Section 2.3 Systems of Linear Equations in Two Variables Solving a System of Equations Graphically: 1. Solve both equations for y and graph in Y1 and Y2. 2. Find the point of intersection. Example: Solve graphically: 2x - 4y = 6 3x + 5y = 20 Solving Systems of Equations by Substitution: 1. Solve one equation for one variable. 2. Substitute the expression into the other equation to give an equation with one variable. 3. Solve for the variable. 4. Substitute into one of the original equations to find the value of the other variable. Example: Solve by substitution 2x - 3y = 2 y = 5x - 18
Example: Solve by substitution: 4x - 5y = -17 3x + 2y = - 7 Solving Systems of Equations by Elimination 1. If necessary, multiply one or both equations to make the coefficients of one variable equal and of opposite sign. 2. Add the equations to eliminate one variable. 3. Solve for the variable in the resulting equation. 4. Substitute into one of the original equations to find the other variable. Example: Solve by elimination 3x + 4y = 10 4x - 2y = 6 Example: Solve by elimination 3y = 5-3x 2x + 4y = 8
Dependent Systems Solve: 3x - 2y = 4 6x - 4y = 8 Inconsistent Systems Solve: 3x - 2y = 4 6x - 4y = 20 This system is dependent. Both equations represent the same line. There are many solutions. This system is inconsistent. The two equations represent parallel lines. There is no solution. Applications: A jewelry maker has a total revenue given by R = 89.75x and incurs a total cost of C = 23.50x + 1192.50, where x is the number of bracelets produced and sold. Applications: Joe has 39 coins, all nickels or dimes. The value of the coins is $2.70. How many nickels are there and how many dimes? Use graphical methods to find the number of units that gives break-even for the product.
Section 3.1 Quadratic Functions and Parabolas What is a quadratic function? A quadratic function can be written in the form: f(x) = ax 2 + bx + c where a, b, and c are real numbers and a is not zero. Is it quadratic? a) - 2x 2 + 7x b) 3x 3 + 4x 2-9 Parabolas and concavity The graph of a quadratic function is a parabola. y = ax 2 + bx + c If a is positive: the parabola is concave up and the vertex is a. If a is negative: the parabola is concave down and the vertex is a. Determining concavity: For each of the following, a) determine if the graph is concave up or concave down, and b) determine if the vertex of the graph is a maximum point or a minimum point. 1. f(x) = x 2 + 4x + 4 2. g(x) = -5x 2-6x + 8 3. h(x) = -2x 2-4x + 6
Finding the vertex of a parabola For the graph of y = ax 2 + bx + c, The x-coordinate of the vertex is The y-coordinate of the vertex can be found by evaluating the function at the x-coordinate. The vertex can also be found graphically using "maximum" or "minimum" Find the vertex: a) f(x) = -2x 2-4x + 6 Examples: Find the vertex: b) y = x 2 + 4x + 4 c) y = 3x + 18x 2 d) y = 0.0034x 2-0.439x + 20.185 Applications: Suppose the monthly revenue from the sale of televisions is given by the function R(x) = -0.1x 2 + 600x dollars a) Find the vertex and determine if it is a maximum or minimum point. b) Interpret the vertex in the context of the application. c) Graph the function. d) For what x-values is the function increasing? Decreasing?
Applications A grapefruit is thrown upward from the top of a building and the height can be modeled by the function: S = -16t 2 + 64t + 80. a) Find the t-coordinate and the S-coordinate of the vertex of the graph of the function. b) Graph the model. c) Explain the meaning of the coordinates of the vertex for this model. Supplement: Factoring Factor the following: x 2-8x + 15 x 2-2x - 35 3x 2-5x - 2 5x 2-6x - 8 4x 2-25
Section 3.2 Solving Quadratic Equations Quadratic Equations A quadratic equation can be written in the form: ax 2 + bx + c = 0 We will solve them using several methods: 1. Factoring 2. Graphing and finding x-intercepts 3. Quadratic Formula Solving by factoring Zero product property: If ab = 0, then a = 0 or b = 0. Solve by factoring: a) 3x 2 + 7x = 6 b) x 2-11x + 28 = 0 c) 5x 2 + 25x - 30 = 0 Solve by Graphing The official method: 1. Make sure the equation equals zero 2. Graph in Y1 and adjust the window 3. Use 2nd > Calc > Zero to find the x-intercepts. Alternate method: 1. Type the function into Y1 2. Let Y2 = 0 (or the other side of the eqn) 3. Graph and find the intersection(s).
Examples Use a graphing utility to find the x-intercepts of the graph of the function: Write answers as integers or reduced fractions. The Quadratic Formula: If ax 2 + bx + c = 0, then: a) y = 5x 2-14x + 8 b) y = x 2-6x - 27 Example: Solve using the quadratic formula: 3x + 2x 2 = -1 The discriminant Discriminant = b 2-4ac If b 2-4ac > 0, there are 2 real solutions. If b 2-4ac = 0, there is 1 real solution. If b 2-4ac < 0, there is no real solution. Example: How many solutions are there to x 2-6x + 9 = 0?
Applications A watermelon is thrown upward and the height of the ball can be modeled by S = 4 + 64t - 16t 2, where t is time in seconds and height is in feet. How long after the watermelon is thrown is the height 32 feet? Applications The number of admissions to hospitals in a country is given by A(t) = 34.231t 2-1025.75t + 40094.034 thousand people where t = number of years after 1983. a) By how much did the number of hospital admissions change between 1992 and 1995? b) Find in what year after 1983 the number of admissions was 32,410,000. Super-secret alternate method to solving quadratic equations! Write a calculator program: PRGM > NEW > Create New Name = QUADR To get input/output commands such as "Prompt" and "Disp", use PRGM > I/O PROGRAM: QUADR : Prompt A : Prompt B : Prompt C : (-B - (B 2-4*A*C)) / (2A) X : (-B + (B 2-4*A*C)) / (2A) Y : Disp X : Disp Y To get an arrow, use the STO button
Section 3.4 Quadratic Models Quadratic Regression To construct a quadratic model that best fits a set of data: 1. Enter the data in L1 and L2. 2. Stat > Calc > QuadReg 3. Type a "Y1" after QuadReg to store the equation in Y1. Mobile Internet Advertising Dollar value of the U.S. mobile Internet advertising market for 2006-2012. Year Advertising Market (billions of dollars) Year Advertising Market (billions of dollars) 2006 0.045 2010 0.545 2007 0.114 2011 0.791 2008 0.273 2012 1.023 2009 0.409 Continued... a) Create a scatter plot of the data. b) Find the quadratic function that best fits the data, with x being the number of years after 2000. c) Graph the aligned data and the model on the same axes. Does this model seem like a reasonable fit?
And more... d) Use the model to find the year in which the mobile Internet advertising market is projected to reach $3 billion. e) Use the model to predict the mobile Internet advertising market in 2013. Average Error Instructions for calculating average error can be found on your green sheet. Calculate the SSE and average error for the Mobile Internet Advertising quadratic model. Find the linear model that best fits the Mobile Internet Advertising data and calculate the SSE and average error. Citadel In-state Tuition The following table gives the cost of tuition and fees at the Citadel from 2000-2010. Year Tuition and fees Year 2000 3404 2006 7168 2001 3727 2007 7735 2002 4067 2008 8428 2003 4999 2009 8735 2004 5900 2010 9871 2005 6522 Tuition and fees Citadel Tuition a) Find the quadratic model that best fits the data. b) Estimate the tuition (using your model) for the year 2027. c) Calculate the SSE and average error for your model.