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3.1 Graphs of Polynomial Functions

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- Pg. 150#25 – 32 all, Pg. 161#1 – 4, 23 – 33 odd, 35 – 37
- #2[-4, 5] by [-10, 100]#5 [-5, 30] by [-5,000, 5,000]
- #8[-4, 6] by [-400, 100]
- #11Zeros = min = (2, 0), (5, 0); max = (3.5, 2.25)
- #14No zeros; no local max; min = (1.5, 5.5)
- #17No zeros; max = (-0.41, 12.59); min = (-1.16, 11.91), (1.57, 3.51)
- #20Zeros = (0, 0), (17, 0), (26.5, 0); max = (6.75, 5465.81);min = (22.25, -1985.81)
- #23Zero = (2.36, 0); max = (0.79, 12.10); min = (0.21, 11.90)
- #50V(x) = x2(100 – x) + (½)x(30 – x)(100 – x)#51 Graph
- #52[0, 30]#53 63,000 ft3 for 30 by 70 ft bldg
- #54R(x) = xp = x(2 + 0.002x – 0.0001x2)#55 (0, 151)
- #5689 glasses at $123.53#57 R(x) drops when x > 89 glasses

Graphing Polynomials

Continuity

A function is considered continuous on a given interval if it is possible to trace it on that given interval without lifting up your pencil.

If the function fails to be continuous, it is discontinuous.

All polynomial functions are continuous!!

- What does a higher degree do to a function?
- You need to memorize and be able to recognize immediately what functions will generally look like from key indicators!!

Points of Discontinuity

Examples

Determine if the following functions are continuous or discontinuous. If they are discontinuous, state why.

- A function f has a point of discontinuity at x = aif one of the following conditions hold:
- The function is not defined at x = a
- The graph has a break at x = a
- The graph has a hole at x = a

Intermediate Value Theorem (Property)

- If a function is continuous on [a, b], then f assumes every value between f(a) and f(b).
- If f(a) < L < f(b), then there is some number c in [a, b] such that f(c) = L.

The cool thing about polynomials… they are continuous EVERYWHERE!!!

Local Extrema

Looking at Graphs

For each of the following graphs:

State the local extrema

State the zeros

State the intervals of increasing and decreasing

- A local maximum occurs on an interval (a, b) if there is a value f(x) such that f(x) ≥ f(c) for all c in (a, b).
- A local minimum occurs on an interval (a, b) if there is a value f(x) such that f(x) ≤ f(g) for all g in (a, b).
- Why are they “Local?”