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

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1 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 o an Equation Graph the ollowing by using a table o 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 ollowing. In each case ind a viewing window that shows all the important characteristics o the graph. 3 y = x 3x + x y = 3x 40x + 50x 45 y = ( x )( x 4)( x 6) How did you determine the windows or each? What things do you think are involved with an analytical approach? sym Number o b ds y- t and x- t(s) o or e degree poly. p tive or n tive leading c cient. As we proceed through this chapter we will be recalling and developing some new analytical tools to help us analyze graphs o equations. AP Calculus Chapter P Page 1

2 Intercepts o a Graph How can I ind the intercepts? Use y = x 3 4 x to demonstrate. Analytically -- Graphically -- Symmetry o 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 o 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-lie applications o mathematics to help predict the uture? Read page 3 irst paragraph and example problem 6 in calculus book HW -- p. 8: 1-4 all, 5-15 odds, (1-57)/3, 64, 65, 68, 7, all, 8, 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 o a Line Calculus Preparation - Section Equations o Lines Point-Slope Form -- Horizontal Lines -- Slope-Intercept Form -- General Form -- Vertical Lines -- Intecept Form -- Ratios and Rates o Change Slopes as ratios occur when x-axis and y-axis have the o. Slopes as rates o change occur when x-axis and y-axis have the o. Average rate o change is always calculated over a interval. Instantaneous rate o change is always calculate at a a given. Graphing Linear Models Many problems in analytical geometry can be classiied 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 i lines are parallel analytically? How do you know i lines are perpendicular graphically? Find the general orm o equation o a line that passes through the point (,-1) and are parallel to the line x 3y = 5. perpendicular to the line x 3y = 5. HW -- p. 16: (3-7)/3, 73, 83 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 Independent variable is usually and the dependent variable is usually. Domain is the set o all numbers or which the equation is. Image is the value o at Range is the set o all numbers that are the o each item in the. Deine a unction: Ways a unction can be speciied: Implicit orm: Explicit orm: Function Notation -- ( x) = x 4x + 1 Note: Function notation has the advantage o clearly identiying the dependent variable (x), ormerly known as y, while at the same time telling you x is the independent variable and that the unction itsel is called. Function notation allows you to be less wordy. Instead o asking What is the value o y that corresponds to x = -? you can ask What is (-)? I ( x) = x + 7, evaluate the ollowing. ( 3a) ( 1) b ( 3) ( 4) ( + ) ( ) x h x h, h 0 The Domain and Range o a Function What is the domain and range? ( x) = x +1 (ind analytically) ( x) = sin x, x is radian measure (ind graphically) ( x) = 1 x, i x < 1 x 1, i x 1 AP Calculus Chapter P Page 5

6 The Graph o a Function The graph y = (x) consists o all points (x, (x)) x is the directed distance rom the - axis (x) is the directed distance rom the - axis How can we graphically determine i something is a unction? Graphs o seven basic unctions. Transormations o Functions Writing Equations or unctions Each o the graphing utility screens shown below has the graph on o the eight basic unctions above. Each screen also has a transormation o the graph. Describe the transormation. Then use your description to write an equation or the transormation. Check them out on your calculator to see i you are correct. Calculus Chapter P Page 6

7 Basic Type o Transormations (c > 0) Original Graph: y = (x) Horizontal shit c units to the right: Horizontal shit c units to the let: Vertical shit c units to the down: Vertical shit c units to the up: Relection (about the x-axis): Relection (about the y-axis): Relection (about the origin): Graph narrows (stretches vert): Graph widens (stretches horz): Classiication and Combination o Functions Elementary Functions Algebraic Transcendental Trigonometric Exponential and Logarithmic Polynomial Functions What is the degree o the polynomial mean? What does the leading coeicient tell us? What are odd and even unctions? Odd -- Even -- Try it both analytically and graphically 3 ( x) = x x g( x) = 1 + cos x, where x is in radian measure. Combining Functions and Composite Functions -- ( ) ( ) I x = x 3 and g x = cos x, ind each o the ollowing. g + ( g )( x) ( x) ( g )( x) HW --p. 7: 1,, (3-33)/3, all, 56, 59, 61, 65, 66, 69, 75, 76, 78, 79, 81, 8, 85, 87, 88, 9, 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 6