Ch 4More Nonlinear Functions and Equations
4.1 More Nonlinear functions and Their Graphs
Constant –
Linear –
Non-Linear –
Monthly average high temps in Daytona Beach
1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 1269 / 70 / 75 / 80 / 85 / 88 / 90 / 89 / 87 / 81 / 76 / 70
Polynomial Functions
A polynomial function f of degree n in variable x can be represented by
where each coefficient is a real number, , and n is a non-negative integer. The leading coefficient is and the degree is n.
Degree Leading
Coefficient
Identifying Extrema
Let c be in the domain of f.
is an absolute maximum if for all x in the domain of f.
is an absolute minimum if for all x in the domain of f.
is an local maximum if for all x near c.
is an local minimum if for all x near c.
Ocean Temperatures at Bermuda
Domain of f
by
min/max
Symmetry
Symmetric wrt y axis
and are both on the graph of f
A function is an even function if for every x in the domain. The graph of an even function is symmetric with respect to the y axis.
A function is an odd function if for every x in the domain. The graph of an odd function is symmetric with respect to the origin.
Even functions contain only even powers.
Odd functions contain only odd powers.
If 0 is in the domain of an odd function, what point must lie on its graph?
-3 / -2 / -1 / 0 / 1 / 2 / 310.5 / 2 / / / / 2 / 10.5
4.2 Polynomial Functions and Models
Unemployment rates in Brazil
Year / 2010 / 2012 / 2014 / 2016 / 2018 / 2020% / 6.75 / 5.48 / 4.84 / 9.19 / 10.4 / 10.0
Polynomial Functions, Equations, Expressions
A turning point occurs whenever the graph of a polynomial function changes from increasing to decreasing or from decreasing to increasing.
Local maximum or minimum
A local extrema is a y-value, not a point, and often corresponds to the y-value of the turning point.
End Behavior of Polynomial Functions
Let f be a polynomial function of degree n with leading coefficient a.
Constant Fcnsdegree zero
No x intercept or turning point
Linear Functionsdegree 1
one x intercept, no turning point
Quadratic Fcns deg 2
Up to 2 x-intercepts, one turning point
Cubic Fcns
Degree 3; 1,2,3 x-int, 0 or 2 turning pts
drops left, rises right
rises left, drops right
Quartic Fcns
Degree 4; 0,1,2,3,4 x-int, up to 3 turning pts
Quintic Fcns
Degree 5; 1,2,3,4,5 x-int, up to 4 turning pts
The graph of a polynomial of degree has at most nx-intercepts, and turning points.
1)If is even
- implies the graph of f rises both to the left and right.
- implies the graph of f falls both to the left and right.
2)If is odd
- implies the graph of f falls to the left and rises to the right.
- implies the graph of f rises to the left and falls to the right.
Degree, x intercepts, and turning points
turning points
x-intercepts
leading coefficient
minimum possible degree?
x-intercepts
turning points
local extrema
degree?Leading coefficient?
End behaviors?
Piecewise-Defined Polynomial Functions
Graphing a piecewise-defined function
Graph
f continuous?
Solve
Diminishing returns and overfishing
Number of fish caught (x00, F000 tons)
, absolute max
Polynomial Regressionopt
12.4 / 21.8 / 20.4 / 19.3 / 24.0
Turning points?Degree?
Falls left, rises right?Leading coeff?
enter points from table
cubic regression
graph
4.3 Division of Polynomials
Long division
Division Algorithm for polynomials
For any polynomial with degree and any number k, there exists a unique polynomial , and a number r such that
The degree of is one less than the degree of and r is called the remainder.
Synthetic Division and long division
Remainder Theorem
If the divisor is , then the division algorithm for polynomials simplifies to .
If a polynomial is divided by , the remainder is .
4.4Real Zeros of Polynomial Functions
Bird Population
Factor Theorem
A polynomial has a factor if and only if .
Zeros with multiplicity
Complete Factored Theorem
Suppose a polynomial
has real zeros , where distinct zeros are listed as many times as their multiplicities. Then can be written in complete factored form as
zeros: and 2
zeros:, 1, and 3
Factoring a polynomial graphically
has a zero of .
Finding the multiplicity of a zero graphically
Depth a ball sinks in water
wood , zeros @ , 7.13, 12
aluminum , zeros @
Water balloon , zeros @ , 10, 10
Polynomial Equations
Solve symbolically
graphically, numerically
Intermediate Value Theoremopt
Let and with and be two points on the graph of a continuous function f. Then, on , f assumes every value between and at least once.
4.5 Fundamental Theorem of Algebra
A polynomial of degree has at least one complex zero.
A polynomial of degree n has at most n distinct zeros.
Represent a polynomial of degree 4 with leading coefficient 2 and zeros of –3, 5, i, and –i in complete factored form and fully expanded form.
Determine the complete factored form of
Conjugate Zeros Theorem
If a polynomial has only real coefficients, and if is a zero of , then is also a zero of .
Find a cubic polynomial with leading coefficient of 2 and zeros of 3 and .
Polynomial Equations with Complex Solutions
Solutions are
Find the zeros of , given that one zero is .
4.6Rational Functions and Models
A function , where and are polynomials and , is a rational function.
Rational? Domain?
Vertical Asymptotes
The line is a vertical asymptote of if or as x approaches k from the left or the right.
Horizontal Asymptotes
The line is a horizontal asymptote of the graph of if as x approaches either or .
1000 / 100 / 10 / 1 / 0.1 / 0.01 / 0.001
-1000 / -100 / -10 / -1 / -0.1 / -0.01 / -0.001 / 0
-0.001 / -0.01 / -0.1 / -1 / -10 / -100 / -1000
Finding vertical and horizontal asymptotes
Let be a rational function in lowest terms.
To find a vertical asymptote, solve for x. If is a zero of , then is a vertical asymptote. (if k is a zero of both and , then is not in lowest terms. Factor out from both.
Horizontal asymptote
1)If the degree of is less than the degree of , is a horizontal asymptote.
2)If the degree of is equal to the degree of , then is a horizontal asymptote, where a and b are the leading coefficients of and .
3)If the degree of is greater than the degree of , there are no horizontal asymptotes.
Slant or Oblique Asymptotes
Graphing with transformations
Graphing a rational function
- Find all vertical asymptotes.
- Find horizontal and slant asymptotes.
- Find y-intercept .
- Find x-intercepts .
- Will graph intersect non-horizontal asymptote? or ?
- Plot selected points. Choose x in each part of domain.
- Connect the dots.
4.7 More Equations and Inequalities
Rational Equations
Designing a box
Volume is 324 cu in, surf area is 342 sq in.
Length is 4x the height
Variation
Let x and y denote two quantities and n be a positive number. Then y is directly proportional to the nth power of x, or y varies directly as the nth power of x, if there is a non-zero k such that.
k is called the constant of proportionality.
The time t required for a pendulum to swing back and forth once is its period. The length l of a pendulum is directly proportional to the square of t. A two-foot pendulum has a 1.57 second period.
Find the constant of proportionality k.
Predict t for .
Let x and y denote two quantities and n be a positive number. Then y is inversely proportional to the nth power of x, or y varies inversely as the nth power of x, if there is a non-zero k such that . If , then y is inversely proportional to x or y varies inversely as x.
Intensity of light
At a distance of 3 meters, a 100 watt bulb produces an intensity of 0.88 watt per square meter.
Find the constant of proportionality k.
Determine the intensity at a distance of 2 meters.
Polynomial and Rational Inequalities
1)If necessary, rewrite equation as, where > can be any inequality symbol.
2)Solve for the boundary numbers.
3)Use boundary numbers to separate the number line into disjoint intervals. On each interval, will be always positive, or always be negative.
4)Evaluate at a test value in each interval to determine if it is part of the solution. Alternatively, graph, and observe the graph.
Modeling customers in a line
Solving rational inequalities
1)If necessary, rewrite equation as , where > can be any inequality symbol.
2)Solve and for boundary values.
3)Use boundary values to separate the number line into disjoint intervals. will be always positive, or always negative on each interval.
4)Evaluate at a test value in each interval to determine if it is part of the solution.
4.8Power Functions and Radical Equations
Evaluate radical functions
, x = 27
, x = 4
, x = 8
The number N of different plant species that live on a Galapagos island can be related to the island’s area A by the function:
Approximate N for A = 100, 200
Does N double when A doubles?
Power Functions
A function , b constant, is a power function. If for some integer , then f is a root function given by , or equivalently, .
Domain
- q odd
q even
- b irrational
Graphing power functions
Wing size
Heavier birds have larger wings. For some species, the surface area to weight can be modeled by
Suppose surface area is square meter
Planetary OrbitsJohannes Kepler (1571-1630)
Mercury / Venus / Earth / Mars / Jupiter / Saturn0.387 / 0.723 / 1.00 / 1.52 / 5.2 / 9.54
0.241 / 0.615 / 1.00 / 1.88 / 11.9 / 29.5
Scatterplot on p 322
closest for
Power function?
? ?
Domain? Range?
Interval where f(x) increasing?
?
Even/Odd function?
Power regressionopt
w
/ 0.5 / 1.5 / 2.0 / 2.5 / 3.0l / 0.77 / 1.10 / 1.22 / 1.31 / 1.40