2 2.1- Linear and Quadratic Functions and Modeling Polynomial Functions-Let n be a nonnegative integer and let a0, a1, a2,…, an-1, an be real numbers with an≠0. The functions given byf(x)=anxn + an-1xn-1+…+a2x2+a1x+a0Is a polynomial function of degree n. The leading coefficient is an.f(x)=0 is a polynomial function.*it has no degree or leading coefficient.
3 Identify degree and leading coefficient for functions: F(x) = 5x3-2x-3/4G(x) = √(25x4+4x2)H(x) = 4x-5+6xK(x)=4x3+7x7
4 Polynomial Functions of No and Low Degree NameFormDegreeZero FunctionF(x) = 0UndefinedConstant FunctionF(x)=a (a≠0)Linear FunctionF(x)=ax+b (a≠0)1Quadratic FunctionF(x)=ax2+bx+c (a≠0)2
5 Linear Functions F(x) = ax+b Slope-Intercept form of a line: Find an equation for the linear function f such that f(-2) = 5 and f(3) = 6
6 Average Rate of ChangeThe average rate of change of a function y=f(x) between x=a and x=b, a≠b, is[F(b)-F(a)]/[b-a]
7 Modeling Depreciation with a Linear Function Weehawken High School bought a $50,000 building and for tax purposes are depreciating it $2000 per year over a 25-yr period using straight-line depreciation.What is the rate of change of the value of the building?Write an equation for the value v(t) of the building as a linear function of the time t since the building was placed in services.Evaluate v(0) and v(16)Solve v(t)=39,000
8 Characteristics of Linear Functions y=mx+b Point of ViewCharacterizationVerbalPolynomial of degree 1AlgebraicF(x)=mx+b (m≠0)GraphicalSlant line with slope m and y-intercept bAnalyticalFunction with constant nonzero rate of change m, f is increasing if m>0, decreasing if m<0
9 Quadratic Functions and their graphs: Sketch how to transform f(x)=x2 into:G(x)=-(1/2)x2+3H(x)=3(x+2) 2-1If g(x) and h(x) and in the form f(x)=ax2+bx+c, what do you notice about g(x) and h(x) when a is a certain value (negative or positive)?
10 Finding the Vertex of a Quadratic Function: f(x)=ax2+bx+cWe want to find the axis of symmetry, which is x=-b/(2a).Then:The graph of f is a parabola with vertex (x,y), where x=-b/(2a). If a>0, the parabola opens upward, and if a<0, it opens downward.
11 Use the vertex form of a quadratic function to find the vertex and axis of the graph of f(x)=8x+4x2+1:x=-b/(2a)
12 Find the vertex of the following functions: F(x)=3x2+5x-4G(x)=4x2+12x+4H(x)=6x2+9x+3f(x)=5x2+10x+5
13 Vertex Form of a Quadratic Function: Any Quadratic Function f(x)=ax2+bx+c, can be written in the vertex form:F(x)=a(x-h)2+kWhere (h,k) is your vertexh=-b/(2a) and k=is the y
14 Using Algebra to describe the graph of quadratic functions: F(x)=3x2+12x f(x)=a(x-h)2+k=3(x2+4x) Factor 3 from the x term=3(x2+4x+() - () ) Prepare to complete the square.=3(x2+4x+(2)2-(2)2) Complete the square.=3(x2+4x+4)-3(4) Distribute the 3.=3(x+2)2-1
15 Find vertex and axis, then rewrite functions in vertex form: f(x)=a(x-h)2+k F(x)=3x2+5x-4F(x)=8x-x2+3G(x)=5x2+4-6x
16 Characteristics of Quadratic Functions: y=ax2+bx+c Point of ViewCharacterizationVerbalPolynomial of degree ___AlgebraicF(x)=______________ (a≠0)Graphicala>0a<0
17 2.2 Power Functions With Modeling HW: Pg.189 #1-10
18 Power function F(x)=k*xa a is the power, k is the constant of variationEXAMPLES:FormulasPowerConstant of VariationC=2∏r12∏A=∏r22∏F(x)=4x3G(x)=1/2x6H(x)=6x-2
19 What is the power and constant of variation for the following functions: F(x) = ∛x1/(x2)What type of Polynomials are these functions? (HINT: count the terms)
20 Determine if the following functions are a power function Given that a,h,and c represent constants,, and for those that are, state the power and constant of variation:6cx-5h/x44∏r23*2xax7x8/9
28 Division Algorithm for Polynomials F(x) = d(x)*q(x)+r(x)F(x) and d(x) are polynomials where q(x) is the quotient and r(x) is the remainder
29 Fraction Form: F(x)/d(x)=q(x)+r(x)/d(x) (3x3+5x2+8x+7)/(3x+2)Write (2x4+3x3-2)/(2x2+x+1) in fraction form
30 Special Case: d(x)=x-k D(x)=x-k, degree is 1, so the remainder is a real numberDivide f(x)=3x2+7x-20 by:(a) x (b) x (c) x+5
31 We can find the remainder without doing long division! Remainder Theorem: If a polynomial f(x) is divided by x-k, then the remainder is r=f(k) Ex: (x2+3x+5)/(x-2) k=2 So, f(k)=f(2)=(2)2+3(2)+5=15=remainder
32 Lets test the Remainder Theorem with our previous example: Divide f(x)=3x2+7x-20 by:(a) x (b) x (c) x+5
33 PROVE: If d(x)=x-k, where f(x)=(x-k)q(x) + r Then we can evaluate the polynomial f(x) at x=k:
34 Use the Remainder Theorem to find the remainder when f(x) is divided by x-k F(x)=2x2-3x+1; k=2F(x)=2x3+3x2+4x-7; k=2F(x)=x3-x2+2x-1; k=-3
35 Synthetic DivisionNow we can use this method to find both remainders and quotients for division by x-k, called synthetic division.(2x3-3x2-5x-12)/(x-3)K becomes zero of divisor
36 STEPS: 3 | 2 -3 -5 -12 _____________ * Since the leading coefficient of the dividend must be the leading coefficient , copy the first “2” into the first quotient position.* Multiply the zero of the divisor (3) by the most recent coefficient of the quotient (2). Write the product above the line under the next term (-3).* Add the next coefficient of the dividend to the product just found and record sum below the line in the same column.* Repeat the “multiply” and “add” steps until the last row is completed.3 |_____________
37 Use synthetic division to solve: (x3-5x2+3x-2)/(x+1)(9x3+7x2-3x)/(x-10)(5x4-3x+1)/(4-x)
38 Rational Zero TheoremSuppose f is a polynomial function of degree n1 of the form f(x)=anxn+…+a0with every coefficient an integer. If x=p/q is a rational zero of f, where p and q have no common integer factors other than 1, thenP is an integer factor of the constant coefficient a0, andQ is an integer factor of the leading coefficient an.Example: Find rational zeros of f(x)=x3-3x2+1
39 Finding the rational zeros: F(x)=3x3+4x2-5x-2Potential Rational Zeros:
41 Upper and Lower Bound Tests for Real Zeros Let f be a polynomial function of degree n≥1 with a positive leading coefficient. Suppose f(x) is divided by x-k using synthetic division.If k≥0 and every number in the last line is nonnegative (positive or zero), then k is an upper bound for the real zeros of f.If k≤0 and the numbers in the last line are alternately nonnegative and nonpositive, then k is a lower bound for the real zeros of f.
42 Example:Lets establish that all the real zeros of f(x)=2x4-7x3-8x2+14x+8 must lie in the interval [-2,5]
43 Now we want to find the real zeros of the polynomial function f(x)=2x4-7x3-8x2+14x+8
44 Steps to finding the real zeros of a polynomial function: Establish bounds for real zerosFind the real zeros of a polynomial functions by using the rational zeros theorem to find potential rational zerosUse synthetic division to see which potential rational zeros are a real zeroComplete the factoring of f(x) by using synthetic division again or factor.
45 Find the real zeros of a polynomial function: F(x)=10x5-3x2+x-6
46 Find the real zeros of a polynomial function: F(x)=2x3-3x2-4x+6F(x)=x3+x2-8x-6F(x)=x4-3x3-6x2+6x+8F(x)=2x4-7x3-2x2-7x-4
48 F(x)=x2+1 has no real zeros In the 17th century, mathematicians extended the definition of √(a) to include negative real numbers a.i =√(-1) is defined as a solution of (i )2 +1=0For any negative real number √(a) = √|a|*i
49 Complex Number- is any number written in the form: a +bi , where a, b are real numbersa+bi is in standard form
56 DO NOW:Find all zeros:f(x) = x4 + x3 + x2 + 3x - 6
57 2.6 Complex Zeros and The Fundamental Theorem of Algebra HW: Pg #2-10e, 28-34e
58 Two Major TheoremsFundamental Theorem of Algebra – A polynomial function of degree n has n complex zeros (real and nonreal).Linear Factorization Theorem – If f(x) is a polynomial function of degree n>0, then f(x) has n linear factors andF(x) = a(x-z1)(x-z2)…(x-zn)Where a is the leading coefficient of f(x) and z1, z2, …, zn are the complex zeros of the function.
59 Fundamental Polynomial Connections in the Complex Case X=k is a…K is aFactor of f(x):
60 Exploring Fundamental Polynomial Connections Write the polynomial function in standard form and identify the zeros :F(x)=(x-2i)(x+2i)F(x)=(x-3)(x-3)(x-i)(x+i)
61 Complex Conjugate Zeros Suppose that f(x) is a polynomial function with real coefficients. If a+bi is a zero of f(x), then the complex conjugate a-bi is also a zero of f(x)
62 EXPLORATION (with your partner): What can happen if the coefficients are not real?Use substitution to verify that x=2i and x=-i are zeros of f(x)=x2-ix+2. Are the conjugates of 2i and –i also zeros of f(x)?Use substitution to verify that x=i and x=1-i are zeros of g(x)=x2-x+(1+i). Are the conjugates of i and 1-i also zeros of g(x)?What conclusions can you draw from parts 1 and 2? Do your results contradict the theorem about complex conjugates?
63 Find a Polynomial from Given Zeros Given that -3, 4, and 2-i are zeros, find the polynomial:Given 1, 1+2i, 1-i, find the polynomial:
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