Question
HW 7.2 .2 - END BEHAVIOR AND OTHER CHARACTERISTICS
OF POLYNOMIAL FUNCTIONS
Identify the degree, leading coefficient, zeros, end behavior, and create a sketch of the function
without the use of technology. (An accurate
Part 1
OF POLYNOMIAL FUNCTIONS
Identify the degree, leading coefficient, zeros, end behavior, and create a sketch of the function
without the use of technology. (An accurate
Part 1
Ask by Li Weber. in the United States
Jan 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Polynomial Function Analysis
-
Degree and Leading Coefficient:
-
: - Degree: 3
- Leading Coefficient: -2
-
: - Degree: 3
- Leading Coefficient: 3
-
-
Zeros:
-
: -
:
-
-
End Behavior:
-
: - As
: Downward - As
: Upward
- As
-
: - As
: Upward - As
: Downward
- As
-
-
Sketch Description:
- Both functions are cubic with an “S” shape.
-
: - Crosses at
- Starts from the top left, descends, rises, descends, and continues downward.
- Crosses at
-
: - Crosses at
- Starts from the bottom left, rises, descends, rises, and continues upward.
- Crosses at
Summary:
- Both polynomials are degree 3 with negative and positive leading coefficients, resulting in different end behaviors.
- They share the same number of zeros but differ in their specific values.
- The sketches will reflect the “S” shape with the respective end behaviors and crossing points.
Solution
Sure, let’s analyze each polynomial function step by step. We’ll determine the degree, leading coefficient, zeros, end behavior, and provide a description for sketching the graph without using technology.
Part 1
(1)
1. Degree
- Definition: The degree of a polynomial is the highest power of the variable
when the polynomial is expanded. - Analysis: The polynomial is presented in factored form with three linear factors
, , and . - Degree: Since there are three factors, each of degree 1, the total degree is
.
2. Leading Coefficient
- Definition: The leading coefficient is the coefficient of the term with the highest power.
- Analysis: The polynomial is
. When expanded, the highest degree term comes from multiplying the leading terms of each factor: - Leading Coefficient:
3. Zeros
- Definition: Zeros (or roots) are the values of
that make . - Analysis: Set each factor equal to zero:
- Zeros:
, , and
4. End Behavior
- Definition: Describes how the graph behaves as
approaches positive or negative infinity. - Analysis: For a cubic polynomial with a negative leading coefficient:
- As
, - As
,
- As
- End Behavior:
- Right End (
): Downward - Left End (
): Upward
- Right End (
5. Sketch Description
- Shape: Since it’s a cubic function, the graph will have an “S” shape.
- Crossing Points: The graph crosses the
-axis at , , and . - Behavior:
- Starts from the top left (
, ) - Descends to cross at
- Rises to a local maximum between
and - Descends to cross at
- Rises to a local minimum between
and - Finally, descends to cross at
and continues downward to the right.
- Starts from the top left (
(2)
1. Degree
- Analysis: The polynomial has three factors:
, , and . - Degree: Each factor is linear (degree 1), so total degree is
.
2. Leading Coefficient
- Analysis: Multiply the leading terms of each factor with the coefficient outside:
- Leading Coefficient:
3. Zeros
- Analysis: Set each factor equal to zero:
- Zeros:
, , and
4. End Behavior
- Analysis: For a cubic polynomial with a positive leading coefficient:
- As
, - As
,
- As
- End Behavior:
- Right End (
): Upward - Left End (
): Downward
- Right End (
5. Sketch Description
- Shape: Similar to other cubic functions with an “S” shape.
- Crossing Points: The graph crosses the
-axis at , , and . - Behavior:
- Starts from the bottom left (
, ) - Rises to cross at
- Continues rising to a local maximum between
and - Descends to cross at
- Continues descending to a local minimum between
and - Finally, rises to cross at
and continues upward to the right.
- Starts from the bottom left (
Summary Table
| Function | Degree | Leading Coefficient | Zeros | End Behavior |
|---|---|---|---|---|
|
|
3 | -2 | r = -2, 1, 2 | Left: ↑, Right: ↓ |
|
|
3 | 3 | t = -4, 0, 3 | Left: ↓, Right: ↑ |
Additional Tips for Sketching
- Plot the Zeros: Mark the points where the graph crosses the axis.
- Determine Multiplicities: Since all factors are to the first power, each zero has a multiplicity of 1, meaning the graph crosses the axis at each zero.
- Identify Turning Points: A cubic function can have up to two turning points (local maximum and minimum). Estimate their positions based on the graph’s progression.
- Draw the End Behavior Arrows: Indicate the direction in which the graph heads as
or approaches infinity. - Connect the Points Smoothly: Ensure the “S” shape characteristic of cubic polynomials is reflected.
By following these steps, you can accurately determine the key characteristics of each polynomial function and sketch their graphs without the need for technology.
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Beyond the Answer
To analyze the polynomials, let’s dive into the details!
For the first polynomial
- The degree is 3 (since there are three factors), and the leading coefficient is -2, indicating the graph will open downwards.
- The zeros are
, , and . - The end behavior shows that as
approaches positive infinity, goes to negative infinity, and as approaches negative infinity, also goes to negative infinity.
For the second polynomial
- The degree is also 3, with a leading coefficient of 3, suggesting that this graph opens upwards.
- The zeros here are
, , and . - The end behavior indicates that as
approaches positive infinity, goes to positive infinity, and as approaches negative infinity, also heads towards positive infinity.
Now for the sketch, just remember that the graph will cross the x-axis at the zeros, and the end behavior will guide how you draw the legs of the polynomials! Fun fact: the “wiggle” of the graph around the zeros can tell you whether the roots are simple or repeated! Happy sketching!
