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Pre-Calculus Chapter 1 notes Name:_____________ Period:_____________ Teacher: Mr. Staley A special thanks to Mrs. Sohalski and Mrs. Nesbitt for sharing their PC materials. Pre-Calculus/Trig – 1 Mr. Staley Assignment Schedule for August 11 to September 22 Date Reading Chapter Problems Assigned Blue Gold Assignment p. 9-11: 2, 4,14, 15, 20, 1.1 Graphs of p. 2-8 21,22, 25, 50, 58, 61, 66, 70 8/11 8/12 equations p. 12-20 p. 21-26: 10, 16, 19, 22, 24, Tues Wed 1.2 Linear equations in 40, 50, 61, 62, 72, 87, 92, Two Variables 116, 136 p. 27-34 p. 35-40: 24, 27, 28, 32, 34, 1.3 Functions p. 41-46 35, 47, 48, 52, 56, 64, 77, 8/13 8/14 1.4: Analyzing Graphs 78, 80 Thurs Fri of Functions p. 47-50: 3-24(multiples of 3), 32, p. 47-50: 34, 41, 44, 50, 53, 8/17 8/18 Wrap up 1.1 to 1.4 58, 62, 64, 66, 68, 69 Mon Tues Worksheet 1.1 to 1.4 Quiz 1.1 to 1.4 p. 52-55 p. 56-58: 2, 3, 13, 30, 32, 8/19 8/20 1.5: Library of 44, 47, 64, 74, 76 Wed Thurs Functions 1.6: Shifting, p. 59-63 p. 64-68: 2, 6(a, c, d, e), 10, 8/21 8/24 Reflecting and 20, 25, 28, 34. 40, 41, 44, Fri Mon Stretching Graphs 48, 80, 82 8/25 8/26 Finish 1.6 Worksheet 1.1 to 1.6 Tues Wed p. 69-73 Read: 1.7: Combinations of 8/27 8/28 Functions Test 1.1 to 1.6 Thurs Fri p. 74: 1, 2, 7, 35 - just these four questions. Labor Day is Monday, September 7th. This is a tentative schedule for the first grading period. Subject to change, if needed. p. 69-73 p. 74-75: 35-38, 40, 41, 44, 8/31 9/1 1.7 Composition of p. 77-82 45, 51, 54, 55, 67, 70 Mon Tues functions p. 87-92 p. 83-86: 1-4, 5, 6, 23, 9/2 9/3 26,30,32,54,63,81,84 1.8 Inverse Functions Wed Thurs 9/4 9/8 1.9: Mathematical p. 110-115 p. 93-98: 2, 3, 7, 13, 14, 22, Fri Tues Modeling 23, 27, 30, 31 p. 121- 129 p. 93-98: 8, 15, 32, 38, 42, 45, 48, 50, 54, 55, 57, 58, 60, 9/9 9/10 Finish 1.9 62 Wed Thurs Quiz 1.7 and 1.8 We will do 65, 71, and 73 in class. Review p. 116-120: 1-8 matching,12, 2.1: Quadratic packet for 14, 15, 17, 23, 24, 26, 33, 34, Functions 9/11 9/14 Test 2 on 39, 40, 42, 46, 50, 55, 59, 67, 2.2: Polynomial Fri Mon Monday 68, 71,78, 84 Functions of Higher Sept. 18 p. 130-133: 1-8, 28, 34, 40, Degree 41, 50, 53, 55, 59 Synthetic Division section: p. 9/15 9/16 Test 1.7 to 2.2 140-142: 5, 8, 13, 21, 25, 34, Tues Wed 37, 40, 45, 49, 50, 58, 59 p. 134 – 139 (section 2.2 ) p. 130: 9, 12, 9/17 9/18 2.3 Synthetic Division 14, 17, 23, 24, Thurs Fri ,64,68,70,77,87,90 2.3 Synthetic Read section p. 140-142: 5, 8, 13, 21, 24, 9/21 9/22 Division 2.3 25, 34, 37, 40, 45, 49, 50, 58, Fri Mon 59 September 18 is the end of the First Grading Period 1.1 PC Notes What you should learn: How to sketch graphs of equations How to find x- and y- intercepts of graphs of equations How to use symmetry to sketch graphs of equations How to use graph of equations in solving real-life problems Vocabulary Equation in Two Variables Solution (or solution point) Graph of an equation Intercepts Symmetry Example 1 – Graphing with a table of values (Point Plotting Method) Graph y 2 x 5 using a table of values 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 Graph y x 3 using a table of values 2 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 What might be a problem with this method? -4 -6 -8 -10 Finding Intercepts To find the x-intercept To find the y-intercept Example 2 – Finding the x- and y-intercepts Find the x- and y-intercepts for y 3x 2 3 Find the x- and y-intercepts for y x 3 2 Symmetry with respect to the x-axis Symmetry with respect to the y-axis Symmetry with respect to the origin How could knowing the symmetry of a graph beforehand be helpful in sketching the graph? Tests for Symmetry Symmetric with respect to the x-axis Symmetric with respect to the y-axis Symmetric with respect to the origin Example 3 – Testing for Symmetry Describe the symmetry of the graph of y x 3 2 Describe the symmetry of the graph of y 4 x 12 2 Example 4 – Using symmetry to graph Use symmetry to sketch the graph of y x 5 x 3 4 2 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 Example 5 – Graphing an absolute value equation Sketch the graph of y x 3 2 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 Circle: Equation (from the definition) Example 6 – Finding the Equation of a Circle The point (2,6) lies on a circle whose center is at (-1,5). Write the standard form of this circle. Three Common Approaches to solving an application Example 7 – Application A rectangle of length x and width w has a perimeter of 12 meters. a. Draw a rectangle that gives a visual representation of the problem. Use the specified variables to label the sides of the rectangle. b. Show that the width of the rectangle is w 6 x and its area is A x(6 x) . c. Use your graphing calculator to graph the area equation, sketch the graph below. 10 8 6 4 d. Use the graph to estimate the dimensions of the rectangle that yield the maximum area 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 1.2 PC Notes What you should learn: How to use slope to graph linear equations in two variables How to find slopes of lines How to write linear equations in two variables How to use slope to identify parallel and perpendicular lines How to use linear equations in two variables to model and solve real-life problems Vocabulary Linear Equation in Two Variables Slope Slope-Intercept Form Ratio Rate (Rate of Change) Point-Slope Form Two-Point Form Linear Extrapolation Linear Interpolation General Form of the Equation Parallel Perpendicular Depreciation Linear Depreciation Book Value Example 1 – Sketching the Graph of a Linear Equation Sketch the graph of each of the equations below y 3x 4 x 8 y 2 3x 2 y 6 10 10 10 10 8 8 8 8 6 6 6 6 4 4 4 4 2 2 2 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -10 -8 -6 -4 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -2 -2 -4 -4 -4 -4 -6 -6 -6 -6 -8 -10 -8 -8 -8 -10 -10 -10 Example 2 – Application of Slope Road Grade: From the top of a mountain road, a surveyor takes several horizontal measurements x and several vertical measurements y, as shown in the table (x and y are measured in feet). a. Sketch a scatter plot of the data. x y 300 -25 600 -50 900 -75 1200 -100 1500 -125 b. Use a straightedge to sketch the 1800 -150 best-fitting line through the points. 2100 -175 c. Find an equation for the line. d. Interpret the meaning of the slope of the line in part (c) in the context of the problem. e. The surveyor needs to put up a road sign that indicates the steepness of the road. For instance, a surveyor would put up a sign that states “8% grade” on a road with a downhill grade that has a slope of -8/100. What should the sign state for the road in this problem? Example 3 – Finding the slope of a line through two points Find the slope of the line passing through (4,5) and (2,6) . Example 4 – Finding Equations for Lines Find the equation for each of the following lines in slope-intercept form, point-slope form and two-point form. a. Passes through (2,6) with a slope of -3. b. Passes through (1,5) and (-2,7) c. Has an x-intercept of -4 and a y-intercept of 3. Equations of Lines General Form Vertical Line Horizontal Line Slope-intercept form Point-slope form Two-point form Example 5 – Finding Parallel and Perpendicular Lines Find the slope-intercept form of the line that passes through (1,4) and is parallel to 3x 2 y 8 . Find the slope-intercept form of the equation of the line that passes through (-2,6) and is perpendicular to 2 x 5 y 3 . Example 6 – Depreciation You purchase a car valued at $35,000. 3 years later you decide to sell the car and when you look up the book value you discover that it is now worth $22,000. Write a linear equation that describes the book value of the car in any given year. Use the equation to determine the book value of the car after 1 year. Is this interpolation or extrapolation? Use the equation to determine the book value of the car after 5 years. Is this interpolation or extrapolation? 1.3 PC Notes What you should learn: How to determine whether relations between two variables are functions How to use function notation and evaluate functions How to find the domains of functions How to use functions to model and solve real-life problems Vocabulary Relation Function: Domain: Range: Independent Variable: Dependent Variable: Function Notation: Piecewise-defined Function: Implied Domain: Example 1 – Testing for Functions Determine if each of the following relations is a function. a. (3,5), (4,8), (1,-3), (7,-3) b. (2,6), (5,7), (2,9), (-3,13) Example 2 – Testing for Functions Represented Algebraically Determine whether the equation represents y as a function of x. a. 2 x 3 y 4 b. x y 2 c. x y 4 2 2 2 Function Notation: Example 3 – Evaluating a Function If h( x) 3 x 2 x 5 find each of the following 2 a. h(3) b. h(2) c. h( x 2) Example 4 – Evaluating a Piecewise-defined Function x2 1 x2 Given f (x) evaluate the function when x 0,3,2,6 3x 4 x2 Example 5 – Finding the Domain of a Function Find the domain of each of the functions below. 1 a. f: {(3,5), (4,7), (8,0), (11,15)} b. g ( x) 3x 2 c. f ( x) 3x 2 2 x 8 d. h( x) x 2 9 Example 6 – Evaluating a Difference Quotient Difference Quotient Find the difference quotient for f ( x) x 3 x 6 2 Example 7 – Application Cost, Revenue, and Profit: A company produces a product for which the variable cost is $12.30 per unit and the fixed costs are $98,000. The product sells for $17.98. Let x be the number of units produced and sold. a. The total cost for a business is the sum of the variable cost and the fixed costs. Write the total cost C as a function of the number of units produced. b. Write the revenue R as a function of the number of units sold. c. Write the profit P as a function of the number of units sold. (Note: P = R – C.) 1.4 PC Notes What you should learn: How to use the Vertical Line Test for functions How to find the zeros of functions How to determine intervals on which functions are increasing or decreasing How to identify even and odd functions Vertical Line Test: Zeros of a Function: Increasing (function): Decreasing (function): Constant (function): Relative Minimum: Relative Maximum: Even (function): Odd (function): To find the domain of a function from its graph: To find the range of a function from its graph: Example 1 - Using the Vertical Line Test Determine if the graph represents y as a function of x If the graph of a function of x has an x-intercept at (a,0), then a is a _____________ of the function. Example 2 - Finding the Zeros of a Function 3x 5 Find the zeros of f ( x) 4 x 19 x 5 2 Find the zeros of f ( x) x6 Find the zeros of f ( x) 12 x 2 The point at which a function changes from increasing to decreasing is a relative ____________. The point at which a function changes from decreasing to increasing is a relative ________________. Example 3 - Determining Information from a Graph Use the graph below to determine each of the following Determine the interval(s) over which 10 the following function is increasing. 8 6 Determine the interval(s) over which 4 the following function is decreasing. 2 -10 -8 -6 -4 -2 2 4 6 8 10 Determine the interval(s) over which -2 the following function is constant. -4 -6 Determine any relative maximum(s). -8 -10 Determine any relative minimum(s). Approximate any zeros A function whose graph is symmetric with respect to the y-axis is a(n) _____________ function. A function whose graph is symmetric with respect to the origin is a(n) ______________ function. Think! Can the graph of a nonzero function be symmetric with respect to the x-axis? Why or why not?? Example 4 - Determining Even and Odd Determine if the function f ( x) 4 x 3x 1 is even, odd or neither. Explain your 2 reasoning. 1.5 PC Notes What you should learn: How to identify and graph linear and squaring functions How to identify and graph cubic, square root, and reciprocal functions How to identify and graph step and other piecewise-defined functions How to recognize graphs of common functions Linear Function: Constant Function: Squaring Function: Identity Function: Cubic Function: Square Root Function: Reciprocal Function: Step Function: Greatest Integer Function: Example 1 - Common Functions and Their Graphs Identify each of the following common functions and then sketch their graphs. f ( x) c f ( x) x f ( x) x 10 10 10 8 8 8 6 6 6 4 4 4 2 2 2 -10 -8 -6 -4 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -2 -2 -4 -4 -4 -6 -6 -6 -8 -8 -8 -10 -10 -10 f ( x) x f ( x) x 2 f ( x) x 3 10 10 10 8 8 8 6 6 6 4 4 4 2 2 2 -10 -8 -6 -4 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -2 -2 -4 -4 -4 -6 -6 -6 -8 -8 -8 -10 -10 -10 f ( x) x 1 f ( x) x 10 10 8 8 6 6 4 4 2 2 -10 -8 -6 -4 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 The graph of a linear function f ( x) ax b is a line with slope ________ and y-intercept _______. Important features of the graph of a linear function f ( x) ax b Important features of the graph of the constant function f ( x) b Important features of the graph of the identity function f ( x) x Important features of the graph of the squaring function f ( x) x . 2 Important features of the graph of the cubic function f ( x) x . 3 Important features of the graph of the square root function f ( x ) x. 1 Important features of the graph of the reciprocal function f ( x) . x Important features of the graph of the greatest integer function f ( x) x . Example 2 - Writing a Linear Function Write a linear function for which f(2)=6 and f(-1)=3. Example 3 - Evaluating a Step Function 10 1 Evaluate the function when x 1,3, , 8 2 6 then sketch the graph. f ( x) x 2 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 Example 4 - Graphing a Piecewise-defined Function x2 1 x0 Graph f(x)= 2x 1 x0 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 1.6 PC Notes What you should learn: How to use vertical and horizontal shifts to sketch graphs of functions How to use reflections to sketch graphs of functions How to use nonrigid transformations to sketch graphs of functions Vertical Shift: Horizontal Shift: Reflection: Rigid Transformation: Nonrigid Transformation: Vertical Stretch: Vertical Shrink: Horizontal Stretch: Horizontal Shrink: Shifting Graphs Graph the three graphs in the same viewing window (hint use different types of graphs for each!) 1. f ( x) x 2 g ( x) ( x 3) 2 h( x) ( x 3) 2 4 2. f ( x) x 3 g ( x ) ( x 2) 3 h( x) ( x 2) 3 3 3. f ( x) x g ( x) x 3 h( x) x 3 5 Given the graph of y f (x) , write the equation that would give the following transformations Vertical shift of c units upward: Vertical shift of c units downward: Horizontal shift of c units to the right: Horizontal shift of c units to the left: Example 1 - Shifts of the Graph of a Function Let f ( x) x . Write the equation for the function resulting from a vertical shift of 3 units downward and a horizontal shift of 2 units to the right. Use the graph of f ( x) x to sketch the graph of h( x) ( x 2) 3 2 2 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 A reflection in the x-axis is a type of transformation of the graph of y f (x) represented by h( x) _____________ . A reflection in the y-axis is a type of transformation of the graph of y f (x) represented by h( x) _______________ . Example 2 - Reflections of Graphs Let f ( x) x . Describe the graph of g ( x) x in terms of f . Example 3 - Finding Equations from Graphs Given the following graph of y f (x) , write the equation for y ' . y f (x) y' 10 10 8 8 6 6 4 4 2 2 -10 -8 -6 -4 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 3 Types of Rigid Transformations 1. 2. 3. 4 Types of Non-Rigid Transformations 1. 2. 3. 4. Use your graphing calculator to graph each of the following and then fill in the blanks below. 1. f ( x) x 2 g ( x) 3x 2 1 2. f ( x) x 3 g ( x) x 3 3 3. f ( x) x g ( x) 2x 2 1 4. f ( x) x 2 g ( x) x 4 For y f (x) and the real number c. A vertical stretch is represented by ________________ where ______________. A vertical shrink is represented by _________________ where ______________. A horizontal shrink is represented by _______________ where ______________. A horizontal stretch is represented by _______________ where ______________. Example 4 - Nonrigid Transformations 1 3 Compare the graph of y x to the graph of 3 y x 4 Compare the graph of f ( x) 2 x to f (2 x) . 2 1.7 PC Notes What you should learn: How to add, subtract, multiply, and divide functions How to find the composition of one function with another function How to use combinations of functions to model and solve real-life problems. Arithmetic Combination of functions: Composition of functions: The domain of an arithmetic combination of functions f and g consists of…. Arithmetic Combination For two functions f and g with __________________________ Sum: ( f g )(x) = Difference: ( f g )(x) Product: ( fg)(x) f Quotient: (x) g Example 1 - Arithmetic Combinations of Functions Let f ( x) 7 x 5 and g ( x) 3 2 x , find ( f g )(4) . Let f ( x) 3x 2 and g ( x) x 3 x 2 , find ( f g )(x) . 2 Let f ( x) 2 x 1 and g ( x) 3x 5 , find ( fg)(x) . Example 2 - Finding the Domain of Arithmetic Combinations of Functions f Let f ( x ) x 2 and g ( x) 3 x 2 , determine the domain of (x) and g g (x) . f Example 3 - The Composition of Functions Let f ( x) 3x 4 and g ( x) 2 x 1 . Find: 2 a. ( f g )(x) b. ( g f )( x) c. ( f g )(3) Example 4 - Finding the Domain of Composite Functions Let f ( x) x and 2 g ( x) 4 x 2 , find the domain of ( f g )(x) . To "decompose" a composite function, look for an ___________ function and an ______________ function. Example 5 - Finding Components of Composite Functions Express the function g ( x) 3( x 1) 2( x 1) 6 as a composition of two 2 functions. Example 6 - Application Health Care Costs: The table shows the total amount (in billions of dollars) spent on health services and supplies in the U.S. (including Puerto Rico) for the years 1993 through 1999. The variables y1 , y2 , and y3 represent out-of-pocket payments, insurance premiums, and other types of payments, respectively. (Source: Centers of Medicare and Medicaid Services) Year y1 y2 y3 1993 148.9 295.7 39.1 1994 146.2 308.9 40.8 1995 149.2 322.3 44.8 1996 155 337.4 47.9 1997 165.5 355.6 52 1998 176.1 376.8 54.8 1999 186.5 401.2 58.9 a. Use the regression feature of a graphing utility to find a quadratic model for y1 and linear models for y2 and y3. Let t = 3 represent 1993. b. Find y1 + y2 + y3. What does the sum represent? c. Use a graphing utility to graph y1 , y2, y3 and y1 + y2 + y3 in the same viewing window. d. Use a model from (b) to estimate the total amount spent on health services and supplies in the years 2003 and 2005. 1.8 PC Notes What you should learn: How to find inverse functions informally and verify that two functions are inverse functions of each other How to use graphs of functions to determine whether functions have inverse functions How to use Horizontal Line Test to determine if functions are one-to-one How to find inverse functions algebraically Inverse Function: Horizontal Line Test: One-to-One Function: Notation for Inverse Function 1 For a function f and its inverse f , the domain of f is equal to ________________ , and the range of f is equal to ________________. To verify that two functions, f and g , are inverse functions of each other….. Example 1 - Verifying that Functions are Inverses x3 Verify that the functions f ( x) 2 x 3 and g ( x) are inverse functions of 2 each other. If the point (a,b) lies on the graph of f , then the point (_____, ______) must lie on the 1 graph of f and vice versa. 1 The graph of f is a reflection of the graph of f in the line ___________. Example 2 - Sketching the graph of the Inverse Function Using the graph of f ( x) 4 x 3 , sketch the graph of its inverse. 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 To tell whether a function has an inverse from its graph…. A function f has an inverse if and only if f is …… Example 3 - Determining if a function has an inverse Does the graph of the function have an inverse? Explain. To find the inverse of a function algebraically.. 1. 2. 3. 4. 5. Example 4 - Finding the Inverse Find the inverse (if it exists) of f ( x) 4 x 5 . 4 2x Find the inverse (if it exists) of f ( x) 3 Find the inverse (if it exists) of f ( x ) x4 3 1.9 PC Notes What you should learn: How to use mathematical models to approximate sets of data points How to write mathematical models for direct variation How to write mathematical models for direction variation to the nth power How to write mathematical models for inverse variation How to write mathematical models for joint variation How to use the regression feature of a graphing utility to find the equation of the least squares regression line Varies Directly (Directly Proportional to): Constant of Variation (Constant of Proportionality): Varies directly as the nth Power (Directly Proportional to the nth power): Varies Inversely (Inversely Proportional to): Varies Jointly (Jointly Proportional to): Sum of Squares Differences: Least Squares Regression Line: Correlation Coefficient: Example 1 - Direct Variation If y varies directly as x, and y is 6 when x is 4, find the value of y when x is 20. Example 2 - Direct Variation, Application In Pennsylvania, the state income tax is directly proportional to gross income. You are working in Pennsylvania and your state income tax deduction is $42 for a gross monthly income of $1500. Find a mathematical model that gives the Pennsylvania state income tax in terms of gross income. Example 3 - Direct Variation as an nth Power If y is directly proportional to the third power of x, and y is 750 when x is 10, find the value of y when x is 8. Example 4 - Direct Variation as an nth Power, Application The distance a ball rolls down an inclined plane is directly proportional to the square of the time it rolls. During the first second, the ball rolls 8 feet. (See Figure 1.87 on page 89) a. Write an equation relating the distance traveled to the time. b. How far will the ball roll during the first 3 seconds? Example 5 - Inverse Variation If y varies inversely as x, and y is 4 when x is 16, find the value of y when x is 10. Example 6 - Joint Variation If z varies jointly as x and y, and if z=10 when x=4 and y=15, find the value of z when x=12 and y=7. Example 7 - Putting it Together Suppose r varies directly as the square of m and inversely as s. If r=12 when m=6 and s=4, find r when m=4 and s=10. Let a vary directly as m and n and inversely as y . If a=9 when m=4, n=9 and y=3, 2 3 find a if m=6, n=2 and y=5. Correlation Coefficient The closer r is to ______, the better. Example 8 - Finding a Least Squares Regression Line The numbers of U.S. Air Force personnel, p, on active duty for the years 1995 through 1999 are shown in the table. Use the regression capabilities of a graphing utility to find a linear model for the data. Let t represent the year with t=5 corresponding to 1995. Find the correlation coefficient and explain what this tells us about our equation. Year 1995 1996 1997 1998 1999 P 400 389 379 363 358