means it is very close to 0. Signup and view all the answers

How many decimal places should you keep after a number?

1 or 2 and round it up. Signup and view all the answers

What does a graph of increasing and decreasing intervals show?

It illustrates how the function behaves with respect to its intervals. Signup and view all the answers

What are x-intercepts and y-intercepts when x or y equals 0 called?

X-intercepts when y=0; y-intercepts when x=0. Signup and view all the answers

How do you find the x-intercepts on the graphing calculator?

Hit 2nd trace, then calculate, and then find zeros. Signup and view all the answers

How do you find the y-intercept on a graphing calculator?

Hit 2nd trace for value, type in 0 for x, then hit enter. Signup and view all the answers

What does a picture example of graphing x and y intercepts show?

It illustrates the intercepts on a graph. Signup and view all the answers

How can you determine the y-intercept without the calculator?

It is the constant because all the x's are 0. Signup and view all the answers

What does a picture example of graphing all key features with the calculator show?

It demonstrates the overall behavior of the polynomial function. Signup and view all the answers

Study Notes

Polynomial Basics

Standard Form combines like terms and arranges them in descending order by degree.
Leading Coefficient is the non-zero factor multiplying the highest power of x in a polynomial.
Degree of a polynomial refers to the highest degree among its terms.

Types of Polynomials by Degree

Constant: A polynomial with a degree of 0 (e.g., 3).
Linear: A polynomial with a degree of 1 (e.g., 5x + 4).
Quadratic: A polynomial with a degree of 2 (e.g., -x² + 11x - 5).
Cubic: A polynomial with a degree of 3 (e.g., 4x³ - x² + 2x - 3).
Quartic: A polynomial with a degree of 4 (e.g., 9x⁴ - 3x³ + 4x² - x + 3).
Quintic: A polynomial with a degree of 5 (e.g., -3x⁵ + 3x⁴ + x³ - 3x² - 2x + 4).

End Behavior of Polynomials

End behavior describes the function's behavior as x approaches positive or negative infinity.
Odd degree polynomials display a one up-one down behavior if the leading coefficient is positive.
If the leading coefficient is negative, the behavior is one up in the second quadrant and one down in the fourth quadrant.
Even degree polynomials have both ends pointing up if the leading coefficient is positive, and both down if negative.

Writing End Behavior Notation

Notation includes stating x approaching ± infinity and correlating with f(x) approaching ± infinity.

Key Features of Polynomial Functions

Turning Points occur where the function changes from increasing to decreasing or vice versa.
Relative Minimum is the point where the function reaches the least value in an interval.
Relative Maximum is the point where the function reaches the greatest value in an interval.

Analyzing Intervals

Intervals of Increasing and Decreasing determine where the graph goes up or down, often represented in interval notation.
Ensuring that numbers cover the entire number line helps verify intervals of change.

Finding Intercepts

X-intercepts occur when y = 0; Y-intercepts occur when x = 0.
Use graphing calculators to find x-intercepts by setting bounds around zeros and confirming with the enter key.
To find the y-intercept, set x to 0 and evaluate using the calculator.

Function Characteristics Without a Calculator

The y-intercept can be determined without a calculator since all x terms become 0, leaving the constant.

Graphing Key Features

Graphing involves evaluating both x and y intercepts and understanding the behavior of the function at critical points.

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Description

This quiz focuses on graphing polynomial functions and includes key concepts such as standard form, leading coefficients, and the degree of a polynomial. Perfect for 10th-grade Algebra 2 topics, it aids in understanding essential definitions and their applications in polynomial functions.