Unit 1 Functions and Graphs 1.1: Modeling and Equation Solving Focus: Factor completely. = ( ) 2 – ( ) 2 5r4 What goes here? = ( + )( – ) 44 5r = ( ) 2 + 2( )( ) + ( ) 2 5x-6 = ( + ) 2 -65x What goes here? 5x-6 = ( 5x – 6 ) 2

A Numerical Model is the most basic kind of model in which numbers (or data) are analyzed to gain insights into phenomena. YearTotalMaleFemale % Male % Female 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 in 2 Solution: Round pizza: A =  r 2 =  (12in) 2 = 144  in 2 = in 2 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) = x 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) = x

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 x x = 0.05x 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 tF

Understanding the Viewing Rectangle [-2, 3] by [-10, 20] x min x max y min y max

Complete Student Checkpoint Choose the correct viewing rectangle and label the tick marks. [-8,10] by [-8,16] [-8,12] by [-8,16]

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 tF Day 2 This pattern looks linear. Use a line of best fit to to find an algebraic model by finding the equation of the line. Using two coordinates we can write the equation. 03.8

Use the point-slope formula and calculate the slope from the two coordinates (0,3.8) and (20,6.7) This does a very nice job of modeling the data.

Solving an equation algebraically. Find all real numbers x for which x26x x 4x4x x 3x3x

Solve the equation algebraically and graphically. and graphically

Looking at the graph, this is the only x-intercept, zero or root Solve the equation algebraically and graphically. and graphically make right side =0

Grapher Failure Graph the equation The graph never intercepts the x-axis. Why? y cannot equal zero Where is the graph undefined?

Modeling and Equation Solving