Chapter Implicit Function Graphs You can graph any one of the following types of implicit functions using the calculator’s built-in functions.
14-1 Before Graphing an Implicit Function k Entering the CONICS Mode 1. In the Main Menu, select the CONICS icon and enter the CONICS Mode. When you do, the following built in function menu appears on the screen. 2. Use f and c to highlight the built-in function you want, and then press w. The following nine functions are built in.
14-2 Graphing an Implicit Function Example 1 To graph the circle (X – 1)2 + (Y – 1)2 = 22 Use the following View Window parameters. Xmin = –6.3 Ymin = –3.1 Xmax = 6.3 Ymax = 3.1 Xscale = 1 Yscale = 1 1. Select the function whose graph you want to draw. cccc 2. Press w and the variable input screen appears. w Graph function Function variables • The values that appear are the values currently assigned to each variable, which are general variables used by the calculator.
14 - 2 Graphing an Implicit Function P.155 • Certain View Window parameters can make a circle graph come out looking like an ellipse. When this happens, you can use the graph correction function (SQR) to make corrections and produce a perfect circle. Example 2 (Y – 1)2 (X – 3)2 To graph the hyperbola –––––––––– – –––––––––– =1 2 2 22 Use the following View Window parameters. Xmin = –8 Ymin = –10 Xmax = 12 Ymax = 10 Xscale = 1 Yscale = 1 1. Select the function whose graph you want to draw.
Graphing an Implicit Function 14 - 2 • A parabola is the locus of points equidistant from fixed line l and fixed point F not on the line. Fixed point F is the “focus,” fixed line l is the “directrix,” the horizontal line that passes through the focus directrix is the “axis of symmetry,” the length of a straight line that intersects the parabola, passes through the locus, and is parallel to fixed line l is the “latus rectum,” and point A where the parabola intersects the axis of symmetry is the “vertex.
14-3 Implicit Function Graph Analysis You can determine approximations of the following analytical results using implicit function graphs. • Focus/vertex calculation • Latus rectum calculation • Center/radius calculation • x-/y-intercept calculation • Directrix/axis of symmetry drawing and analysis • Asymptote drawing and analysis After graphing an implicit function, press 5 (G-Solv) to display the Graph Analysis Menu. • Parabolic Graph Analysis 1 (FOCS) .... Determines the focus. 2 (SYM) .......
Implicit Function Graph Analysis 14 - 3 u To calculate the focus and vertex Example To determine the focus and vertex for the parabola X = (Y – 2)2 + 3. Use the following View Window parameters. Xmin = –1 Ymin = –5 Xmax = 10 Ymax = 5 Xscale = 1 Yscale = 1 5 (G-Solv) 1 2 3 4 5 6 1 (FOCS) (Calculates the focus.) 5 (G-Solv) 1 2 3 4 5 6 4 (VTX) (Calculates the vertex.) • When calculating two foci for an ellipse or hyperbolic graph, press e to calculate the second focus.
14 - 3 Implicit Function Graph Analysis 5 (LEN) (Calculates the latus rectum.) u To calculate the center and radius Example To determine the center and radius for the circle X2 + Y2 – 2X – 2Y – 3 = 0 Use the following View Window parameters. Xmin = –6.3 Ymin = –3.1 Xmax = 6.3 Ymax = 3.1 Xscale = 1 Yscale = 1 5 (G-Solv) 1 2 3 4 5 6 1 (CNTR) (Calculates the center.) 5 (G-Solv) 1 2 3 4 5 6 2 (RADS) (Calculates the radius.
Implicit Function Graph Analysis 14 - 3 u To calculate the x- and y-intercepts Example To determine the x- and y-intercepts for the hyperbola (X – 1)2 (Y – 1)2 –––––––––– – –––––––––– =1 22 22 Use the following View Window parameters. Xmin = –6.3 Ymin = –3.1 Xmax = 6.3 Ymax = 3.1 Xscale = 1 Yscale = 1 5 (G-Solv) 1 2 3 4 5 6 2 (X-IN) (Calculates the x-intercept.) 5 (G-Solv) 1 2 3 4 5 6 3 (Y-IN) (Calculates the y-intercept.) • Press e to calculate the second set of x-/y-intercepts.
14 - 3 Implicit Function Graph Analysis 2 (SYM) (Draws the axis of symmetry.) 5 (G-Solv) 1 2 3 4 5 6 3 (DIR) (Draws the axis of directrix.) u To draw and analyze the asymptotes Example To draw the asymptotes for the hyperbola (X – 1)2 (Y – 1)2 –––––––––– – –––––––––– = 1 22 22 Use the following View Window parameters. Xmin = –6.3 Ymin = –5 Xmax = 6.3 Ymax = 5 Xscale = 1 Yscale = 1 5 (G-Solv) 1 2 3 4 5 6 5 (ASYM) (Draws the asymptotes.
14-4 Implicit Function Graphing Precautions • Assigning the following types of values to variables contained in built-in function produces an error. (1) Parabola graph A=0 (2) Circle graph R = 0 for (X – H)2 + (Y – K)2 = R2 A = 0 for AX2 + AY2 + BX + CY + D = 0 (3) Ellipse/hyperbola graph A = 0 or B = 0 • You cannot overwrite implicit function graphs. • The calculator automatically clears the screen before drawing a new implicit function graph.
