1.2 - Investigating Polynomial Functions MCB4U - Santowski

1.2 - Investigating 1.2 - Investigating Polynomial FunctionsPolynomial Functions

MCB4U - SantowskiMCB4U - Santowski

(A) Terminology(A) Terminology

PolynomialsPolynomials: an expression in the form of a: an expression in the form of annxxnn + a + an-n-

11xxn-1n-1 + a + an-2n-2xxn-2n-2 + ...... + a + ...... + a22x² + ax² + a11x + ax + a00 where a where a00, , aa11, ...a, ...ann are real numbers and n is a natural number are real numbers and n is a natural number

Polynomial FunctionsPolynomial Functions: a function whose equation is : a function whose equation is defined by a polynomial in one variable: defined by a polynomial in one variable:

ex: f(x) = aex: f(x) = annxxnn + a + an-1n-1xxn-1n-1 + a + an-2n-2xxn-2n-2 + ...... + a + ...... + a22x² + ax² + a11x x + a+ a00

leading coefficientleading coefficient: the coefficient of the term with : the coefficient of the term with the highest powerthe highest power

degreedegree: the value of the highest exponent on the : the value of the highest exponent on the variablevariable

Standard FormStandard Form: the function is expressed such that : the function is expressed such that the terms are written in descending order of the the terms are written in descending order of the exponentsexponents


DomainDomain: the set of all possible : the set of all possible xx values (independent values (independent variable) in a functionvariable) in a function

RangeRange: the set of all possible function values : the set of all possible function values (dependent variable, or (dependent variable, or yy values) values)

to to evaluate a functionevaluate a function: substituting in a value for the : substituting in a value for the variable and then determining a function value. Ex f(3)variable and then determining a function value. Ex f(3)

finite differencesfinite differences: subtracting consecutive : subtracting consecutive yy values or values or subsequent subsequent yy differences differences

zeroes, roots, x-interceptszeroes, roots, x-intercepts: where the function crosses : where the function crosses the the xx axes axes

y-interceptsy-intercepts: where the function crosses the : where the function crosses the yy axes axes direction of openingdirection of opening: in a quadratic, curve opens up or : in a quadratic, curve opens up or

downdown symmetrysymmetry: whether the graph of the function has : whether the graph of the function has

"halves" which are mirror images of each other"halves" which are mirror images of each other


turning pointturning point: points where the direction of the function : points where the direction of the function changeschanges

maximummaximum: the highest point on a function: the highest point on a function minimumminimum: the lowest point on a function: the lowest point on a function local vs absolutelocal vs absolute: a max can be a highest point in the entire : a max can be a highest point in the entire

domain (absolute) or only over a specified region within the domain (absolute) or only over a specified region within the domain (local). Likewise for a minimum.domain (local). Likewise for a minimum.

increaseincrease: the part of the domain (the interval) where the : the part of the domain (the interval) where the function values are getting larger as the independent variable function values are getting larger as the independent variable gets higher; if f(xgets higher; if f(x11) < f(x) < f(x22) when x) when x11 < x < x22; the graph of the ; the graph of the function is going up to the right (or down to the left)function is going up to the right (or down to the left)

decreasedecrease: the part of the domain (the interval) where the : the part of the domain (the interval) where the function values are getting smaller as the independent variable function values are getting smaller as the independent variable gets higher; if f(xgets higher; if f(x11) > f(x) > f(x22) when x) when x11 < x < x22; the graph of the ; the graph of the function is going up to the left (or down to the right)function is going up to the left (or down to the right)

""end behaviourend behaviour": describing the function values (or appearance ": describing the function values (or appearance of the graph) as of the graph) as xx values getting infinitely large positively or values getting infinitely large positively or infinitely large negativelyinfinitely large negatively

(B) Types of (B) Types of Polynomial FunctionsPolynomial Functions (i) (i) LinearLinear: Functions that generate graphs of straight : Functions that generate graphs of straight

lines, polynomials of degree one => f(x) = alines, polynomials of degree one => f(x) = a11xx11 + a + a00 or or more commonly written as y = mx + b, or Ax + By + C more commonly written as y = mx + b, or Ax + By + C = 0, or y = k(x - s)= 0, or y = k(x - s)

(ii) (ii) QuadraticQuadratic: Functions that generate graphs of : Functions that generate graphs of parabolas; polynomials of degree two parabolas; polynomials of degree two f(x) = a f(x) = a22x² + x² + aa11xx11 + a + a00 or y = Ax² + Bx + C or y = a(x-s)(x-t) or y = or y = Ax² + Bx + C or y = a(x-s)(x-t) or y = a(x - h)² + ka(x - h)² + k

(iii)(iii) Cubic Cubic polynomials of degree 3 polynomials of degree 3

(iv) (iv) QuarticQuartic: polynomials of degree 4: polynomials of degree 4

(v) (v) QuinticQuintic: polynomials of degree 5: polynomials of degree 5

(C) Investigating (C) Investigating Characteristics of Polynomial Characteristics of Polynomial FunctionsFunctions We can complete the following We can complete the following

analysis for polynomials of degrees 1 analysis for polynomials of degrees 1 through 5 and then make some through 5 and then make some generalizations or summaries:generalizations or summaries:

In order to carry out this investigation, In order to carry out this investigation, use either WINPLOT, a GDCuse either WINPLOT, a GDC

You may also use the following prograYou may also use the following program from m from AnalyzeMathAnalyzeMath

(i) Linear functions(i) Linear functions

Determine the following for the linear functions Determine the following for the linear functions f(x) = 2x – 1 f(x) = 2x – 1 g(x) = -½x + 3 g(x) = -½x + 3

(1) Leading coefficient(1) Leading coefficient (2) degree (2) degree (3) domain and range (3) domain and range (4) evaluating f(-2) (4) evaluating f(-2) (5) zeroes or roots (5) zeroes or roots (6) y-intercept(6) y-intercept (7) Symmetry(7) Symmetry (8) turning points(8) turning points (9) maximum values (local and absolute)(9) maximum values (local and absolute) (10) minimum values (local and absolute)(10) minimum values (local and absolute) (11) intervals of increase and intervals of (11) intervals of increase and intervals of

decreasedecrease (12) end behaviour (+x) and end behaviour (-x)(12) end behaviour (+x) and end behaviour (-x)

(ii) Quadratic (ii) Quadratic FunctionsFunctions

For the quadratic functions, determine the following:For the quadratic functions, determine the following: f(x) = x² - 4x – 5f(x) = x² - 4x – 5 f(x) = -½x² - 3x - 4.5f(x) = -½x² - 3x - 4.5 f(x) = 2x² - x + 4f(x) = 2x² - x + 4

(1) Leading coefficient(1) Leading coefficient (2) degree (2) degree (3) domain and range (3) domain and range (4) evaluating f(-2) (4) evaluating f(-2) (5) zeroes or roots (5) zeroes or roots (6) y-intercept(6) y-intercept (7) Symmetry(7) Symmetry (8) turning points(8) turning points (9) maximum values (local and absolute)(9) maximum values (local and absolute) (10) minimum values (local and absolute)(10) minimum values (local and absolute) (11) intervals of increase and intervals of decrease(11) intervals of increase and intervals of decrease (12) end behaviour (+x) and end behaviour (-x)(12) end behaviour (+x) and end behaviour (-x)

(iii) Cubic Functions(iii) Cubic Functions

For the cubic functions, determine the following:For the cubic functions, determine the following: f(x) = xf(x) = x33 - 5x² + 3x + 4 - 5x² + 3x + 4 f(x) =-2xf(x) =-2x33 + 8x² - 5x + 3 + 8x² - 5x + 3 f(x) = -3xf(x) = -3x33-15x² - 9x + 27-15x² - 9x + 27


Conclusions for Cubic Conclusions for Cubic FunctionsFunctions

1. Describe the general shape of a cubic function1. Describe the general shape of a cubic function 2. Describe how the graph of a cubic function with a positive 2. Describe how the graph of a cubic function with a positive

leading coefficient is different than a cubic with a negative leading coefficient is different than a cubic with a negative leading coefficientleading coefficient

3. What does a3. What does a00 represent on the graph of a cubic? represent on the graph of a cubic? 4. How many real roots do/can cubic functions have?4. How many real roots do/can cubic functions have? 5. How many complex roots do/can cubic functions have?5. How many complex roots do/can cubic functions have? 6. How many turning points do/can cubic functions have?6. How many turning points do/can cubic functions have? 7. How many intervals of increase do/can cubic functions 7. How many intervals of increase do/can cubic functions

have?have? 8. How many intervals of decrease do/can cubic functions 8. How many intervals of decrease do/can cubic functions

have?have? 9. Describe the end behaviour (9. Describe the end behaviour (++xx) of a cubic with a (i) ) of a cubic with a (i)

positive (ii) negative leading coefficientpositive (ii) negative leading coefficient 10. Are cubic functions symmetrical? (You may need to 10. Are cubic functions symmetrical? (You may need to

investigate further)investigate further)

(iv) Quartic Functions(iv) Quartic Functions

For the quartic functions, determine the following:For the quartic functions, determine the following: f(x)= -2xf(x)= -2x44-4x-4x33+3x²+6x+9+3x²+6x+9 f(x)= xf(x)= x44-3x-3x33+3x²+8x+5+3x²+8x+5 f(x) = ½xf(x) = ½x44-2x-2x33+x²+x+1+x²+x+1


Conclusions for Quartic Conclusions for Quartic FunctionsFunctions

1. Describe the general shape of a quartic function1. Describe the general shape of a quartic function 2. Describe how the graph of a quartic function with a 2. Describe how the graph of a quartic function with a

positive leading coefficient is different than a quartic with positive leading coefficient is different than a quartic with a negative leading coefficienta negative leading coefficient

3. What does a3. What does a00 represent on the graph of a quartic? represent on the graph of a quartic? 4. How many real roots do/can quartic functions have?4. How many real roots do/can quartic functions have? 5. How many complex roots do/can quartic functions have?5. How many complex roots do/can quartic functions have? 6. How many turning points do/can quartic functions have?6. How many turning points do/can quartic functions have? 7. How many intervals of increase do/can quartic functions 7. How many intervals of increase do/can quartic functions

have?have? 8. How many intervals of decrease do/can quartic 8. How many intervals of decrease do/can quartic

functions have?functions have? 9. Describe the end behaviour (9. Describe the end behaviour (++xx) of a quartic with a (i) ) of a quartic with a (i)

positive (ii) negative leading coefficientpositive (ii) negative leading coefficient 10. Are quartic functions symmetrical? (You may need to 10. Are quartic functions symmetrical? (You may need to


(v) Quintic Functions(v) Quintic Functions

For the quintic functions, determine the following:For the quintic functions, determine the following: f(x)= xf(x)= x55+7x+7x44-3x-3x33-18x²-20-18x²-20 f(x)= -¼xf(x)= -¼x55+ 2x+ 2x44-3x-3x33+3x²+8x+5+3x²+8x+5 f(x)=(x²-1)(x²-4)(x+3) f(x)=(x²-1)(x²-4)(x+3)


Conclusions for Quintic Conclusions for Quintic FunctionsFunctions

1. Describe the general shape of a quintic function1. Describe the general shape of a quintic function 2. Describe how the graph of a quintic function with a positive 2. Describe how the graph of a quintic function with a positive

leading coefficient is different than a quintic with a negative leading coefficient is different than a quintic with a negative leading coefficientleading coefficient

3. What does a3. What does a00 represent on the graph of a quintic? represent on the graph of a quintic? 4. How many real roots do/can quintic functions have?4. How many real roots do/can quintic functions have? 5. How many complex roots do/can quintic functions have?5. How many complex roots do/can quintic functions have? 6. How many turning points do/can quintic functions have?6. How many turning points do/can quintic functions have? 7. How many intervals of increase do/can quintic functions 7. How many intervals of increase do/can quintic functions

have?have? 8. How many intervals of decrease do/can quintic functions 8. How many intervals of decrease do/can quintic functions

have?have? 9. Describe the end behaviour (9. Describe the end behaviour (++xx) of a quintic with a (i) ) of a quintic with a (i)

positive (ii) negative leading coefficientpositive (ii) negative leading coefficient 10. Are quintic functions symmetrical? (You may need to 10. Are quintic functions symmetrical? (You may need to


(D) Examples of Algebraic Work (D) Examples of Algebraic Work with Polynomial Functionswith Polynomial Functions

ex 1. Expand & simplify h(x) = (x-1)ex 1. Expand & simplify h(x) = (x-1)(x+3)²(x+2). (x+3)²(x+2).

ex 2. Where are the zeroes of h(x)?ex 2. Where are the zeroes of h(x)? ex 3. Predict the end behaviour of h(x).ex 3. Predict the end behaviour of h(x). ex 4. Predict the shape/appearance of h(x).ex 4. Predict the shape/appearance of h(x). ex 5. Use a table of values to find additional ex 5. Use a table of values to find additional

points on h(x) and sketch a graph.points on h(x) and sketch a graph. ex 6. Predict the intervals of increase and ex 6. Predict the intervals of increase and

decrease for h(x).decrease for h(x). ex 7. Estimate where the turning points of h(x) ex 7. Estimate where the turning points of h(x)

are. Are the max/min? and local/absolute if are. Are the max/min? and local/absolute if domain was [-4,1]domain was [-4,1]

(D) Examples of Algebraic (D) Examples of Algebraic Work with Polynomial Work with Polynomial FuntionsFuntions ex 8. Sketch a graph of the polynomial function ex 8. Sketch a graph of the polynomial function

which has a degree of 4, a negative leading which has a degree of 4, a negative leading coefficient, 3 zeroes and 3 turning pointscoefficient, 3 zeroes and 3 turning points

ex 9. Equation writing: Determine the equation ex 9. Equation writing: Determine the equation of a cubic whose roots are -2, 3,4 and f(5) = 28of a cubic whose roots are -2, 3,4 and f(5) = 28

ex 10. Prepare a table of differences for f(x) = -ex 10. Prepare a table of differences for f(x) = -2x2x33 + 4x² - 3x - 2. What is the constant + 4x² - 3x - 2. What is the constant difference and when does it occur? Is there a difference and when does it occur? Is there a relationship between the equation and the relationship between the equation and the constant difference? Can you predict the constant difference? Can you predict the constant difference for g(x) = 4xconstant difference for g(x) = 4x44 + x + x33 - x² + 4x - x² + 4x - 5?- 5?

(E) Internet Links(E) Internet Links

Polynomial functions from The Math PaPolynomial functions from The Math Pagege

Polynomial Functions from Calculus QuPolynomial Functions from Calculus Questest

Polynomial Tutorial from WTAMUPolynomial Tutorial from WTAMU

Polynomial Functions from Polynomial Functions from AnalyzeMathAnalyzeMath

(E) Homework(E) Homework

Nelson Text, page 14-17, Q1-3; Nelson Text, page 14-17, Q1-3; zeroes and graphs 6,7; zeroes and graphs 6,7; increase/decrease 9,10; equation increase/decrease 9,10; equation writing11,14writing11,14

Nelson text page 23-26, Q1, end Nelson text page 23-26, Q1, end behaviour 5,6; combinations 7,8,9; behaviour 5,6; combinations 7,8,9; graphs 11,12; finite differences 14,15graphs 11,12; finite differences 14,15

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1.2 - Investigating Polynomial Functions MCB4U - Santowski