Graphing Polynomial Functions: The Leading Coefficient Test and End Behavior: For an n th n. degree polynomial function a 0, 0, then

Transcription

1 Graphing Polynomial Functions: The Leading Coefficient Test and End Behavior: For an n th n n 1 degree polynomial function a 0, n If n is even and an 0, then f x a x a x a x a with n n x

2 If n is even and a 0, then n x

3 If n is odd and an 0, then 3 x

4 If n is odd and a 0, then n 3 x

5 Determine the end behavior of the following polynomial functions f x 4x x. 4 f x x 1x f x x x 8x f x 4x x 4. 6

6 5. f x x x 3 f x x x f x x 1 x 3 8. f x x x

7 Behavior at the x-intercepts: If x c k number, then is the highest power of x c that is a factor of f x, with c a real If k is even, then the graph touches the x-axis at c but doesn t cross the axis. x 1 x 1

8 If k is 1, then the graph crosses the x-axis at c with a non-zero angle. x 1 x 1

9 If k is odd and greater than 1, then the graph crosses the x-axis at c with a zero angle(flat). x 1 3 x 1 3

10 Steps for sketching graphs of polynomial functions: 1. Determine the end behavior, and indicate it on the graph with arrows.. Find all the real zeros(x-intercepts) of f x, and indicate them on the graph with points. 3. Find the y-intercept, and indicate it on the graph with a point. 4. Use the end behavior and x-intercept behavior to connect the previous points and arrows into a reasonable graph.

11 Sketch the graphs of the following polynomial functions f x 7 x 4 x 3

12 . f x x x

13 3. 3 f x x x

14 4. f x x x 3 {Factor first.}

15 3 5. f x x x {Factor first.}

16 6. 3 f x x x 8x {Factor first.}

17 f x x 1x 8x 48x {Factor first.}

18 4 8. f x x x {Factor first.}

19 Finding zeros of polynomials is not always an easy task. For quadratic polynomials, you have the quadratic formula, but sometimes it gives results like 3 4, and you still have to approximate the zeros because of the radical,. There s also a cubic formula for cubic polynomials, and a quartic formula for quartic polynomials, but they still can lead to results requiring approximations because of the presence of radicals. It was proven in the 1800 s that there are no equivalent formulas for the zeros of fifth degree and higher polynomials in terms of radicals of the coefficients.

20 Approximating real zeros of polynomials: Intermediate Value Theorem: For f x a polynomial function, if f a and f b have opposite signs, then there is at least one value c between a and b with f c 0. f a a b f b

21 The Bisection Method: If f a and f b have opposite signs with a b, then there is at least one zero in the a b interval a,b, so our first approximation is the midpoint,. The error of our b a first approximation is less than half the width of the interval,. Next, examine the sign at the midpoint If a b f a and a b. f have opposite signs, then there is at least one zero in the interval a b a a,, and our next approximation is the midpoint of this interval, b a error bound of. 4 a b If f and a b, with an f b have opposite signs, then there is at least one zero in the interval a b a b b,b, and our next approximation is the midpoint of this interval,, with an b a error bound of. The process continues. 4

22 a b a a a b b Left Endpoint(sign) a(+) a(+) Midpoint(sign) Right Endpoint(sign) Error Bound a b b a b(-) a b a a b b a 4

23 Example: 1. Use the Bisection Method to approximate the zero of 3 and 1. Left Endpoint(sign) 1 0(+) 0(+) 1 f x x 3x 1 between 0 Midpoint(sign) Right Endpoint(sign) Error Bound 1 1(-) See the link Bisection Worksheet to get a fast expansion of the table.

24 . Use the Bisection Method to approximate the zero of 3 and. Left Endpoint(sign) 3 1(-) 1(-) 5 f x x x 1 between 1 Midpoint(sign) Right Endpoint(sign) Error Bound 1 (+) See the link Bisection Worksheet to get a fast expansion of the table.