Transcript 1.3 Functions, Continued

§ 1-3 Functions, Continued
The student will learn about:
polynomial functions,
rational functions,
exponential functions,
piecewise functions,
and difference quotients.
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Introduction to Polynomials
A polynomial function is of the form
f (x) = an xn + an-1 xn-1 + … + a1 x + a0
and graphs as a curve that wiggles back and
forth across the x-axis.
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Polynomials
f (x) = an xn + an-1 xn-1 + … + a1 x + a0
The coefficients a0, a1, … , an are real
numbers with an ≠ 0.
The domain of a polynomial function is the
set of real numbers.
The degree of the polynomial is the highest
power of the variable.
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Polynomial Functions
Polynomials are used to model many
situations in which change occurs
at different rates.
For example, the polynomial graphed
on the right might represent the total
cost of manufacturing x units of a product.
At first, costs rise steeply because of high start-up
expenses, then more slowly as the economies of mass
production come into play, and finally more steeply as
new production facilities need to be built.
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Solving a Polynomial
Solutions may be found algebraically.
Let f (x) = 0 and solve for x :
a. Factor – a bit of work that sometimes requires
synthetic division (whatever that is).
b. Complete the square – No thank you!
c. Use the quadratic formula – Next slide please!
d. We will use a graphing calculator and use the
calc and zero buttons.
       I love my calculator!       5
SOLVING A POLYNOMIAL EQUATION BY FACTORING
Solve 2x 4 + 4x 3 - 6x 2 = 0 by calculator.
Solution:
Find the zeros by using
your calculator under
calc and zeros.
Or use table.
Introduction to Rational Functions
Def: A rational function is any function of the
n ( x)
form:
f ( x) 
d ( x)
where n (x) and d (x) are polynomials. The
domain is the set of all real numbers such that
d (x) ≠ 0.
x2  4x
y 2
x 4
Aren’t they
pretty?
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Rational Functions
The graphs of these functions are shown below.
Notice that these graphs have asymptotes, lines
that the graphs approach but never actually
reach.
The Exponential Function
Definition: The equation f (x) = b x for b > 0,
b ≠ 1, defines an exponential function. The
number b is the base. The domain of f is the set
of all real numbers and the range of f is the set
of all positive numbers.
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Some Examples
y = 3x
-4  x  4
0  y  30.
-4  x  4
0  y  30.
y = 3-x
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Piecewise Linear Functions
For this example assume that the YCP
Tuition is a continuous function.
Number of Credits
Less than 12
12 to 18
Greater than 18
Cost
$505 credit
$8505
$8505 plus $505 per
credit over 18.
Graph this function. Note the points of
discontinuity.
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Graph
$8505
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0 ≤ x ≤ 22 by 1
and
Y1 = 445X/(x<12) etc.
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0 ≤ y ≤ $10,000 by $500
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Absolute Value Function
The absolute value function;
f (x) = |x| =
x
if x ≥ 0
- x if x < 0
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Difference Quotients
In this course it will be important to evaluate
the difference quotient for a function:
f (x  h )  f ( x )
h
Given a function - evaluate:
1. f (x + h)
( - ) 2. f (x)
3. f (x + h) – f (x), and finally
4. f (x  h )  f ( x )
h
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Difference Quotients - Example
Find the difference quotient for the function
f (x  h )  f ( x )
f (x) = 4x + 3. Remember
h
I will call this the 4-step procedure.
1. f (x + h) = 4 (x + h) + 3 = 4x + 4h + 3
( - ) 2. f (x) =
4x
+3
3. f (x + h) – f (x) = step 1 – f (x) = 4h
4. f (x  h )  f ( x )  4h  4
h
h
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Difference Quotients - Example
Find the difference quotient for the function
f (x) =
x2
+ 3x - 4. Remember
f (x  h )  f ( x )
h
1. f (x + h) = (x + h) 2 + 3(x + h) - 4
( - ) 2.
= x 2 + 2xh + h 2 + 3x + 3h - 4
f (x) = x 2
+ 3x
-4
3. f (x + h) – f (x) = 2xh + h 2 + 3h
2
f
(
x

h
)

f
(
x
)
2xh  h  3h
4.

 2x  h  3
h
h
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Summary.
• We learned about polynomial functions.
• We learned about the rational functions.
• We learned about the exponential functions.
• We learned about piecewise linear functions.
• We learned about difference quotients.
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ASSIGNMENT
§1.3 Homework on my website
15, 16, 17, 18, 19.
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