Podcast
Questions and Answers
What is the standard form of a polynomial?
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?
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?
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?
What is the name of a polynomial with 0 as the degree?
What is the name of a polynomial with 1 as the degree?
What is the name of a polynomial with 1 as the degree?
What is the name of a polynomial with 2 as the degree?
What is the name of a polynomial with 2 as the degree?
What is the name of a polynomial with 3 as the degree?
What is the name of a polynomial with 3 as the degree?
What is the name of a polynomial with 4 as the degree?
What is the name of a polynomial with 4 as the degree?
What is the name of a polynomial with 5 as the degree?
What is the name of a polynomial with 5 as the degree?
What do we usually call a polynomial after the quartic (4th degree) function?
What do we usually call a polynomial after the quartic (4th degree) function?
What does end behavior describe?
What does end behavior describe?
When there is an odd degree, what is the way of the end behavior?
When there is an odd degree, what is the way of the end behavior?
When there is an odd degree and a positive leading coefficient, what does the end behavior look like?
When there is an odd degree and a positive leading coefficient, what does the end behavior look like?
When there is an odd degree and a negative leading coefficient, what does the end behavior look like?
When there is an odd degree and a negative leading coefficient, what does the end behavior look like?
When there is an even degree, what is the way of the end behavior?
When there is an even degree, what is the way of the end behavior?
When there is an even degree and a positive leading coefficient, what does the end behavior look like?
When there is an even degree and a positive leading coefficient, what does the end behavior look like?
When there is an even degree and a negative leading coefficient, what does the end behavior look like?
When there is an even degree and a negative leading coefficient, what does the end behavior look like?
What does a picture example of all end behaviors with the even and odd degrees show?
What does a picture example of all end behaviors with the even and odd degrees show?
How do you write the notation for end behavior?
How do you write the notation for end behavior?
What does a picture example of finding and writing the end behavior notation show?
What does a picture example of finding and writing the end behavior notation show?
What are turning points?
What are turning points?
What is a relative minimum?
What is a relative minimum?
What is a relative maximum?
What is a relative maximum?
What does a picture example of finding the turning points and maximum and minimums show?
What does a picture example of finding the turning points and maximum and minimums show?
How can you ensure your intervals of increasing and decreasing are correct?
How can you ensure your intervals of increasing and decreasing are correct?
What does the term 'intervals of increasing and decreasing' mean?
What does the term 'intervals of increasing and decreasing' mean?
What do the two problems related to finding turning points and maximum and minimums represent?
What do the two problems related to finding turning points and maximum and minimums represent?
What does an E in the decimal on the graphing calculator indicate?
What does an E in the decimal on the graphing calculator indicate?
How many decimal places should you keep after a number?
How many decimal places should you keep after a number?
What does a graph of increasing and decreasing intervals show?
What does a graph of increasing and decreasing intervals show?
What are x-intercepts and y-intercepts when x or y equals 0 called?
What are x-intercepts and y-intercepts when x or y equals 0 called?
How do you find the x-intercepts on the graphing calculator?
How do you find the x-intercepts on the graphing calculator?
How do you find the y-intercept on a graphing calculator?
How do you find the y-intercept on a graphing calculator?
What does a picture example of graphing x and y intercepts show?
What does a picture example of graphing x and y intercepts show?
How can you determine the y-intercept without the calculator?
How can you determine the y-intercept without the calculator?
What does a picture example of graphing all key features with the calculator show?
What does a picture example of graphing all key features with the calculator show?
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.
