The Graph of an Equation Graph the following by using a table of values and plotting points.

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1 Precalculus - Calculus Preparation - Section 1 Graphs and Models Success in math as well as Calculus is to use a multiple perspective -- graphical, analytical, and numerical. Thanks to Rene Descartes we can represent ordered pairs in a plane by equations. The Graph of an Equation Graph the following by using a table of values and plotting points. 3x + y 7 y x 1 y x( 39 10x + x 4 ) x + y 7 y x y x( 39 10x + x 4 ) 30 Comparing Graphical and Analytical Approaches Using your graphing calculator graph the following. In each case find a viewing window that shows all the important characteristics of the graph. 3 y x 3x + x y 3x 40x + 50x 45 y ( x )( x 4)( x 6) How did you determine the windows for each? What things do you think are involved with an analytical approach? ym Number of h ps y- t and x- t(s) o or e deree poly. p tive or n tive leading c cient. relative max and relative min. As we proceed through this chapter we will be recalling and developing some new analytical tools to help us analyze graphs of equations. AP Calculus Chapter P Page 1

2 Intercepts of a Graph How can I find the intercepts? Use y x 3 4 x to demonstrate. Analytically -- Graphically -- Symmetry of a Graph Types Symmetrical to y-axis -- Analytical Test -- Graphical -- Symmetrical to the x-axis -- Analytical Test -- Graphical -- Symmetrical to the origin -- Analytical Test Graphical -- Try it with x y 1. Is it symmetrical to the x-axis, y-axis, origin, or none? Use this to help you graph the equation using values. AP Calculus Chapter P Page

3 Points of Intersection How can I solve x Analytically -- y 3 x y 1 analytically and geometrically? Geometrically -- Modeling Models How has Descartes revolutionary idea help in real-life applications of mathematics to help predict the future? Read page 3 first paragraph and example problem 6 in calculus book HW -- worksheet all, 5-53 odds, 54, 55, 59-6 all. Note: Only the ones that have calculators beside the problem indicate that you are allowed to use the graphing utility on your calculator. AP Calculus Chapter P Page 3

4 The Slope of a Line Precalculus - Calculus Preparation - Section Equations of Lines Point-Slope Form -- Slope-Intercept Form -- General Form -- Rates of Change How are rates of change related to lines? Graphing Linear Models Many problems in analytical geometry can be classified into two categories: Given a graph what is the equation? Given the equation, what is the graph? Sketch the equations y x + 1 y 3y + x 6 0 Parallel and Perpendicular Lines How do you know if lines are parallel analytically? How do you know if lines are perpendicular graphically? Find the general form of equation of a line that passes through the point (,-1) and are parallel to the line x 3y 5. (verify graphically) perpendicular to the line x 3y 5. (verify graphically) HW -- worksheet --- (3-73)/3, 74 Note: Only the ones that have calculators beside the problem indicate that you are allowed to use the graphing utility on your calculator. AP Calculus Chapter P Page 4

5 Precalculus - Calculus Preparation - Section 3 Functions and Function Notation Independent Variable -- Dependent Variable -- Function Notation -- f ( x) x 4x + 1 Note: Function notation has the advantage of clearly identifying the dependent variable f (x), formerly known as y, while at the same time telling you x is the independent variable and that the function itself is called f. Function notation allows you to be less wordy. Instead of asking What is the value of y that corresponds to x -? you can ask What is f (-)? If f ( x) x + 7, evaluate the following. f ( 3a) f ( b 1) f ( x + x) f ( x), x 0 x The Domain and Range of a Function Types of functions Implicit -- Explicit -- Domain -- Range -- What is the domain and range? f ( x) x +1 (find analytically) f ( x) sin x, x is radian measure (find geometrically) f ( x) { 1 x, if x< 1 x 1, if x 1 AP Calculus Chapter P Page 5

6 The Graph of a Function The graph y f (x) consists of all points (x, f (x)) How can we graphically determine if something is a function? Graphs of seven basic functions. Transformations of Functions Writing Equations for functions Each of the graphing utility screens shown below has the graph on of the eight basic functions above. Each screen also has a transformation of the graph. Describe the transformation. Then use your description to write an equation for the transformation. Check them out on your calculator to see if you are correct. Calculus Chapter P Page 6

7 Basic Type of Transformations (c > 0) Original Graph: y f (x) Horizontal shift c units to the right: Horizontal shift c units to the left: Vertical shift c units to the down: Vertical shift c units to the up: Reflection (about the x-axis): Reflection (about the y-axis): Reflection (about the origin): Classification and Combination of Functions Elementary Functions Algebraic -- Trigonometric -- Exponential and Logarithmic -- Polynomial Functions What is the degree of the polynomial mean? What does the leading coefficient tell us? What are odd and even functions? Odd -- Even -- Try it both analytically and graphically 3 f ( x) x x g( x) 1 + cos x, where x is in radian measure. Combining Functions and Composite Functions -- If f ( x) x 3 and g( x) cos x, then find the following. f ( x) + g( x) ( f g)( x) f ( g( x)) g( x) f ( x) HW -- worksheet -- (3-7)/3, 9-33 all, (36-48)/3, 49, 60 Note: Only the ones that have calculators beside the problem indicate that you are allowed to use the graphing utility on your calculator. Calculus Chapter P Page 7