Unit 1 Functions and Graphs1.1: Modeling and Equation Solving

Focus: Factor completely.25r2 16

25x2 60x 36

7x3 28x

= ( )2 – ( )2 5r 4

What goes here? What goes here?

= ( + )( – ) 445r 5r

= ( )2 + 2( )( ) + ( )2 5x -6

= ( + )2-65xWhat goes here? What goes here?

5x -6

= ( 5x – 6 )2

7x(x2 4)

7x(x 2)(x 2)

2x2 11x 5

(2x 1)(x 5)

A Numerical Model is the most basic kind of model in which numbers (or data) are analyzed to gain insights into phenomena.

Year Total Male Female%

Male%

Female

1980 316 304 12 96.2 3.8

1985 480 459 21 95.6 4.4

1990 740 699 41 94.5 5.5

1995 1085 1021 64 94.1 5.9

2000 1382 1290 92 93.3 6.7

U.S. Prison Population (Thousands)

Is the proportion of female prisoners over the years increasing?

Yes

A pizzeria sells a rectangular 18” by 24” pizza for the same price as its large round pizza (24” diameter). If both pizzas have the same thickness, which option gives the most pizza for the money?

We need to compare the areas of the pizzas.Rectangular pizza:

A = lw = (18in)(24in) = 432 in2

Solution:

Round pizza: A = r2

= (12in)2 = 144in2 = 452.4 in2

The round pizza is larger and therefore gives more for the money.

An Algebraic Model uses formulas to relate variable quantities associated with the phenomena being studied.

You are choosing between two long-distance telephone plans. Plan A has a monthly fee of $20 with a charge of $0.05 per minute for all long-distance calls. Plan B has a monthly fee of $5 with a charge of $0.10 per minute for all long-distance calls.

Express the monthly cost for plan A, f, as a function of the number of minutes of long-distance calls in a month, x.

f(x) = 20 + 0.05x

Express the monthly cost for plan B, g, as a function of the number of minutes of long-distance calls in a month, x.

g(x) = 5 + 0.10x

For how many minutes of long-distance calls will the costs for the two plans be the same?We are interested in how many minutes of long-distance calls, x, result in the same monthly costs, f and g, for the two plans. Thus, we must set the equations for f and g equal to each other. We then solve the resulting linear equation for x.

0.05x + 20 = 0.10x + 5-0.05x - 5 - 0.05x - 5

15 = 0.05x

0.05 0.05300 minutes = x

A Graphical Model is a visual representation of a numerical model or an algebraic model that gives insight into the relationships between variable quantities.

From the data table of prison populations, let t be the number of years after 1980 and let F be the percentage of females in the prison population from year 0 to year 20. Create a scatter plot of the data.

t

F t F

0 3.85 4.4

10 5.5

15 5.9

20 6.7

Understanding the Viewing Rectangle

-2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3

-5

-10

5

10

15

20

[-2, 3] by [-10, 20]

x minx max

y miny max

Complete Student CheckpointChoose the correct viewing rectangle and label the tickmarks.[-8,10] by [-8,16][-8,12] by [-8,16]

-8 -6 -4 -2 2 4 6 8 10 12

-8

-4

4

8

12

16

From the data table of prison populations, let t be the number of years after 1980 and let F be the percentage of females in the prison population from year 0 to year 20. Create a scatter plot of the data.

t

F t F

5 4.4

10 5.5

15 5.9

20 6.7

Day 2

This pattern looks linear. Use a line of best fit to to find analgebraic model by finding the equation of the line.

Using two coordinates we can write the equation.

0 3.8

Use the point-slope formula and calculate the slope from the two coordinates (0,3.8) and (20,6.7)

y y1 m(x x1)

y (3.8) 6.7 3.8

20 (0)(x (0))

y 3.8 0.145x3.8 3.8

y 0.145x 3.8

This does a very nice job of modeling the data.

Solving an equation algebraically.Find all real numbers x for which

6x3 11x2 10x 11x2 10x 11x2 10x

6x3 11x2 10x 0

x 6x2 11x 10 0 4-15-11

-60 6x2

-10-15x

4x

2

-5

2x

3x

x 2x 5 3x 2 0

x 0 or 2x 5 0 or 3x 2 0

x 0 or x 5

2or x

2

3

Solve the equation algebraically and graphically.

x2 10 4x 10 4x 10 4x

x2 4x 10 0

x (4) (4)2 4(1)( 10)

2(1)

x 4 56

2x 1.74 or x 5.74

and graphically x2 4x 10 y

x 1.74 or x 5.74

Looking at the graph, thisis the only x-intercept, zero or root

Solve the equation algebraically and graphically.

x x 1 x x

x 1 x

x 2 1 x 2

x 1 2x x2

x x

0 x2 3x 1

x ( 3) ( 3)2 4(1)(1)

2(1)

x 3 5

2

x 2.62 or x 0.38

and graphicallymake right side =0

x x 1x x 1 0

x x 1 y

Grapher FailureGraph the equation y 3 / (2x 5)

The graph never intercepts the x-axis. Why?

0 3

2x 50 2x 5 3

0 3

y cannot equal zero

Where is the graph undefined?

y 3

2x 5 when denominator equals 0

2x 5 0

x 2.5

Modeling and Equation Solving