Download + (+1)

Standard 4, 9, 17
PARABOLAS
DEFINITION OF A PARABOLA
PARTS OF A PARABOLA
SUMMARY OF FORMULAS
PROBLEM 1
PROBLEM 2
PROBLEM 3
PROBLEM 4
PROBLEM 5
PROBLEM 6
PROBLEM 7
PROBLEM 8
END SHOW1
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
ALGEBRA II STANDARDS THIS LESSON AIMS:
STANDARD 4:
Students factor polynomials representing the difference of squares, perfect square trinomials,
and the sum and difference of two cubes
ESTÁNDAR 4:
Los estudiantes factorizan polinomios representando diferencia de cuadrados, trinomios
cuadrados perfectos, y la suma de diferencia de cubos.
STANDARD 9:
Students demonstrate and explain the effect that changing a coefficient has on the graph of
quadratic functions; that is, students can determine how the graph of a parabola changes as a,
2
b, and c vary in the equation y = a(x-b) + c.
ESTÁNDAR 9:
Los estudiantes demuestran y explican los efectos que tiene el cambiar coeficientes
en la
2
gráfica de funciones cuadráticas; esto es, los estudiantes determinan como la gráfica de una
parabola cambia con a, b, y c variando en la ecuación y=a(x-b)2 + c
STANDARD 17:
2
Given a quadratic equation of the form ax 2 + by + cx + dy + e = 0, students can use the method
for completing the square to put the equation into standard form and can recognize whether
the graph of the equation is a circle, ellipse, parabola, or hyperbola. Students can then graph
the equation.
Estándar 17:
2
Dada una equación cuadrática de la forma ax +by 2 + cx + dy + e=0, los estudiantes pueden
usar el método de completar al cuadrado para poner la ecuación en forma estándar y pueden
reconocer si la gráfica es un círculo, elipse, parábola o hiperbola. Los estudiantes pueden
2
graficar la ecuación
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Standard 4, 9, 17
DEFINITION OF A PARABOLA:
A parabola is the set of all points in a plane that are the same distance from
a given point called the focus and a given line called directrix.
y
x
focus
directrix
3
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Standard 4, 9, 17
PARTS OF A PARABOLA:
y
axis of symmetry
latus rectum
x
focus
vertex
directrix
4
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Standard 4, 9, 17
PARABOLAS: SUMMARY
2
Form of equation
y = a(x-h) + k
x = a(y-k)2 + h
Axis of symmetry
x= h
y=k
Vertex
(h,k)
(h,k)
Focus
1
(h, k + 4a
)
(h + 4a , k)
Directrix
Direction of
openning
Length of latus
rectum
y=k-
1
4a
Upward if a>0
Downward if a<0
1
a
1
units
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
1
x = h - 4a
Right if a>0
Left if a<0
1
a
units
5
Standard 4, 9, 17
Find the vertex, axis of symmetry, focus, directrix and Latus
2
rectum from y= 1 (x+5) + 2 and graph it.
6
1
1
Focus: (h, k + 4a ) =( -5, 2 +
1 )
4
2
1
6
y= (x+5) + 2
6
1
=( -5, 2 +
)
4
Rewriting the equation:
1
1
6
2
1
1
1
3
1
y= (x- -5) + (+2)
3
=
=4 = 2
6
=(
-5,
2
+
)
2
4
2
3
1
6
6
2
=( -5, 3
)
y = a(x-h) + k
2
2 2+ 3 = 4 + 3
1
2
2 2
h= -5
2
Directrix: y = k 4a
7
=
k= 2
1
2
y
=
2
3
1
1 )
a=
4(
1
2
7
6
3
6
6
2
Vertex: (h,k) = (-5, 2)
y=2- 3
1
2
Axis of symmetry: x= -5
1
y=
2 2- 3 = 4 - 3
2
1
2
2
2
2
1
1
1
=
1
=6
2
1 = 1
Latus rectum:
a
6
6
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Standard 4, 9, 17
Summarizing the information about the parabola and graphing it:
x= -5
y
10
Vertex: (-5, 2)
8
6
Axis of symmetry: x= -5
4
1
Focus: ( -5, 3
)
2
1
Directrix: y =
2
y=
2
-10 -8
-6 -4
-2
2
-2
4
6
8
1
2
10
x
-4
Latus rectum = 6
-6
a>0 so it opens upward.
-8
7
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Standard 4, 9, 17
Find the vertex, axis of symmetry, focus, directrix and Latus
rectum from y= - 1 (x+6)2 + 3 and graph it.
8
1
1
)
Focus: (h, k + 4a ) =( -6, 3 +
1
4
2
1
8
y= - (x+6) + 3
8
1
=( -6, 3 +
)
1
1
4
Rewriting the equation:
1
1
1
+2
8
=1
2
=
=
1
1
y= - (x- -6) + (+3)
4
8
- 4
=( -6, 3 + -2 )
2
8
8
= -2
2
=( -6, 3 - 2)
y = a(x-h) + k
h= -6
k= 3
Directrix: y = k -
1
4a
a= - 1
8
Vertex: (h,k) = (-6, 3)
1
4( - 1 )
8
y = 3 - (-2)
Axis of symmetry: x= -6
y=3+2
=( -6, 1 )
y=3 -
y=5
1
Latus rectum:
a
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
1
-1 =
8
1
1
-1
8
=8
Standard 4, 9, 17
Summarizing the information about the parabola and graphing it:
y
Vertex: (-6, 3)
x=- 6
20
16
Axis of symmetry: x= -6
12
8
Focus: ( -6, 1)
y= 5
4
Directrix: y = 5
-20 -16 -12 -8
-4
4
-4
Latus rectum =8
8
12 16
20
x
-8
a<0 so it opens downward
-12
-16
9
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Standard 4, 9, 17
2
Write 4x= y 2 -4y + 8 in the form x = a(y-k) + h and graph it.
Changing the form of the equation:
Vertex: (h,k) = (1, 2)
2
4x = y -4y + 8
2
4
4
4x = (y 2 -4y + 2 )+ 8 - 2
2
Axis of symmetry: y= 2
1
Focus: (h +
, k) =(1 +
4a
2
2
4x = (y - 4y + (2) )+ 8 - (2)
2
2
4x =(y -4y + 4 )+ 8 - (4)
2
4x = (y -2) + 4
4
4
4
2
1
x= (y-2) +
4
4
2
1
x= (y-2) + 1
4
Rewriting the equation:
2
1
x= (y- +2) + (+1)
4
2
x = a(y-k) + h
Directrix: x = h - 1
4a
1
x=14 1
4
1
x=11
h= 1
x= 0
k= 2
1
1
a=
Latus
rectum:
4
a
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
1 , 2)
4 1
4
= (1 + 1 , 2)
4
4
= ( 1 + 1 , 2) =(2,2)
1
1
1
1
1 = 1
4
4
=4
Standard 4, 9, 17
Summarizing the information about the parabola and graphing it:
directrix
y
Vertex: (1,2)
latus rectum
5
4
Axis of symmetry: y= 2
3
Focus: (2,2)
axis of symmetry
focus
2
vertex
1
Directrix: x =0
-5
-4
-3
-2
1
-1
-1
Latus rectum = 4
2
3
4
5
x
-2
a>0 so it opens rightward.
-3
-4
11
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Standard 4, 9, 17
2
Write 4x= y 2 -6y + 3 in the form x = a(y-k) + h and graph it.
Vertex: (h,k) = ( - 3 , 3)
2
2
4x = y -6y + 3
Axis of symmetry: y= 3
2
1
6
6 2
Focus: (h +
, k) =(- 3 + 1 , 3)
4a
2
4 1
4x = (y 2 -6y + 2 )+ 3 - 2
4
2
2
3
=
(
+ 1 , 3)
4x = (y 2 - 6y + (3) )+ 3 - (3)
3
1 2
3 2
2
4
+
=
+
2
2
1
2
2
2
(9)
9
4x =(y -6y +
)+ 3 4
=- 1
3 + 1 , 3)
2
=(
2
4x = (y -3) - 6
2 1
4
4
1
=( - ,3)
1
2
Directrix: x = h 6
2
1
4a
x= (y-3) 4
4
3
1
x
=
2
1
2 4 1
x= (y-3) - 3
4
2
4
3 1 2
x
=
Rewriting the equation:
3
2
1 2
h=
1
2
1
3 2
2
5
x=
(y- +3) + ( - 3 )
x= - - = 1
4
2 2
2
k= 3
2
1
=4
1
1
1
1
=
2
a=
Latus rectum:
x = a(y-k) + h
4
4
a
4
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Changing the form of the equation:
Standard 4, 9, 17
Summarizing the information about the parabola and graphing it:
x= - 5
2
y
Vertex: (h,k) = ( - 3 , 3)
2
Axis of symmetry: y= 3
5
4
y= 3
3
Focus: ( - 1 ,3)
2
2
1
Directrix: x = - 5
2
Latus rectum = 4
-5
-4
-3
-2
1
-1
-1
2
3
4
5
x
-2
a>0 so it opens rightward.
-3
-4
1
2
13
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Standard 4, 9, 17
2
Write y= 4x2+ 24x + 16 in the form y = a(x-h) + k and graph it.
Changing the form of the equation:
Vertex: (h,k) = (-3,-20)
2
y = 4x + 24x + 16
Axis of symmetry: x= -3
1
1
)
Focus: (h, k + 4a ) =(-3, -20 +
4(
)
4
2
2
2
(3)
(3)
y = 4(x + 6x +
)+ 16 -4
1
=(-3, -20 + 16 )
2
y = 4(x + 6x + 9 )+ 16 -4 (9)
15
-19
=(-3,
2
16 )
y = 4(x + 3) + 16 - 36
1
Directrix: y = k 2
4a
y = 4(x + 3) - 20
y = -20- 1
4( 4 )
Rewriting the equation:
1
2
y
=
-20
y = 4(x - -3) + (-20)
16
1
2
y
=
-20
y = a(x-h) + k
16
h= -3
a>0 so it opens upward.
k= -20
14
1
1
Latus
rectum:
a= 4
a
4 =.25
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
2
6
6
2
y = 4(x + 6x + 2 )+ 16 -4 2
2
Standard 4, 9, 17
Summarizing the information about the parabola and graphing it:
y
Vertex: (-3,-20)
axis of symmetry
Axis of symmetry: x= -3
-5
-4
-3
-2
1
-1
2
3
4
x
5
-3
15
Focus: (-3, -19 )
16
1
Directrix: y = -20
16
-6
-9
Latus rectum = .25
-12
a>0 so it opens upward.
-15
latus rectum
focus
-18
vertex-21
-20
directrix
-24
-27
15
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Standard 4, 9, 17
The coordinates of the focus and equation of the directrix of a parabola
are as follows. Write an equation for the parabola and graph it.
(-3, -2); y = -6
Since the directrix is a horizontal line the parabola
is a vertical parabola. And the following applies:
1
If focus formula is (h, k +
4a )
then h = -3 and
k+
1
4a = -2 Equation 1
Solving both equations by substitution:
-6 +
1
4a
+
1
= -2
4a
2
= -2
4a
+6
+6
2
(4a) = 4 (4a)
4a
-6 +
2 = 16a
1
If directrix formula is y = k 4a
then
1
k4a = -6
1
1
+
+ 4a 4a
k=-6+
2 = 16a
16 16
a= 2
16
a= 1
8
.
. 2
.
.2
1
Equation 2
4a
Substituting a in equation 2:
1
1
1
1
k = -6 +
=
-6
+
=
-6
+
4
4
4 1
8
8
8
8
k = -6 +
= -6 + 2
4
16
k = -4
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Standard 4, 9, 17
Summarizing information about the parabola:
h= -3
k= -4
1
a=
8
Vertex: (-3,-4)
So the equation is:
2
1
y= (x- -3) + (-4)
8
2
y = 1 (x+3) - 4
8
Axis of symmetry is x = -3
Focus: (-3, -2)
Directrix: y = -6
1
Latus rectum:
a
1
1
1
1 = 1
8
8
y
10
x=- 3
8
=8
6
4
2
-10 -8
-6 -4
-2
2
4
6
8
10
-2
-4
-6
x
y= -6
-8
17
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Standard 4, 9, 17
Find the equation of the parabola shown in the graph.
y
5
We know that the graph pass
through point (x,y)=(-2,3). We use
this point to substitute it on the
equation to find a:
4
3
2
2
1
-5
-4
-3
-2
x=a(y-1) -3
1
-1
-1
-2
-3
2
3
4
5
x
2
( -2 )= a (( 3 )-1) -3
-2 = a(3-1)2 - 3
2
-2 = a(2) - 3
-4
From the graph:
Vertex = (-3,1)
This is a horizontal parabola
x = a(y-k)2 + h
so:
2
x= a (y- +1 ) + ( -3 )
-2 = a(4) - 3
-2 = 4a - 3
+3
+3
1 = 4a
4 4
Now we substitute in the
parabola’s equation:
1
a=
4
2
x=a(y-1) -3
x=
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
2
1
(y-1) - 3
4
18
Standard 4, 9, 17
Find the equation of the parabola shown in the graph.
y
5
We know that the graph pass
through point (x,y)=(0,-3). We use
this point to substitute it on the
equation to find a:
4
3
2
2
1
-5
-4
-3
-2
y=a(x-2) -4
1
-1
-1
-2
-3
2
3
4
5
x
2
( -3 )= a (( 0 )-2) -4
-3 = a(0-2)2 - 4
2
-3 = a(-2) - 4
-4
From the graph:
Vertex = (2,-4)
This is a vertical parabola
2
so:
y = a(x-h) + k
2
y= a (x- +2 ) + ( -4 )
-3 = a(4) - 4
-3 = 4a - 4
+4
+4
1 = 4a
4 4
1
a=
4
Now we substitute in the
parabola’s equation:
2
1
y= (x-2) - 4
4
2
y=a(x-2) -4
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
19