What is a Polynomial?
Definition, leading term, degree, and leading coefficient
A polynomial function is any function equivalent to the analytical form:
p(x) = aโxโฟ + aโโโxโฟโปยน + aโโโxโฟโปยฒ + ยทยทยท + aโxยฒ + aโx + aโ
where n is a positive integer, each aแตข is a real number, and aโ โ 0.
Leading Term
aโxโฟ
The term with the highest power of x
Degree
n
The highest exponent of x
Leading Coefficient
aโ
The coefficient of the leading term
๐ Example 1 โ Find the leading coefficient and degree.
f(x) = 3xโด + 2x โ 7
Leading Term3xโด
Leading Coeff.3
Degree4
y = 12x โ 7xยณ + 11
Leading Termโ7xยณ
Leading Coeff.โ7
Degree3
g(x) = 4 = 4xโฐ
Leading Term4xโฐ
Leading Coeff.4
Degree0
Leading term โ first term written
Always find the term with the highest exponent. In y = 12x โ 7xยณ + 11, the first term written is 12x (degree 1), but the leading term is โ7xยณ (degree 3). The negative sign belongs to the leading coefficient.
Extrema โ Two Types
Local (Relative) and Global (Absolute) โ minimums and maximums
๐ Two Types of Extrema
๐ Relative (Local) Extrema
A polynomial has a relative minimum or relative maximum where it switches between decreasing and increasing.
At a relative min: function switches from decreasing โ increasing (a valley).
At a relative max: function switches from increasing โ decreasing (a peak).
Can also occur at an endpoint if the polynomial has a restricted domain.
๐ Absolute (Global) Extrema
Of all local maxima, the greatest output value is the absolute maximum.
Of all local minima, the least output value is the absolute minimum.
An absolute extremum may not exist โ for example, if a polynomial has no upper or lower bound (extends to ยฑโ).
Write: Absolute Maximum = [value] at x = [input]. Always give both the output value and the x-value.
Finding and Classifying Extrema โ Example 2
Read x-values of peaks and valleys, then identify the greatest/least y-values
a) Downward parabola-shaped
Rel. Min at x =N/A
Rel. Max at x =1
Abs. Min =N/A
Abs. Max =3 at x = 1
b) S-curve with local max at x = 3
Rel. Min at x =โ2
Rel. Max at x =โ4 and 3
Abs. Min =N/A (โ โโ)
Abs. Max =2 at x = 3
c) Full wave โ multiple extrema
Rel. Min at x =โ3 and 3
Rel. Max at x =โ4, 0, and 5
Abs. Min =โ3 at x = โ3
Abs. Max =4 at x = 5
๐ Fun Facts About Polynomials
Fact 1: Between any 2 real zeros of a polynomial, there must be at least one local maximum or local minimum.
Fact 2: Polynomials of even degree must have either a global (absolute) maximum or a global (absolute) minimum.
Points of Inflection
Where concavity changes โ and what happens to the rate of change
๐ Definition: Point of Inflection
A point of inflection occurs when a function changes from concave up to concave down, or from concave down to concave up.
At a point of inflection, the rate of change changes from increasing to decreasing or from decreasing to increasing.
The ROC is neither at a max nor min โ it is transitioning at an inflection point, just as a function transitions between increasing/decreasing at an extremum.
Extrema vs. Inflection Points
Extremum: the function's output transitions (going up โ going down). ROC = 0 here.
Inflection point: the ROC itself transitions (increasing โ decreasing). The concavity changes.
ROC on Intervals โ Example 3
Graph of g(x) โ W-shape with inflection points at x = โ1 and x = 1
๐ Example 3 โ a) Find the inflection points of g.
Inflection points at x = โ1 and x = 1 (where concavity changes)
๐ Example 3 โ b) For each interval, is the rate of change of g increasing or decreasing?
i. (3, 4)
ROC Increasing
g is concave up on (3,4). Going uphill and getting steeper โ slope is increasing (positive and growing).
ii. (โ4, โ3)
ROC Increasing
g is concave up on (โ4,โ3). Going downhill but getting less steep โ negative slopes are increasing (becoming less negative).
iii. (โ1, 1)
ROC Decreasing
g is concave down on (โ1,1) โ between the two inflection points. Slopes are decreasing (becoming more negative).
iv. (1, 2)
ROC Increasing
g is concave up on (1,2). Going downhill but getting less steep โ negative slopes are increasing (less negative).
Real-World Application โ Example 4
Parking lot model โ when does the count change from increasing to decreasing?
๐ Example 4 โ For 0 โค t โค 3, the number of cars in a parking lot is modeled by C(t) = โ1.37tโต + 4.218tโด โ 0.357tยฒ + 3. At what time t does the number of cars change from increasing to decreasing?
C(t) = โ1.37tโต + 4.218tโด โ 0.357tยฒ + 3, 0 โค t โค 3
1
The number of cars changes from increasing to decreasing at a local maximum of C(t).2
Use a graphing calculator to find the maximum of C(t) on [0, 3].3
The calculator gives a local maximum at approximately t โ 2.4456.At t โ 2.4456 hours, the number of cars reaches its maximum and changes from increasing to decreasing.
Increasing โ Decreasing = Local Maximum
Any time a function changes from increasing to decreasing, that point is a local maximum. Any time it changes from decreasing to increasing, that's a local minimum. Use your calculator to find these x-values on polynomial problems.
Quick Reference
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๐ Key Rules
Leading term = highest power of x
Not necessarily the first term written
Local max: inc โ dec ยท Local min: dec โ inc
Abs max/min: give both value AND x
Inflection: concavity changes โ ROC transitions
Even degree poly โ global max OR global min
โ ๏ธ Common Mistakes
โ Leading coeff = coefficient of first term written
Always find the term with the HIGHEST exponent first. In 12x โ 7xยณ + 11, the leading coefficient is โ7 (from โ7xยณ), not 12.
โ Absolute max always exists
If a polynomial extends to +โ or โโ, there is NO absolute max (or min). Write N/A. Odd-degree polynomials with unrestricted domain have NO absolute extrema.
โ Inflection point = local extremum
Inflection points are about concavity changing, NOT the function going up/down. They are NOT local mins or maxes.
โ Concave up = ROC positive
Concave up means ROC is INCREASING โ the slope could still be negative (decreasing function that gets less steep is concave up).