Section 6

Section 6.1

Linear Modeling

The linear model

Example: Converting temperature in Celsius to Fahrenheit

Given the equation

1) Sketch a graph of the function

Temperature in Celsius	Temperature in Fahrenheit
0
20
80

2) Find the slope: Use which makes the slope

3) Find the Fahrenheit equivalent of 40° C\

Example 2

The speed of sound has been calculated to be approximately 1090 ft/s, when the temperature is 32° F. However, as the temperature rises above 32° F, the speed of sound is about 1110 ft/s. Find the linear equation that relates the sped of sound to the Fahrenheit temperature and determine the speed of sound at 100° F

First use the slope formula

Use the points (32,1090) and (50,1100) to find the slope of the function.

find

Find the y-intercept

Thus, the linear equation would be

Use this equation to calculated the speed of sound at 100° F

Examples for the book exercises

Make a table of values

C	I
0	0
2	2.54(2) = 5.08
10	2.54(10) = 25.4

Plot the given points

12)

K	M
0	0
10
20

The equation C = 1.25 x + 6 gives the cost C is dollars when x French wine bottles are produced.

What is the slope of the equation

Answer; m=1.25

c) Find the cost of making 100 bottles of wine

d) How many wine bottles could be made with $1000

Quadratic Models

Graph of Quadratic Models

The parabola

A quadratic function is a function where the graph is a parabola and an equation of the

form: where

The x coordinate vertex is given by the equation:

Examples

Find the vertex and x-intercepts, then make a sketch of the parabola.

Graph

Graph of the function

Using the quadratic formula to solve an equation

The Quadratic Formula

The solution to the equation is given by

1) Solve

2) Solve

Page 301

8) Find the vertex, graph, and x intercepts of each parabola

x-intercepts

Vertex

12) At a local frog jumping contest. Rivet’s jump can be approximated by the equation and Croak’s jump can be approximate by, where x = the length of jump in feet and y = the height of the jump in feet.

a) Which frog can jump higher

Rivet’s vertex: Height:

Croak’s vertex: Height:

Croak can jump higher at 8 feet

b) Which frog can jump farther

Rivet’s can jump farther at 2(6 ft) = 12 feet

Graph of both frog jumps (Rivet’s jump and Croak’s jump)

Exponential models

The exponential function

“The Euler number”

Examples

The graph of the exponential function

1) Graph

x	y
-2
-1
0
1
2

2) Graph

x	y
-2
-1
0
1
2

Graph of

Exponential Models

Exponential Growth

Examples

1) The population of the United States is 290 million, what would be the population of the U. S. be in 20 years if its population would growth at a steady rate of .7 % for 20 years?

2) The population of Blacksburg, Virginia is 41,000, what would be in 10 years Blacksburg would grow at a rate of 1.1 % per year?

3) In 1995 the United States had greenhouse emissions of about 1400 million tons, where as China had greenhouse emissions of about 850 million tons. If in the next 25 years China greenhouse emission grew by 4 percent and the U. S. greenhouse emission grew by 1.3 percent, what would the emissions in tons for both countries in 2020?

4) Using the exponential growth formula, find the amount of money that you would have in a bank account if you deposited $3,000 in the account for 15 years at 1.1 % interest rate.

Exponential decay

5) A certain population of black bears in the eastern United States has been decreasing by 3.1 percent per year. If this trend keeps up, what will be the population of bears in 20 years if there is currently 1000 bears.

6) A certain isotope decreases at a rate of 5% per years. It there is currently 340 grams of the isotope, how many grams of the isotope will there be in 20 years?