Transcript 1 5 Analyze Graphs

1.
2.
3.
4.
Objectives:
To calculate average
rate of change
To determine whether
a function is even,
odd, or neither
To use a graphing
utility to find minima
and maxima
To find the zeros of a
function algebraically
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•
•
•
•
Assignment:
P. 62: 16, 18, 22-24
P. 63: 49, 51, 53, 54,
63-69 odd
P. 63: 71-76 some
P. 64: 88
P. 65: 91-95 odd
Rate of Change
Secant Line
Tangent Line
Minimum Point
Maximum Point
Relative Minimum
Relative Maximum
Slope is often referred to as rate of change. Why
is the rate of change for any given line always
constant?
C
4
B
2
-10
-5
A
5
-2
10
Would the rate of change be constant for other
graphs, like circles or parabolas?
-5
6
6
4
4
2
2
5
-5
5
For these graphs, the rate of change is different
at every point along the curve.
-5
6
6
4
4
2
2
5
-5
5
It was the search for this rate of change, called the
instantaneous rate of change, that eventually
lead to the discovery of differential calculus.
-5
6
6
4
4
2
2
5
-5
5
Calculus tells us that the rate of change at any
given point on a graph is equal to the slope of
the tangent line at that point.
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6
6
4
4
2
2
5
-5
5
A line is a tangent if
and only if it
intersects a curve in
one point.
Finding the slope of a
tangent line to a circle
is fairly easy, even
though you only have
one point on the line.
You simply find the
slope of the radius,
and then take the
negative reciprocal.
6
4
2
5
-2
But what about other
curves? For
example, shown is
the graph of
y = 6 – x2. How
would we find the
slope of the tangent
line at, say, (1, 5)?
6
4
2
-5
5
-2
The problem is that a
parabola, or most
other curves, do not
have a radius that is
perpendicular to the
tangent line at any
given point, and we
only have one point
on the line.
6
4
2
-5
5
-2
It was the resolution
of this problem, by
Fermat, Newton,
and Leibniz that led
to the discovery of
differential calculus.
It begins with
another line, called
the secant line.
6
4
2
-5
5
-2
A line is a secant if and
only if it intersects a
curve in two points.
Watch how a series of
secants can get
closer and closer to
the tangent line.
Watch how a series of
secants can get
closer and closer to
the tangent line.
Finding the slope of a
secant line gives us
the average rate of
change.
The average rate of change between any two
points is the slope of the secant line through
the two points:
f ( x2 )  f ( x1 )
Average rate of change =
x2  x1
y
=
x
=msec
Find the average rate of change of f (x) = x2 – 2x
for x1 = −1 and x2 = 3.
Find the average rate of change of
for x1 = 3 and x2 = 8.
f ( x)   x  1  3
In physics, the position of something (i.e., a
thrown rock) is given by the equation
s  16t 2  v0t  s0
where s = position, t = time, v0 = initial velocity,
and s0 = initial position.
If we knew calculus, we could calculate the
instantaneous rate of change at any point; but since
we don’t, we’ll have to settle for the average rate of
change.
In physics, the position of something (i.e., a
thrown rock) is given by the equation
s  16t 2  v0t  s0
Write a function that represents the situation,
then find the average rate of change.
1. An object (rock) is thrown upward from a
height of 6.5 feet at a velocity of 72 fps.
– t1 = 0, t2 = 4
In physics, the position of something (i.e., a
thrown rock) is given by the equation
s  16t 2  v0t  s0
Write a function that represents the situation,
then find the average rate of change.
2. An object (rock) is dropped from a height of
80 feet.
– t1 = 1, t2 = 2
You will be able to
determine whether a
function is even, odd, or
neither
Shown below are two types of symmetry that the graph
of a function can have.
If the graph has y-axis
symmetry, the function is said
to be even.
If the graph has origin
symmetry, the function is said
to be odd.
You don’t always want to look at a graph to see if a
function is even or odd, so you can perform these
simple algebraic tests instead.
1. A function f (x) is even 2. A function f (x) is odd
if, for each x in the
if, for each x in the
domain, f (x) = f (−x).
domain, f (x) = −f (x).
Plugging in −x for x in f (x)
doesn’t change the
function.
Plugging in −x for x in f (x)
gives you −f (x).
Let n be an even
number:
x  x
n
n
For example:
Let m be an odd
number:
m

x


x
 
m
For example:
2
x  x
2
3

x


x
 
x  x
4
5

x


x
 
4
3
5
Determine whether each of the following
functions are even, odd, or neither.
f ( x)  x 2  x 4
h( x )  x 5  2 x 3  x
k ( x)  x 5  2 x 4  x  2
g ( x)  2 x 3  1
j ( x )  2  x 6  x8
p ( x)  x 9  3x 5  x 3  x
What do you notice about all of even functions?
What do you notice about all of the odd
functions? Can you come up with a shortcut?
f ( x)  x 2  x 4
h( x )  x 5  2 x 3  x
k ( x)  x 5  2 x 4  x  2
g ( x)  2 x 3  1
j ( x )  2  x 6  x8
p ( x)  x 9  3x 5  x 3  x
You don’t look at the graph or perform an algebraic test
to see if a function is even or odd. Just look at the
powers.
1. A function f (x) is even 2. A function f (x) is odd
if, for each x in the
if, for each x in the
domain, f (x) = f (−x).
domain, f (x) = −f (x).
This happens when all the
powers of x are even.
This happens when all the
powers of x are odd.
Caveats:
1. The previous shortcuts only work on
polynomial functions.
2. You have to think of a constant as k∙x0 (a
number times x0) which is an even power of
x.
The vertex of a parabola
marks the turning point
of the graph of a
quadratic function.
– A turning point is a point
at which the function
values “turn” from
increasing to decreasing or
vice versa.
The y-coordinate of a
parabola’s turning
point marks the
absolute minimum or
maximum of the
function since there
are no other points
above or below it.
Other polynomial functions also have various
turning points that mark minima and maxima;
however, they may not be absolute.
Extrema (min/max values) come in two
varieties:
1. Absolute
2. Relative (Local)
• Relative (Local) Minimum:
The y-coordinate of a turning
point that is lower on a graph
than its surrounding points
• Relative (Local) Maximum:
The y-coordinate of a turning
point that is higher on a
graph than its surrounding
points
It’s a fairly easy exercise to approximate the location of
relative extrema using your graphing utility.
1. Press Y= and enter the function.
2. Choose your favorite ZOOM setting.
3. Press 2nd TRACE for the CALC menu.
4. Choose minimum or maximum.
5. Set the left bound, right bound, and a guess.
6. Magic!
Use a graphing utility to find the extrema of
each of the following functions.
1. h(x) = 0.5x3 +x2 – x + 2
2. j(x) = x4 + 3x3 – x2 – 4x – 5
A square piece of sheet
metal is 10 inches by 10
inches. Squares of side
length x are cut from the
corners and the
remaining piece is folded
to make an open box.
What size square(s) can
be cut from the corners
to create a box with a
maximum volume?
You will be able to find the
zeros of a function using algebra
A zero of a function is the x-value that makes
the function equal zero.
Zeros = Roots
Roots = 𝑥-intercepts
To find the find the zeros of a function, set it
equal to zero and solve for x.
How many zeros should you expect the function
below to have?
f ( x)  5 x  3 x  2 x  9
7
4
How many zeros should you expect the function
below to have?
f ( x)  5 x  3 x  2 x  9
7
4
For polynomial functions, the degree (highest
power) determines the number of zeros.
– However, some may be repeated
Find the zeros of each function algebraically.
2
1. f ( x)  3x  x  10
2
g
(
x
)

81

x
2.
3x  7
3. h( x) 
x5
1.
2.
3.
4.
Objectives:
To calculate average
rate of change
To determine whether
a function is even,
odd, or neither
To use a graphing
utility to find minima
and maxima
To find the zeros of a
function algebraically
•
•
•
•
•
Assignment:
P. 62: 16, 18, 22-24
P. 63: 49, 51, 53, 54,
63-69 odd
P. 63: 71-76 some
P. 64: 88
P. 65: 91-95 odd