Graph of a Curve

Continuity

This curve is _____________ These curves are _____________

Smoothness

This curve is _____________ These curves are _____________

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Graph of a Curve (cont’d)

IncreasingA function f is increasing on an open interval I if, for any x1 and x2 in I, with x1 < x2, we have f(x1) < f(x2).

DecreasingA function f is decreasing on an open interval I if, for any x1 and x2 in I, with x1 < x2, we have f(x1) > f(x2).

ConstantA function f is constant on an open interval I if, for any x1 and x2 in I, we have f(x1) = f(x2).

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f(x1)

f(x2) y

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f(x1) f(x2)

Local (or Relative) Extrema A function f has a local maximum at x = x0 if locally, f(x0) is greater than all the surrounding values of f(x). We call this f(x0) a local maximum of f.

A function f has a local minimum at x = x0 if locally, f(x0) is less than all the surrounding values of f(x). We call this f(x0) a local minimum of f.

x1 x2 x3

f(x3) is a _____________

f(x1) is a ___________________________

f(x2) is a ______________

Global (or Absolute) ExtremaA function f has a global maximum at x = x0 if f(x0) is greater than or equal to all values of f(x). We call this f(x0) the global maximum of f.

A function f has a global minimum at x = x0 if f(x0) is less than or equal to all values of f(x). We call this f(x0) the global minimum of f.

Use each graph to find the domain and range of the function:

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-5 -4 -3 -2 -1 1 2 3 4 5

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Domain = _______________Range = _______________

Domain = _______________Range = _______________

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Domain = _______________Range = _______________

Domain = _______________Range = _______________

1) Find f(5), f(2), and f(–6): f(5) = ___, f(2) = ___, f(–6) = ___

Use the graph of the function f given below to answer the following questions:

2) Is f(–1) positive? ___ f(5)? ___

3) Is f(2) negative? ___ f(–4)? ___

4) What is the domain of f ? ________

5) What is the range of f ? _________

6) What are the x-intercepts? _______

7) What is the y-intercept? ___

11) How often does the line y = 1 intersect the graph? ___ y = ½? ___ y = –2? ___

10) For what value(s) of x does f(x) = 0? _______ f(x) = 1? _____

8) Is f continuous? _______

9) Is f smooth? _______

12) List the interval(s) on which f is decreasing? ______________

13) List the interval(s) on which f is increasing? ______________

14) List the interval(s) on which f constant? _______________

15) If any, list all the local maxima? ______ At which x values? ______

16) If any, list all the local minima? ______ At which x values? ______

17) Is there any global maximum? _____ global minimum? _____

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