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§ 1-3 Functions, Continued

The student will learn about:

piecewise functions,exponential functions,

and difference quotients.

polynomial functions,

rational functions,

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Introduction to PolynomialsA polynomial function is of the form

f (x) = an xn + an-1 x

n-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 x

n + an-1 xn-1 + … + a1 x + a0

The degree of the polynomial is the highest power of the variable.

The coefficients a0, a1, … , an are real

numbers with an ≠ 0.

The domain of a polynomial function is the set of real numbers.

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. 4

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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!

SOLVING A POLYNOMIAL EQUATION BY FACTORING

Solve 2x 4 + 4x 3 - 6x 2 = 0 by calculator.

Find the zeros by using your calculator under calc and zeros.

Or use table.

Solution:

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Introduction to Rational FunctionsDef: A rational function is any function of the form:

where n (x) and d (x) are polynomials. The domain is the set of all real numbers such that d (x) ≠ 0.

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42

2

x

xxy

)x(d

)x(n)x(f

Aren’t they pretty?

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.

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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.

y = 3-x

-4 x 4 0 y 30.

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Piecewise Linear Functions

For this example assume that the YCP Tuition is a continuous function.

Graph this function. Note the points of discontinuity.

Number of Credits Cost

Less than 12 $505 credit

12 to 18 $8505

Greater than 18 $8505 plus $505 per credit over 18.

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Graph

0 ≤ x ≤ 22 by 1 and 0 ≤ y ≤ $10,000 by $500

12 18

$8505

Y1 = 445X/(x<12) etc.

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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 QuotientsIn this course it will be important to evaluate the difference quotient for a function:

h

)x(f)hx(f

Given a function - evaluate:1. f (x + h)

4. h

)x(f)hx(f

2. f (x)

3. f (x + h) – f (x), and finally

( - )

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Difference Quotients - ExampleFind the difference quotient for the function

f (x) = 4x + 3. Rememberh

)x(f)hx(f

1. f (x + h) =

3. f (x + h) – f (x) = step 1 – f (x)

4.

h

)x(f)hx(f

2. f (x) = 4x + 3

1. f (x + h) = 4 (x + h) + 3 =1. f (x + h) = 4 (x + h) + 3 = 4x + 4h + 3

= 4h

4h

h4

I will call this the 4-step procedure.

( - )

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Difference Quotients - ExampleFind the difference quotient for the function

f (x) = x 2 + 3x - 4. Rememberf (x h ) f ( x )

h

1. f (x + h) =

3. f (x + h) – f (x) =

4.

h

)x(f)hx(f

2. f (x) = x 2 + 3x - 4= x 2 + 2xh + h 2 + 3x + 3h - 4

1. f (x + h) = (x + h) 2 + 3(x + h) - 4

2xh + h 2 + 3h

22xh h 3h2x h 3

h

( - )

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Summary.

• We learned about piecewise linear functions.

• We learned about the rational functions.

• We learned about polynomial functions.

• We learned about the exponential functions.

• We learned about difference quotients.

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ASSIGNMENT

§1.3 Homework on my website

15, 16, 17, 18, 19.