GRAPH 4.4 By Megha K. Raman APRIL 22, 2015
1. Preface... 4 2. Introduction:... 4 3. Plotting a function... 5 Sample funtions:... 9 List of Functions:... 10 Constants:... 10 Operators:... 11 Functions:... 11 Properties of Graph... 13 4. Middle School Math... 19 Middle School Functions... 19 Plotting Scatter Plots... 20 Plotting inequalities... 21 Parabolas... 23 Plotting Shifts... 26 Exponential and Logarithmic Function... 28 Sample Middle School Math Problems... 30 5. Calculus... 31 Calculus and Pre Calculus Functions:... 31 Plotting Derivatives... 32 Calculating Definite Integrals... 35 Plotting a Tangent to a Function.... 36 Plotting Polar Functions... 38 How to plot a vertical line... 40 Calculating Arc Lengths between 2 values.... 41 Taylor Polynomials... 44 1
Figure 1 Inserting a Function... 5 Figure 2 example of entering a function... 6 Figure 3 Graphed Function 1... 7 Figure 4 Modify Function... 7 Figure 5 Edit Function... 8 Figure 6 Evaluating the Function... 8 Figure 7 Sample of Evaluating a Function... 9 Figure 8 Line Style/Type... 14 Figure 9 Adjusting Color of Functions... 14 Figure 10 Zoom... 15 Figure 11 Reset Zoom... 16 Figure 12 Editing Axes... 17 Figure 13 Editing x axis... 17 Figure 14 Editing y-axis... 18 Figure 15 Insert title... 18 Figure 16 Editing Style of Axes... 19 Figure 17 Menu icon for scatter Plot... 20 Figure 18 Scatter Plot Sample data... 20 Figure 19 Scatter Plot Sample Graph... 21 Figure 20 Inserting Inequality... 21 Figure 21 sample inequalities... 22 Figure 22 Sample Inequality Graph... 22 Figure 23Upward facing parabola... 23 Figure 24 Downward facing parabola... 24 Figure 25 Right facing parabola... 25 Figure 26 Left facing parabola... 26 Figure 27 Shifts and Translation sample 1... 27 2
Figure 28Stretch and Compression... 28 Figure 29 Inserting a Derivative... 33 Figure 30 Inserting Equation of Derivative... 33 Figure 31 Sample Graph of derivative and original function... 34 Figure 32 Multiple derivatives... 35 Figure 33 Calculating definite Integral... 35 Figure 34 Results of the Integration of definite integrals... 36 Figure 35 Option to inset tangent... 37 Figure 36 Select options for the tangent... 37 Figure 37results of plotting tangent to the chosen function.... 38 Figure 38 Function type... 38 Figure 39 Polar Equation options.... 39 Figure 40 Sample polar graph cos(2t)... 40 Figure 41 Vertical lines using Graph 4.4... 41 Figure 42 Selecting range for the function.... 42 Figure 43 r(t) = 3 5 cos(t)... 42 Figure 44Menu to choose Arc length... 43 Figure 45 Arc Length calculation... 43 Figure 46 Taylor Polynomials of e^x... 44 Figure 47Taylor Polynomials of cos(x)... 45 Figure 48 Taylor polynomials of sin(x)... 46 3
1. PREFACE I would like to thank both my parents for reviewing this manual countless numbers of times and giving me feedback and help. I have learned a lot by writing this, and I hope you as a reader do as well. This book, covering middle school math concepts as well as calculus concepts, is designed to aid you in solving homework problems. So whether you are actually doing your homework or you are just eager to learn more about Graph 4.4, this manual has you covered. 2. INTRODUCTION: This book was writen to be a guide for Graph 4.4 is a visual graphing tool. It can be used for students undestand function behaviour. The tool gives an adjustable view of function, larger than graphing calculators with equal amount of features. In this book, we have provided visual examples on how to use the tool. We also include typical problems students will encounter in: Algebra I: Equations of lines, absolute valued functions, parabola functions, and scatter plots Algebra II, CC1, CC2, and CC3: Plotting linear functions, regular polynomial functions, rational polynomial functions, functions of conics, exponential functions and logrithmic functions, Inequalities Pre calculus: trignometric functions, polar functions and parametric functions Calculus AB/BC Finding and explaining limits, Ccntinuity, derivatives, tangents, integrals, Taylor polynomials, and arc lengths. Graph 4.4 can be downloaded from: Windows: https://www.padowan.dk/download/ Mac: https://www.padowan.dk/mac/ 4
After you download the application, run the application. It is a few clicks install and does not have any config options to choose. If the links do not work, you can google Graph 4.4 and a link to instal it, should pop up. 3. PLOTTING A FUNCTION Graph 4.4 can be used to plot: 1. Standard functions y = f(x) 2. Parametric functions 3. Polar functions Once Graph 4.4 is installed, double click on the icon on your desktop Choose menu option Function -> Insert Function Figure 1 Inserting a Function 5
The Function Tab at the top has a pull down menu will show up as seen in Figure 1. Choose the Insert Function option from the pull down menu to type in the function you would like to graph. Choose the type of function and enter in the function. Remember to use parentheses when needed to make sure you have accurately inserted your function. You can also get a list of functions using Help -> List of Functions. We have included a List of Functions: In Figure 2, we have shown an example of a function. We type in our f(x), in the text box provided for Function Equation. Figure 2 example of entering a function 6
You can choose color, domain, and other properties for the graph and by clicking the ok button, your graph will plot as shown in Figure 3. Figure 3 Graphed Function 1 You can edit the function by Right click -> edit as shown in Figure 4. You may also do this by selecting the function and pressing enter. Figure 4 Modify Function 7
Producing a dialogue as shown in Figure 5 Figure 5 Edit Function To evaluate the function value, choose from the icon menu, Evaluate or trace the selected funtion. Ctrl E as shown in Figure 6. If the desired function is not selected, you can highlight the function you need to evaluate. Figure 6 Evaluating the Function This menu option produces, a dialogue in the left bottom corner where you can get the value of function for chosen value of x (input). Sample shows the function value at x = 1. Figure 7. 8
Figure 7 Sample of Evaluating a Function This option can be used to find zeros of funtion (especially non Integer roots), local maxima and minima of functions. You can calulate the function value and also where it occurs. SAMPLE FUNTIONS: Linear Functions: Y = f(x) = 3x + 5 Y = f(x) = -8 Y = f(x) = 9-2x Y = f(x) = Abs(x) Quadratic Functions: Y = f(x) = 4x^2 + 3x 10 Y = f(x) = 10 (x-5)^2-30 Ploynomial Functions: Y = f(x) = 6x^4-3x^3 + 18 x^2 2000 Y = f(x) = (x+5) (x-10) (x-1)^2 Rational Functions: Y = f(x) = (x^3-27) / ( (x-1) (x +5) (x +8)^2) 9
Complex Functions; Y = f(x) = ((x + 1) ^2 / (sin (3x)) * e^x) Y = f(x) = sin^2(x)/(x^2) Y = f(x) = sin (2x) Y = f(x) = sqrt (sin(x^2)) Y = f(x) = 2(x+3) x = 2*(x+3)*x Y = f(x) = asin(x) LIST OF FUNCTIONS: The following is a list of all variables, constants, operators and functions supported by the program. The list of operators shows the operators with the highest precedence first. The precedence of operators can be changed through the use of brackets. (), {} and [] may all be used alike. Notice that expressions in Graph are case insensitive, i.e. there are no difference between upper and lower case characters. The only exception is e as Euler's constant and E as the exponent in a number in scientific notation. CONSTANTS: x The independent variable used in standard functions. t The independent variable called parameter for parametric functions and polar angle for polar functions. e pi Euler's constant. In this program defined as e=2.718281828459045235360287 The constant π, which in this program is defined as pi=3.141592653589793238462643 undef Always returns an error. Used to indicate that part of a function is undefined. i The imaginary unit. Defined as i^2 = -1. Only useful when working with complex numbers. inf The constant for infinity. Only useful as argument to the integrate function. rand Evaluates to a random number between 0 and 1. 10
OPERATORS: Exponentiation (^) Rise to the power of an exponent. Example: f(x) =2^x Negation (-) The negative value of a factor. Example: f(x) =-x Logical NOT (not) not a evaluates to 1 if a is zero, and evaluates to 0 otherwise. Multiplication (*) Multiplies two factors. Example: f(x) =2*x Division (/) Divides two factors. Example: f(x) =2/x Addition (+) Adds two terms. Example: f(x) =2+x Subtraction (-) Subtracts two terms. Example: f(x) =2-x Greater than (>) Indicates if an expression is greater than another expression. Greater than or equal to (>=) Indicates if an expression is greater or equal to another expression. Less than (<) Indicates if an expression is less than another expression. Less than or equal to (<=) Indicates if an expression is less or equal to another expression. Equal (=) Indicates if two expressions evaluate to the exact same value. Not equal (<>) Indicates if two expressions does not evaluate to the exact same value. Logical AND (and) a and b evaluates to 1 if both a and b are 0, and evaluates to 0 otherwise. Logical OR (or) a or b evaluates to 1 if either a or b are 0, and evaluates to 0 otherwise. Logical XOR (xor) a xor b evaluates to 1 if either a or b, but not both, are 0, and evaluates to 0 otherwise. FUNCTIONS: Trigonometric sin Returns the sine of the argument, which may be in radians or degrees. cos Returns the cosine of the argument, which may be in radians or degrees. tan Returns the tangent of the argument, which may be in radians or degrees. asin Returns the inverse sine of the argument. The returned value may be in radians or degrees. acos Returns the inverse cosine of the argument. The returned value may be in radians or degrees. atan Returns the inverse tangent of the argument. The returned value may be in radians or degrees. sec Returns the secant of the argument, which may be in radians or degrees. csc Returns the cosecant of the argument, which may be in radians or degrees. cot Returns the cotangent of the argument, which may be in radians or degrees. asec Returns the inverse secant of the argument. The returned value may be in radians or degrees. acsc Returns the inverse cosecant of the argument. The returned value may be in radians or degrees. acot Returns the inverse cotangent of the argument. The returned value may be in radians or degrees. 11
Hyperbolic sinh Returns the hyperbolic sine of the argument. cosh Returns the hyperbolic cosine of the argument. tanh Returns the hyperbolic tangent of the argument. asinh Returns the inverse hyperbolic sine of the argument. acosh Returns the inverse hyperbolic cosine of the argument. atanh Returns the inverse hyperbolic tangent of the argument. csch Returns the hyperbolic cosecant of the argument. sech Returns the hyperbolic secant of the argument. coth Returns the hyperbolic cotangent of the argument. acsch Returns the inverse hyperbolic cosecant of the argument. asech Returns the inverse hyperbolic secant of the argument. acoth Returns the inverse hyperbolic cotangent of the argument. Power and Logarithm sqr exp sqrt root ln log logb Returns the square of the argument, i.e. the power of two. Returns e raised to the power of the argument. Returns the square root of the argument. Returns the nth root of the argument. Returns the logarithm with base e to the argument. Returns the logarithm with base 10 to the argument. Returns the logarithm with base n to the argument. Complex abs arg conj re im Returns the absolute value of the argument. Returns the angle of the argument in radians or degrees. Returns the conjugate of the argument. Returns the real part of the argument. Returns the imaginary part of the argument. Rounding trunc Returns the integer part of the argument. fract Returns the fractional part of the argument. ceil Rounds the argument up to nearest integer. floor Rounds the argument down to the nearest integer. round Rounds the first argument to the number of decimals given by the second argument. 12
Piecewise sign Returns the sign of the argument: 1 if the argument is greater than 0, and -1 if the argument is less than 0 u Unit step: Returns 1 if the argument is greater than or equal 0, and 0 otherwise. min Returns the smallest of the arguments. max Returns the greatest of the arguments. range Returns the second argument if it is in the range of the first and third argument. if Returns the second argument if the first argument does not evaluate to 0; Else the third argument is returned. Special integrate Returns the numeric integral of the first argument from the second argument to the third argument. sum Returns the sum of the first argument evaluated for each integer in the range from the second to the third argument. product Returns the product of the first argument evaluated for each integer in the range from the second to the third argument. fact Returns the factorial of the argument. gamma Returns the Euler gamma function of the argument. beta Returns the beta function evaluated for the arguments. w Returns the Lambert W-function evaluated for the argument. zeta Returns the Riemann Zeta function evaluated for the argument. mod Returns the remainder of the first argument divided by the second argument. dnorm Returns the normal distribution of the first argument with optional mean value and standard deviation. PROPERTIES OF GRAPH Style 13
Figure 8 Line Style/Type You can select the line style of the function you choose. Figure 9 Adjusting Color of Functions You can change the color of each function to differentiate the functions. Zoom 14
During function analysis, you will need to look at function behavior around a point: Either to check for a zero of a function, local maximum or a local minimum, You will use Zoon in feature of the graph. Similarly, to find the end behavior of a function (asymptotic behavior), you will use Zoom out feature. Figure 10 Zoom 15
You can zoom in or out on your function by clicking on the tab labeled zoom. You can also click on the three magnifying glasses. The square option makes the x intervals and y intervals the same. This is handy to view circles. Figure 11 Reset Zoom To go back to the default zoom, click Zoom > Standard, or simply press Ctrl+D Axes Settings 16
Figure 12 Editing Axes To edit the axes, click on the highlighted button or double click the axis on the side bar. This option is very useful if the user wants to vary the x cordinate increments and y coordinate increments different for clarity. Figure 13 Editing x axis You are able to to edit the domain of the x values as well as the unit, or scale of the graph. You can also alter the graph s appearance. 17
Figure 14 Editing y-axis You may also do this with y values. Figure 15 Insert title You can create a title for your graph, place your legend, alter the style of you axis, and change the mode when dealing with trigonometry. 18
Figure 16 Editing Style of Axes Lastly, you can change the colors of the axes and the font of labels, numbers, and legend 4. MIDDLE SCHOOL MATH MIDDLE SCHOOL FUNCTIONS Middle school students should be able to plot functions and study their behavior. They should predict how the graph will look like and verify. Predicting the domain and range of a function and verify them by plotting. Predict zeros, vertical asymptotes, and horizontal asymptotes, along with horizontal and vertical shifts. Verify them. If the function does not have an integer or a rational zero, Graph 4.4 can be used to find the approximate value close to a desired decimal place. Calculate slanting asymptotes and verify them as you can plot multiple functions at a time. You can plot the function and the slanting asymptote to verify them and understand the behavior. Plot straight lines Parabolic Functions Exponential Functions 19
Logarithmic functions. PLOTTING SCATTER PLOTS Figure 17 Menu icon for scatter Plot Figure 18 Scatter Plot Sample data 20
After plugging in the sample scatter plot data, select ok to your scatter plot. It should look like this: Figure 19 Scatter Plot Sample Graph PLOTTING INEQUALITIES Suppose you want to plot a system of inequalities, you can use the Graph 4.4 especially when the number of constraints are only 2. Choose Insert and choose Insert relations: Figure 20 Inserting Inequality 21
In the ensuing dialog type in the inequalities. Y > 2x + 4 as a relation and y <= 5 3x as a constraint Figure 21 sample inequalities The result is: Figure 22 Sample Inequality Graph 22
PARABOLAS Students working on CC1, CC2, and CC3 can use Graph 4.4 to plot a parabola, find its vertex or verify the calculated values, understand the domain and range. They can understand the behavior by marking focus, and plotting the directrix. Plotting parabolas opening upwards or downwards is straight forward. Plotting an Upward Parabola: Y=x^2+5x+6 Figure 23Upward facing parabola Plotting a Downward Parabola: Y=-2x^2+4x+5 Refer to Figure 24 Downward facing parabola where the vertex (1,7) is marked and the axis x = 1; 23
Figure 24 Downward facing parabola Plotting a Sideways Parabola:.e.g. x = 4y^2 + 3 or x = 4y y^2 + 16, you need to plot the top and bottom halves of the parabolas separately. For the first graph, plot: y = (1/4) * sqrt(x-3) and y = (-1/4) * sqrt(x-3) as shown in Figure 25 Right facing parabola 24
Figure 25 Right facing parabola For second graph solve the equation by completing the square: x = 4y y^2 + 16 x = -(y^2-4y ) + 16 x = - (y^2 4y + 4) + 4 + 16 x = - (y-2) ^2 + 20 (y-2)^2 = 20 x y-2 = sqrt(20-x) y = sqrt(20 x) + 2; Plot: y = sqrt(20-x) + 2 and y = -sqrt(20-x) + 2 As shown in: Figure 26 Left facing parabola. 25
Figure 26 Left facing parabola PLOTTING SHIFTS You are often required to find the domain and range of a shifted or translated function. Rules of translation: Vertical Stretch: New function = a * f(x) where a > 1 Vertical compression: New function = a * f(x) where abs (a) < 1 Horizontal Compression: New function = f(a *x) where a > 1 Horizontal Stretch: New function = f(a *x) where abs(a) < 1 Left shift: New function = f(x+a) a > 0 Right shift: New function = f(x-a) a > 0 Vertical shift: New function = f(x) + a Reflection over x axis: New function = (-) f(x) Reflection over y axis: New function = f(-x) Construct and plot equation of the following function and find the transformed function s domain and range: F(x) = 1/x vertically shifted by a factor of 7, reflected over y axis and translated 5 units to the right and 3 units down. 26
vertically shifted by a factor of 7 f(x) = 7/x reflected over Y axis f(x) = -7/x translated 5 units to the right and 2 units down: f(x) = -7/(x-5) + 3 shown in Figure 27 Shifts and Translation sample 1 Figure 27 Shifts and Translation sample 1 New domain (-infinity, 5) U (5, infinity) New range (-infinity, 3) U (3, infinity) F(x) = sin(x) Construct f(x) from this base function, with vertical stretch of 9, horizontal stretch of 2, and right shift of pi/4 vertical shift of -2 Vertical stretch of 9: f(x) = 9 * sin(x) Horizontal stretch of 2: f(x) = 9 * sin(x/2) Right shift of pi/4: f(x) = 9 * sin(x/2 pi/4) Vertical shift of -1: f(x) = 9 * sin(x/2 pi/4) - 2 Please refer to: Figure 28Stretch and Compression the green line shows the shifted axis. 27
Figure 28Stretch and Compression Domain will stay the same, (-infinity, infinity), while The range is [-11, 7]. EXPONENTIAL AND LOGARITHMIC FUNCTION You can use Graph to show that e^x and ln(x) are inverse functions of each other. Similarly, 10^x and log(x) 28
Graph gives option to plot logarithmic functions of different base other than 10 or e. To plot a logarithm with the base 2, use: logb(x,2) You can use the plotted graph to verify the domain, range and asymptotes for the functions. 29
SAMPLE MIDDLE SCHOOL MATH PROBLEMS Describe the translation of y=(x-3) 2 +5 In graph 4.4 you can plot both the parent function (y=x 2 ) and the given function at the same time to recognize the translation. The graph moved five units up and three units to the right. Find the vertical and horizontal asymptotes of y=(1/(x+5))+2 30
Asymptotes are indicated by the red dashed lines. 5. CALCULUS CALCULUS AND PRE CALCULUS FUNCTIONS: Study the increasing and decreasing intervals, 31
Plot derivative of a function. Plot first and second derivative of a function and show the relations between the derivatives and the original function. Plot tangents of the functions at a given point. Calculate the value of a definite integral of a given function. Find zeros, vertical asymptotes, holes, horizontal asymptotes and slanting asymptotes. Calculate arc length of a function for a given interval. Plot polar and parametric equation. Study convergence interval of Taylor Polynomials. PLOTTING DERIVATIVES Derivatives or functions, which give the equation of the slope of the tangent to a given function at any point. Calculus students study derivatives to understand function behavior. Since derivatives are also functions, we can take the derivatives of a derivative which is called the second derivative of the original function. Calculus students predict if the function is increasing or decreasing using the first derivative and they use the second derivative to understand the concavity of the original function. Plotting all of them in the same screen helps the student understand these concepts better. For plotting a derivative, choose the function, choose Function Inset f (x) 32
Figure 29 Inserting a Derivative Figure 30 Inserting Equation of Derivative 33
Figure 31 Sample Graph of derivative and original function The pink is the original function and The blue is the derivative funtion. You can choose any function and plot its derivative: The following shows plotting derivatives continuously. We started with degree 3 function: First derivative is degree 2 Second derivative is degree 1 You can use this to show when a derivative is positive and negative and what it means for a function: 34
Figure 32 Multiple derivatives CALCULATING DEFINITE INTEGRALS Select the function you want to get the definite integral of, by highlighting the function. Choose calculate definite integral over definite interval: Figure 33 Calculating definite Integral 35
Figure 34 Results of the Integration of definite integrals PLOTTING A TANGENT TO A FUNCTION. Highlight the function for which you want to plot the tangent to: Select: Insert a tangent or normal to the selected function. You can select the value of x and select either tangent or normal. You can also name it. 36
Figure 35 Option to inset tangent Figure 36 Select options for the tangent 37
Figure 37results of plotting tangent to the chosen function. PLOTTING POLAR FUNCTIONS This contains information how to create polar and parametric graphs using graph 4.4 Choose Function menu and choose Insert function option. In the ensuing dialogue box, choose Polar function r = f(t) Figure 38 Function type 38
The function option changes to r(t). Choose Function Equation r(t) = cos(2t) Leave Argument range (theta values) as: 0 to 2pi. You should change this to the required range when you want to find the plot of the graph in a restricted domain. The symbol for start and end can be chosen to understand where the graph begins and where it end for the chosen interval. You can customize using colors, thickness, line syle and hit OK to plot the graph. Figure 39 Polar Equation options. 39
Figure 40 Sample polar graph cos(2t) HOW TO PLOT A VERTICAL LINE If you want to show the vertical asymptote (CC3, Pre-Calculus), or axis of a parabola (CC1, CC2, CC3), or optimization problems (CC1, CC2) you need to draw vertical lines. It is not possible to use standard functions, or parametric functions to plot vertical lines. Using Graph 4.4, one can plot horizontal lines: y = K; or f(x) = K, where K is a constant, For vertical lines you need to use Polar functions. Vertical lines are x = K; Converting to polar you use (x/r) = cos(t) x = r * cos(t) = K Solving for r(t) we get: r(t) = K / cos(t) Examples are shown in: Figure 41 Vertical lines using Graph 4.4 40
Figure 41 Vertical lines using Graph 4.4 CALCULATING ARC LENGTHS BETWEEN 2 VALUES. Arc lengths are calculated using integrals. For some functions it may not be easily to integrate. You can use Graph 4.4 to calculate the arc lengths of any functions. You can use this to check your answer or find the answer for a difficult function. Let us find the length of a polar curve r(t) = 3 5 cos(t) from pi to 2 pi. We are going to show how we can plot for a different interval and show how to select a sub range to calculate the length. The example shows how to mark it with a different color. This will help students understand the behavior of the Polar functions. 41
Figure 42 Selecting range for the function. This will plot the entire inter from 0 to 2pi. We could also plot exactly the region we want by selecting the domain of t. Figure 43 r(t) = 3 5 cos(t) You can get the length of the curve using: 42
Figure 44Menu to choose Arc length Choose from the icon menu, Calculate the length of the path between two given points on the curve option. This will open a a dialogue box at the bottom left corner with options to choose points. Figure 45 Arc Length calculation When you select the two points between which you want to calculate the length, the calculated length is shown in the dialogue box and the portion of the curve you selected is color coded. One cal also use 43
this to understand functions better. This can be used to veriy your answers for arc length related problems after manual calculations. TAYLOR POLYNOMIALS For calculus BC sudents, plotting the base function and the Taylor Polynomials. For f(x) = e^x P1 = 1 + x/1! P2 = 1 + x + (x^2)/2! P3 = 1 + x + (x^2)/2 + (x^3)/3! P4 = 1 + x + (x^2)/2 + (x^3)/6 + (x^4)/4! P5 = 1 + x + (x^2)/2 + (x^3)/6 + (x^4)/24 + (x^5)/5! The graph is shown as below in: Figure 46 Figure 46 Taylor Polynomials of e^x Similarly, for: f(x) = Cos(x) 44
P0 = 1 P2 = 1 (x^2)/ 2! P4 = 1 x^2/2 + (x^4) / 4! P6 = 1 x^2/2 + (x^4) / 24 - (x^6)/6! P8 = 1 x^2/2 + (x^4) / 24 - (x^6) / 120 + x^8 / 8! You can see how the higher degree polynomials have larger interval of convergence in the following: Figure 47 Figure 47Taylor Polynomials of cos(x) For the base function sin(x) f(x) sin(x) P1 = x P3 = x x^/3! P5 = x x^3/6 +x^5 / 5! P7 = x x^3 / 6 + x^5 / 120 x^7 / 7! 45
P9 = x x^3 / 6 + x^5 /120 x^7 / 5040 + x^9 / 9! The plot of the base functions and the polynomials is shown in: Figure 48 Figure 48 Taylor polynomials of sin(x) 46