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 .
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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} =r^{2}
b) The general form is as follows: Ax^{2} + By^{2}
+Cx +Dy + E = 0.
c) To convert a formula of this configuration to a radius-center
type, complete the square as follows:
x^{2} +y^{2} -4x
-6y+8 = 0
x^{2} -4x +
y^{2} -6y = -8 (Rearrange to complete the
squares.)
(x^{2} -4x + 4)+
y^{2} -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)
x^{2} -8x +16 =8
(Replace "P" with its value 2.)
x^{2} -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} /a^{2}
+(y-k)^{2} /b^{2} =1
Equation for eccentricity: e = c/a (Where c
is the distance to the focus.)
Equation for "b.": b^{2} = a^{2}
-c^{2 }
Vertical Ellipse: (y-k)^{2} /a^{2}
+ (x-h)^{2} /b^{2} =1
Equation for distance from
vertex to focus: c^{2 }= a^{2} + b^{2}_{ },
therefore, c = √(a^{2} +b^{2})
b) The general for is as follows: Ax^{2} + By^{2} +Cx
+Dy + E = 0.
c) To convert a formula of this configuration to a radius-center
type, complete the square as follows:
4x^{2} –
5y^{2} +
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} /a^{2}
-(y-k)^{2} /b^{2} =1
Equation of Asymptotes: y =
± b/a(x-h) +k
Vertical Hyperbola: (y-k)^{2} /a^{2}
-(x-h)^{2} /b^{2} =1
Equation of Asymptotes: y =
± a/b(x-h) +k
Equation for distance from
vertex to focus: c^{2 }= a^{2} + b^{2}_{ },
therefore, c = √(a^{2} +b^{2})
b) The general for is as follows: Ax^{2} + By^{2} +Cx
+Dy + E = 0.
c) To convert a formula of this configuration to a radius-center
type, complete the square as follows:
4x^{2} –
5y^{2} +
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.)
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