1. Graph of a Curve Continuity This curve is continuous
These curves are discontinuous y x y x y x
hole
gap
GAP
Smoothness
This curve is smooth
These curves are not smooth y x y x y x
corner
cusp

2. Graph of a Curve (cont’d)
Increasing
Decreasing
Constant
A 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).
A 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).
A function f is constant on an open
interval I if, for any x1 and x2 in I,
we have f(x1) = f(x2). x y x1 x2 I
f(x1)
f(x2) y x x1 x2 I
f(x1)
f(x2) y x x1 x2 I
f(x1)
f(x2)
Local (or Relative) Extrema
f(x1) is a local max and also a global max
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.
f(x3) is a local max x1 x2 x3
Global (or Absolute) Extrema
A 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.
f(x2) is a local min

3. Use each graph to find the domain and range of the function:-5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 4 5
Domain = [–5, –2)  (–1, 5]
Domain = [–4, 4)
Range = (–2, 2]
Range = [–3, 1)  [2,4) -5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 4 5
Domain = (–∞, –3)  [–1, 4)
Domain = (–, 5)  (5, )
Range = (–, –2)  (–1, 3]
Range = {–3}  [–2, 1)  (1, )

4. 1) Find f(5), f(2), and f(–6): f(5) = 3, f(2) = 1, f(–6) = –1
Use the graph of the function f given below to answer the following questions: -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6
1) Find f(5), f(2), and f(–6): f(5) = 3, f(2) = 1, f(–6) = –1
12) List the interval(s) on which f is decreasing? (–9, –2)  (0, 1)  (3, 4)
2) Is f(–1) positive? No f(5)? Yes
13) List the interval(s) on which f is increasing? (–2, 0)  (1, 3)  (4, 5)
3) Is f(2) negative? No f(–4)? Yes
14) List the interval(s) on which f constant? (5, 6)
4) What is the domain of f ? (–9,6]
15) If any, list all the local maxima? 1, 4 At which x values? 0, 3
5) What is the range of f ? [–3,4]
6) What are the x-intercepts? –7.5, –1, 1
16) If any, list all the local minima? –3, 0, 1 At which x values? –2, 1, 4
7) What is the y-intercept? 1
8) Is f continuous? Yes
17) Is there any global maximum? Yes, 4 global minimum? Yes, –3
9) Is f smooth? No
10) For what value(s) of x does f(x) = 0? –7.5, –1, 1 f(x) = 1? 0, 2, 4
11) How often does the line y = 1 intersect the graph? 3 times y = ½? 4 times y = –2? 2 times