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Conics APP for TI-83 Plus & TI-84

Content:  This document covers the use of the Conics APP and provides a review of some of the salient features of conics.
Additions: 5/1/15. 
INDEX:

To facilitate lookup, the instructions are divided into the following categories:

         I. Basic Use of Conics APP - circle, parabola, ellipse, hyperbola
       
II. Exploring the Circle – equations for circle
       III. Exploring the Parabola - equations for different types of parapolas
       IV. Exploring the Ellipse – equations for different orientations of ellipses, equation for finding magnitude of b, eccentricity,
        V. Exploring the Hyperbola - equations for different orientations of hyperbolas, equations for asymptotes,

RELEASE DATE:  4/30/15         DATE LAST REVISED: 5/20/15
© 2015 Frank Kizer  NOTE:  See copy restrictions and printing hints at the end of this document.  
Important Note:  The TI-84 Silver Edition and the TI-84 calculators with newer operating systems have
  different keystrokes for some statistical operations and have additional functions for other operations.
  To familiarize yourself with these changes, click here .
FEEDBACK: If you didn't find what you came here looking for, send me an email telling me what you
  wanted to know.

 General: The Conics APP is a simple program that returns different parameters of a conic section when different parameters of
a praticular conic is entered in the program.

Accessing the APP: To access the App on either the TI-83 Plus or the TI-84, press the APPS button located to the right of the
MATH button. Find the Conics entry and either press the number opposite that entry or move the cursor to highlight the entry
and press ENTER.
NB:
It is important to realize that if you don't know which equation applies to a particular configuration of a conic, you can
easily find out by just opening the relevant conic and a sketch of the graphs will be displayed along with their standard equations.

I. Basic Use of Conics APP:
   1.The Circle:
       Standard Equation:
        a) To open the circle portion of the APP, either press ENTER or enter the number 1.On the screen you will see a selection
            of either the standard form of a circle or the general equation of the circle. Press 1 to explore the standard equation.
        b) Enter the numbers from the standard equation. The standard equation is listed as the top item on the screen.
        c) Enter numbers opposite "H," "K," and "R" as follows: H=1; K=2, R=5.
        d) Press ALPHA, ENTER and value of the radius and the x- and y-values will be displayed.
   To move back to the previous screen, press the Y= button which is escape, ESC, for this application.
      General Equation:
        
First realize that you will need numbers from a general equation. If you enter numbers randomly, you may come up with
          a non-real configuration.
        a) To open the circle portion of the APP, either press ENTER or enter the number 1.On the screen you will see a selection
            of either the standard form of a circle or the general equation of the circle. Press 2 to explore the standard equation.
        b) Enter the numbers from the general equation. The general equation is listed as the top item on the screen. Note that "C"
            and "D" are the coefficients of the x- and y-terms respectively.
        c) Enter numbers opposite the letters as follows: A=1; B=2, D=-6, C=-4.
        d) Press ALPHA, ENTER and value of the radius and the x- and y-coordinates for the center will be displayed.
   To move back to the previous screen, press the Y= button which is escape, ESC, for this application.

2.The Ellipse:
       Standard Equation:
        NOTE:
Note that some texts may refer to the axis along the "a" parameter as the transverse axis and the axis along "b" as the
           conjugate axis.
        a) To open the ellipse portion of the APP, either move the cursor to the ellipse entry and press ENTER or enter the number 2.
            On the screen that appears, you will see a selection of the standard form of a vertical or a horizontal ellipse displayed.
        b) Select the horizontal ellipse, and enter the numbers from the horizontal equation. The standard equation is listed as the
            top item on the screen.
        c) Enter numbers opposite  "A", "B:, "H," and "K," as follows: A=3, B=2, H=-2; K=3.
        d) Press ALPHA, ENTER and values of the coordinates for the center and the two foci will be displayed.
   To move back to the previous screen, press the Y= button which is escape, ESC, for this application.

3.The Hyperbola:
       Standard Equation:
        a) To open the hyperbola portion of the APP, either move the cursor to the hyperbola entry and press ENTER or enter the number 3.
            On the screen that appears, you will see a selection of the standard form of a vertical and a horizontal hyperbola displayed.
        b) Select the horizontal hyperbola, and enter the numbers from the horizontal equation. The standard equation is listed as the
            top item on the screen.
        c) Enter numbers opposite  "A", "B:, "H," and "K, as follows: A=3, B=2, H=-2; K=3.
        d) Press ALPHA, ENTER and value of the coordinates for the center, the two vertices, the two foci and the slopes for the two
            asymptotes will be displayed.
   To move back to the previous screen, press the Y= button which is escape, ESC, for this application.

4.The Parabola:
       Standard Equation:
        a) To open the parabola portion of the APP, either move the cursor to the parabola entry and press ENTER or enter the number 4.
            On the screen that appears, you will see a selection of the standard form of a parabola that opens upward and one that opens
            to the right. Note that to get one that opens downward or to the left, put a negative sign before the value for "P."
        b) Select the parabola that opens upward, and enter the numbers from the equation. The standard equation is listed as the
            top item on the screen.
        c) Enter numbers opposite  "H," and "K," and "P" as follows:  H=3 and K=, P=2.
        d) Press ALPHA, ENTER and value of the coordinates for the vertex, focus , and directrix will be displayed.
   To move back to the previous screen, press the Y= button which is escape, ESC, for this application

II. Exploring the Circle
   a) Standard equation of a circle whose center is (h, k) and radius is r.
       (x-h)2 +(y-k)2 =r2
   b) The general form is as follows: Ax2 + By2 +Cx +Dy + E = 0.
   c) To convert a formula of this configuration to a radius-center type, complete the square as follows:
        x2 +y2 -4x -6y+8 = 0
        x2 -4x +    y2 -6y    = -8  (Rearrange to complete the squares.)
        (x2 -4x + 4)+   y2 -6y +9)   = -8+4+9  (Take 1/2 the coefficient of the x-term, square it and add to both sides of the equation.
        (x-2)2 +   (y -3)2 = -8+4+9
        (x-2)2 +   (y -3)2 =5

III. Exploring the Parabola
a) Standard equations of a parabola whose center is (h, k).
       Opens Upward: (x-h)2 =4p(y-k)
       Opens Downward: (x-h)2 =-4p(y-k)
       Opens Right: (y-k)2  =4p(x-h)
       Opens Left: (y-k)2  =-4p(x-h)

         Finding the distance from vertex to focus or directrix:
          With the equation y=12x; observe from the equation y=4px that 4p=12, therefore p=3.
   b) Example: Suppose you have a parabola with center (4, -1) that passes through the point (0, 1). Find the focus.

 
     (x-h)2 =4p(y-k)
       (0-4)2 =4p(1+1)
           16    =8p
             p=2
  c) Find the zeros of the parabala.
         (x-4)2 =4p(0+1)  (Let y = 0)
          x2 -8x +16 =8  (Replace "P" with its value 2.)
          x2 -8x +8 =0
          x=4
±2√6  (Using quadratic formula.)

IV. Exploring the Ellipse
a) Standard equation of a ellipse whose center is (h, k).
       Horizontal Ellipse: (x-h)2 /a2 +(y-k)2 /b2 =1
       Equation for eccentricity: e = c/a (Where c is the distance to the focus.)
       Equation for "b.": b2 = a2 -c2
       Vertical Ellipse: (y-k)2 /a2 + (x-h)2 /b2 =1

         Equation for distance from vertex to focus: c2 = a2 + b2 , therefore, c = √(a2 +b2)

   b) The general for is as follows: Ax2 + By2 +Cx +Dy + E = 0.
   c) To convert a formula of this configuration to a radius-center type, complete the square as follows:
       
4x2 – 5y2 + 40x – 30y – 45 = 0
       
4(x + 5)2 – 5(y + 3)2 = 45 + 100 – 45  (Using the techniques as detailed in the circle.)
       
4(x + 5)2 – 5(y + 3)2 = 100
        
(x + 5)2/25 – 5(y + 3)2 /20= 1(Divide both sides by 100.)
       
       

V. Exploring the Hyperbola
a) Standard equation of a hyperbola whose center is (h, k).
       Horizontal Hyperbola: (x-h)2 /a2 -(y-k)2 /b2 =1
       Equation of Asymptotes: y =
± b/a(x-h) +k
       Vertical Hyperbola: (y-k)2 /a2 -(x-h)2 /b2 =1
       Equation of Asymptotes: y =
± a/b(x-h) +k
         Equation for distance from vertex to focus: c2 = a2 + b2 , therefore, c = √(a2 +b2)

   b) The general for is as follows: Ax2 + By2 +Cx +Dy + E = 0.
   c) To convert a formula of this configuration to a radius-center type, complete the square as follows:
       
4x2 – 5y2 + 40x – 30y – 45 = 0
       
4(x + 5)2 – 5(y + 3)2 = 45 + 100 – 45  (Using the techniques as detailed in the circle.)
       
4(x + 5)2 – 5(y + 3)2 = 100
        
(x + 5)2/25 – 5(y + 3)2 /20= 1(Divide both sides by 100.)
       
     

Copy Restrictions:  You may make single copies of this document for your own personal use and for the use of other students, but inclusion in another document, publication or any use for profit requires my permission.  Teachers may make multiple copies of this document for their students if they first get permission from me.  Merely send me an email (Just click on Webmaster in the navigation bar.) with a one-sentence explanation of what you’re using the document for.  I’ll give you permission in a timely manner. 

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