Section 5.7: Logistic Functions Logistic Functions

When growth begins slowly, then increases rapidly, and then slows over time and almost levels off, the graph is an S-shaped curve that can be described by a "logistic" function.

Logistic growth:--spread of a disease--population of a species in a limited habitat (fish in a lake, fruit flies in a jar)--sales of a new technological product

Logistic Function

For real numbers a, b, and c, the function:

is a logistic function.

c = the limiting value

Example: Population growthSuppose that the size of the population of an island is given by:

thousand people

where t is the number of years after 1988. a) Graph for 0 < t < 30b) Find and interpret P(10)c) Find and interpret P(100)d) What appears to be the upper limit for the size of this population?

Example: Spread of a disease

If the number of students at a school infected by a disease t days after the students are first exposed is given by

a) How many students will be infected after 3 days?b) How many days until at least 500 students are infected?

Example: Internet Usage

The percent of the U.S. population that used the internet during selected years is given.

Year Percent Year Percent

1997 22.2 2003 59.2

2000 44.1 2004 68.8

2001 50.0 2005 68.1

2002 58.0 2007 70.2

Internet Usage (continued)

a) Find the logistic function that models the data, letting x = years after 1995.

b) Is the model a good fit for the data?

c) What does the model predict would be the percent of internet users in 2010?

Algebra Toolbox: Factoring, FOIL, and Rational Expressions

When to FOIL?Simplify the following:

1. (xy)2

2. (x + y)2

3. (3w)2

4. (z - 5)2

When to Factor?

Simplify:5. 6.

7. 8.

Simplifying Rational Expressions

Simplify:9.

10.

Factoring out -1:

Simplify:

11.

12.

Multiplying Rational Expressions

13.

14.

Section 6.1Higher-Degree Polynomial Functions

Cubic FunctionsA cubic function has the form:f(x) = ax3 + bx2 + cx + d (a can't be zero)

Graph the following cubic functions and observe the x-intercepts, turning points, and end behavior.a) f(x) = x3 -5x2 -2x + 5 x-intercepts (approx):

number of turning points: end behavior:

More cubic functions

b) f(x) = -2x3 + 6x2

x-intercepts (approx): number of turning points:

end behavior:

For a cubic function:maximum number of x-intercepts:maximum number of turning points:possible end behavior:

Local Extrema Points

Turning points are also called local extrema points.

Use 2nd > Calc > Minimumor 2nd > Calc > Maximumto find these points on a graph.

Find the local maximum and local minimum for the previous function, f(x) = -2x3 + 6x2

Example: Cubic Graph

a) Using an appropriate window, graph y = x3 - 27xb) Find the local maximum and local minimum, if possible.

Quartic Functions

A quartic function has the form:f(x) = ax4 + bx3 + cx2 + dx + e (a can't be zero)

Graph the following functions, observing end behavior, x-intercepts, and turning points:a) f(x) = x4

b) f(x) = x4 - 3x2

c) f(x) = 3x4 - 4x3

Polynomial Graphs1. The graph of a polynomial function of degree n has at most n - 1 turning points.2. The graph of a polynomial function of degree n has at most n x-intercepts.3. The end behavior of the graph of a polynomial function with odd degree can be described as “one end opening up and one end opening down.”4. The end behavior of the graph of a polynomial function with even degree can be described as “both ends opening up” or “both ends opening down.”

Example 1:

a) number of x-intercepts?

b) number of turning pts?

c) leading coefficient positive or negative?

d) degree of polynomial even or odd?

e) minimum possible degree?

Example 2:

a) number of x-intercepts?

b) number of turning pts?

c) leading coefficient positive or negative?

d) degree of polynomial even or odd?

e) minimum possible degree?

Example 3:

a) number of x-intercepts?

b) number of turning pts?

c) leading coefficient positive or negative?

d) degree of polynomial even or odd?

e) minimum possible degree?

Example 4:

a) number of x-intercepts?

b) number of turning pts?

c) leading coefficient positive or negative?

d) degree of polynomial even or odd?

e) minimum possible degree?

ApplicationsUsing data for 1982-2005, the total number of fatalities in drunk driving crashes in S.C. can be modeled by the function y = -0.0395x4 + 2.101x3 - 35.079x2 + 194.109x + 100.148, where x is the number of years after 1980.a) Graph to show 1980-2005.b) How many fatalities occurred in 2005, according to the model?c) Estimate the year when the number of fatalities was at a maximum.

Section 6.2: Modeling with Cubic and Quartic Functions

Poverty LevelThe numbers of people living below the poverty level in the U.S. are shown in the table.

Poverty level (continued)

a) Construct a scatter plot of the data.b) Find a cubic function that models the data, with x equal to the number of years after 2000 and y in millionsc) Graph to see how well the model fits the data.d) Use the model to estimate the number of people below the poverty level in 2008.e) Calculate the SSE and average error for your model.

U.S. Unemployment

Unemployment (continued)

a) Make a scatter plot of the data with x = 0 in 2000.b) Find the quartic function that models the data.c) Graph the model on the scatter plot.d) Estimate the unemployment rate in 2016 using your model.

Foreign-born population

The percent of the U.S. population that were foreign born from 1900 to 2010:

Year Percent Year Percent

1900 13.6 1970 4.8

1910 14.7 1980 6.2

1920 13.2 1990 8.0

1930 11.6 2000 10.4

1940 8.8 2005 11.7

1950 6.9 2010 12.0

1960 5.4

Foreign-born population (cont)a) Find a cubic function to model the data, using x = # of years after 1900 and rounding coefficients to seven decimal places.b) Calculate the SSE and average error.c) Find the quartic function to model the data, rounding coefficients to seven decimal places.d) Calculate the SSE and average error.e) Which function is a better fit for the data?f) The actual percent of the population that was foreign born in 2008 was 12.2%. What do the two models estimate for 2008?

Section 6.3Solutions of Polynomial Equations

Solving Polynomial Equations by Factoring

The zero-product property: If AB = 0, then A = 0 or B = 0.

Example: a) Solve the equation: (x - 1)(x + 2)(x + 5) = 0.

Examples

Solve the equations by factoring:b) 2x3 - 8x = 0

c) x4 - 6x3 + 9x2 = 0

d) x4 - 3x3 +2x2 = 0

e) 9v3 - 81v = 0

The Root Method

The real solutions of the equation xn = C are found by taking the nth root of both sides:

if n is odd; if n is even.

Examples:

Solve the following equationsa) x3 = 125

b) 5x4 = 80

c) 4x2 = 18

The graph of f(x).

f(x) = x3 - 7x2 +2x +40 Use the graph of f(x) to a) solve f(x) = 0

b) Find the factorization of f(x).

Solving Polynomial Equations Graphically

Example: Solve graphically6x3 - 9x2 - 69x + 36 = 0

Solution: Let Y1 = 6x3 - 9x2 - 69x + 36and let Y2 = 0. Find the intersections!(Alternatively, you could use 2nd > calc > zero)

Application: Maximum volumeA box is to be formed by cutting a square of x inches per side from each corner of a square piece of cardboard that is 24 inches on each side and folding up the sides. Write expressions for each of the following:height:length and width:volume:What input values make sense for this problem?Find the x that gives the maximum volume.

Sections 7.1 and 7.2Systems of Equations and Matrices

Solving Linear Equations with 3 or more variables

# of unknowns = # of equations Methods:--Elimination--Matrices

Method 1: EliminationExample 1:

Method 1: EliminationExample 2: Solve. 4x + 7y = 56x + 5y + 4z = -58x + 3y + 4z = 25

Method 2: Matrices

Solve the system:

First, represent by the augmented matrix: 2 -3 1 -1

1 -1 2 -3 3 1 -1 9

Working with matrices on your calculator:

Entering the matrix into the calculator:2nd > Matrix > Edit > [A] > 3 x 4 (for 3 eqns)Type the values into matrix A.

Reducing the matrix:2nd > Matrix > Math > rref( You will then be on the home screen and need to choose Matrix A:2nd > Matrix > Names > [A]

Row-reduced Echelon Form:

You should now have the reduced matrix:

1 0 0 2 0 1 0 1 0 0 1 -2

This gives the equations: x = 2, y = 1, z = -2.

Example 3: A manufacturer of furniture has three models of chairs: Anderson, Blake, and Colonial. The numbers of hours required for framing, upholstery, and finishing for each type of chair are given in the table. The company has 1500 hours per week for framing, 2100 hours for upholstery, and 850 hours for finishing. How many of each type of chair can be produced under these conditions?

Ex 4: An office supply company with four locations sells TI-82, TI-83, TI-85, and TI-86 calculators. During the month of August, the stores reported the following sales.

TI-82 TI-83 TI-85 TI-86 Receipts#1 4 94 10 32 $14,130#2 8 80 3 29 $12,100#3 6 50 2 4 $ 5,970#4 5 63 7 21 $ 9,655 At what price did the company sell each model calculator?

Example 5: Judy is in charge of ordering 37 pizzas for the office party. She orders three types of pizza: cheese, pepperoni, and supreme. The cheese pizzas cost $6 each, the pepperoni pizzas cost $9 each, and the supreme pizzas cost $12 each. She spent exactly twice as much on the pepperoni pizzas as she did on the cheese pizzas. If Judy spent a total of $324 on pizza, how many many pizzas of each type did she buy?

Example 6: A bank loans $285,000 to a development company to purchase three business properties. One of the properties costs $45,000 more than another, and the third costs twice the sum of these two properties. Find the cost of each property.

Example 7: A theater owner wants to divide a 3600-seat theater into three section, with tickets costing $40, $70, and $100, depending on the section. He wants to have twice as many $40 tickets as the sum of the other two types of tickets, and he wants to earn $192,000 from a full house. Find how many seats he should have in each section.

Example 8: Jane would like to run a mile in 8 minutes on a treadmill. This means she needs to average 7.5 mph (verify this!), and she would like to alternate between running 6 mph and 10 mph. How many minutes does she need to run at each speed?