Polynomial Functions. Linear Graphs and Linear Functions 1.3

Transcription

1 Polynomial Functions Linear Graphs and Linear Functions 1.3

2 Forms for equations of lines (linear functions) Ax + By = C Standard Form y = mx +b Slope-Intercept (y y 1 ) = m(x x 1 ) Point-Slope x = a Vertical Line y = b Horizontal Line Slope, ratio, rate (rate of change) Slope formula CAUTION: m = y 2 y 1 x 2 x 1 = y 1 y 2 x 1 x 2 m y 2 y 1 x 1 x 2

3 Parallel and Perpendicular Lines Two lines are parallel if they have the same slope (m 1 =m 2 ) Two lines are perpendicular if their slopes are negative reciprocals (m 1 = - 1/m 2 )

4 Linear Extrapolation: Prediction based on a linear model An extrapolated point does not lie between the given data points Linear Interpolation: Estimation based on a linear model An interpolated point lies between the given data points

5 Polynomial Functions Quadratic Functions 2.1

6 Forms for equations of parabolas (quadratic functions) ax 2 +bx+c, where a 0 Standard Form a(x-h) 2 +k, where a 0 Vertex Form Axis of symmetry: x=h a vertical line Vertex: the point (h, k) a minimum or a maximum of the function

7 Completing the Square Move the constant term to the right, ignore it for a bit. Make sure the leading term is 1 If a 1 then factor out a. (ax 2 + bx)+c = a(x 2 + b a x)+c Add zero -- in a tricky way ;o) a(x 2 + b a x +(1 2 b a )2 )+c a( 1 2 b a )2 Factor & Simplify (Notice: Now in vertex form) a(x + b 2a )2 +(c b2 4a )

8 Polynomial Functions Polynomials of Higher Degree & Division 2.2 & 2.3

9 Polynomial Function Let a 0, a 1, a 2,, a n-1, a n be real numbers with a n 0, f(x) = a n x n + a n-1 x n a 2 x 2 + a 1 x + a 0 is a polynomial function of x with degree n. Polynomials are continuous with smooth rounded turns.

10 Leading Coefficient Test (LCT) n even; a n >0 n even; a n <0 n odd; a n >0 n odd; a n <0

11 Real Zeros (Equivalent Statements) 1. x = a is a zero of the function f 2. x = a is a solution of the polynomial equation f(x) = 0 3. (x a) is a factor of f(x) 4. (a, 0) is an x-intercept of the graph of f Note: A polynomial function of degree of n, has at most n real zeros and at most n-1 turning points.

12 Repeat Roots (Zeros) A factor (x a) k, k > 1 yield a repeated zero x = a of multiplicity of k. If k is odd, the graph crosses at x = a. If k is even, the graph touches the x-axis (but does not cross) at x = a.

13 Intermediate Value Theorem (IVT) Let a and b be real numbers such that a < b. If f is a polynomial function such that f(a) f(b), then in the interval [a, b] f takes every value between f(a) and f(b).

14 Factor Theorem: A polynomial f(x) has a factor (x k) iif f(k)=0. Remainder Theorem: If a polynomial f(x) is divided by x k, the remainder is r = f(k). Long Division always works! Synthetic Division works only if the divisor is the form (x k)

15 Polynomial Functions Complex Numbers 2.4

16 Complex Numbers i = 1 i 2 = -1 i 3 = -i i 4 = 1 a + bi, where a and b are real numbers, is a complex number a is the real part and b is the imaginary part The (principal) square root of a negative number: If a is a real number, where a > 0, then a = i a

17 Properties of Complex Numbers 1. a + bi = c + di, iif a=c and b=d 2. Real Numbers are a subset of the Complex Numbers 3. Addition: (a+bi) + (c+di) = (a+c) + (b+d)i 4. Subtraction: (a+bi) (c+di) = (a c) + (b d)i 5. Multiplication: (a+bi)(c+di) = ac + adi + bci +bdi 2 = (ac bd) +(ad+bc)i 6. Division: a+bi c+di = a+bi c+di c di c di = ac adi+bci+bdi2 c 2 d 2 i 2 = (ac+bd)+(bc ad)i c 2 +d 2 7. Complex conjugate of a + bi is a bi = ac+bd c 2 +d 2 + (bc ad) c 2 +d 2 i

18 Recall: The Quadratic Formula Let a, b, and c be real numbers with a = 0 If ax 2 + bx + c = 0, then x = b± b 2 4ac 2a.

19 Polynomial Functions Zeros 2.5

20 Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, where n > 0, then f has at least one zero in the Complex number. This implies that: f has precisely n linear factors f(x) = a n (x c 1 )(x c 2 ) (x c n ) where c 1, c 2,, c n are complex numbers Note: Complex zeros come in conjugate pairs!! If (a + bi) is a zero of f(x), then (a bi) is also a zero.

21 Finding Zeros of a Polynomial Rational Zeros Test (RZT) If the polynomial f(x)=a n x n +a n-1 x n-1 + +a 1 x+a 0 has integer coefficients, every rational zero of f has the form: rational zero = p/q where p and q have no common factors other than 1, and p = a factor of the constant term a 0 q = a factor of the leading coefficient a n

22 Finding Zeros of a Polynomial Descartes s Rule of Signs Let f(x)=a n x n +a n-1 x n-1 + +a 1 x+a 0 be a polynomial with real coefficients and a 0 0. The number of positive real zeros of f is either equal to the number of variations in sign of f(x) or less than that number by and even integer. The number of negative real zeros of f is either equal to the number of variations in sign of f(-x) or less than that number by an even integer.

23 Finding Zeros of a Polynomial Upper and Lower Bound Rules Let f(x) be a polynomial with real coefficients and a positive leading coefficient. Suppose f(x) is divided by (x c), using synthetic division. If c >0 and each number in the last row is either positive or zero, c is an upper bound for the real zeros of f. If c < 0 and the numbers in the last row are alternately positive and negative (zero entries count as positive or negative), c is a lower bound for the real zeros of f.

24 Polynomial Functions Rational Functions 2.6

25 Rational Functions A rational function is a ratio (fraction) of polynomials; f(x)=n(x)/d(x). Domain: xr such that D(x) = 0

26 Asymptotes (V. A.) Vertical Asymptote Vertical Line x = a; where D(a) = 0 Find V.A. by looking for restrictions in the domain. The function cannot touch nor cross a vertical asymptote.

27 Asymptotes (H.A.) Horizontal Asymptotes Horizontal Line y = b; where lim x ± f(x) =b Find H.A. by looking at the end behavior of the function. The function can touch or cross a horizontal asymptote. If deg(n(x)) < deg(d(x)), then y=0 is the H.A. If deg(n(x)) = deg(d(x)), then the H.A. is the ratio of leading coefficients If deg(n(x)) > deg(d(x)), then there is no H.A.

28 Asymptotes (S. A.) Slant Asymptote If y = N(x) D(x) = ax + b + r(x) And deg(n(x)) = deg(d(x)) + 1, then the rational function has a slant asymptote; y = ax +b. Find S.A. using long division. A rational function NEVER has both a H.A. and S.A.

29 Sketching the Graph Identify domain. Simplify Find/Plot: vertical, horizontal, and slant asymptotes Find/Plot: x-intercept(s) and y-intercept Plot: At least one point on each side of the x-intercept(s) and vertical asymptote(s) Fill in with smooth curve.