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Title: Equations and Graphs - Differential Calculus
Description: In this study note about the equations and graphs, you will learn the parts of the conic sections such as parabolas, ellipses, and hyperbolas. Given this sections, you will also learn their own equations (standard and general) given such properties and variables. You will also learn on how to identify their graph on the rectangular coordinate system. In addition, you will improve your learnings because there have example problems in this study note and there is also a practice test on the last part so that you'll try to challenge your knowledge and increase your learning! Happy studying!
Description: In this study note about the equations and graphs, you will learn the parts of the conic sections such as parabolas, ellipses, and hyperbolas. Given this sections, you will also learn their own equations (standard and general) given such properties and variables. You will also learn on how to identify their graph on the rectangular coordinate system. In addition, you will improve your learnings because there have example problems in this study note and there is also a practice test on the last part so that you'll try to challenge your knowledge and increase your learning! Happy studying!
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DIFFERENTIAL CALCULUS
Study Notes 5
“EQUATIONS AND GRAPHS”
Prepared By: Tutor Win
EQUATIONS AND GRAPHS
07/31/2021
In the last StudyNote 4, we have learned the underlying concepts
behind circles
...
Part
of that lesson is to convert the standard equation of the circle to its general
form, and vice versa
...
Let’s get into it!
LINES
We have learned that the equation of a line is in the form y = mx +b
where m is the slope, and b is the y-intercept
...
PARABOLAS
Parabolas are the representations of quadratic functions
...
Standard Equation:
Where
(x-h)2 = 4c(y-k)
(h, k) is the vertex of the parabola,
(h, k + c) is the focus of the parabola
y = k – c is the directrix of the parabola
1|Page
EQUATIONS AND GRAPHS
07/31/2021
x = h is the axis of symmetry of the parabola
If c > 0, the parabola opens upward
...
GRAPH OF THE PARABOLA
S
Axis of Symmetry
x= h
Point A
distance d
distance d
focus ( h, k+c)
k
directrix
Vertex (h, k)
y = k—c
h
Note that the distance between the directrix to Point 1 on the parabola
is equal to the distance between this Point 1 on the parabola to the focus
...
Find the vertex, focus, directrix and axis of symmetry of the
parabola whose equation is x2 = 24y
...
Step 1
...
(x)2 = 24(y)
Separate the x and y variable
Step 3
...
(x – 0)2 = (4)( )(y-0)
Add a factor of 4 and divide 24 by another 4
Step 5
...
Identify the value of the variables
...
The focus is (h, k + c), and it is (0, 0 + 6), then (0, 6)
...
The axis of symmetry is the line x = h, so it is x = 0 or the y-axis
...
Find the vertex, focus, directrix and axis of symmetry of the
parabola whose equation is y = x2+4x – 68
...
Step 1
...
Put the y variable alone on the right
...
We need to complete the square of x2 + 4x, so get rid of
-68 by adding 68 on both sides
...
x2+4x = y + 68
Step 5
...
So,
x2+4x + 4= y + 68 + 4
Step 6
...
(x + 2)2 = y + 72
Step 8
...
But what about 4c? (y+72) actually has 1 right
before it as a factor
...
To find 4c, divide 1 by 4
and that’s ¼
...
(x + 2)2 = 1(y + 72)
(x + 2)2 = 4(1/4)(y + 72)
(x – h)2 = 4c(y-k)
Step 9
...
4|Page
EQUATIONS AND GRAPHS
Answer:
07/31/2021
The vertex is (h,k) and it is the origin (-2,-72)
...
75)
...
25
...
EXAMPLE 3
...
Note that the general equation is
the value of y in terms of x
...
(2x+1)2 = 2(y – 2)
Step 2
...
4x2 + 4x + 1 = 2y – 4
Step 4
...
4x2 + 4x + 5= 2y
Step 6
...
2x2 + 2x + = y
Answer:
y = 2x2 + 2x +
𝟓
𝟐
5|Page
EQUATIONS AND GRAPHS
07/31/2021
ELLIPSES
Ellipse might be similar to a circle but there are also differences
between the two
...
Standard Equation:
Where
(
a
...
)
+
(
)
=1
(h, k) is the center of the ellipse, and a > b
2a is the length of the major axis
2b is the length of the minor axis
(h, k ± 𝑎) are the coordinates of the vertices
(h± 𝑏, k) are the coordinates of the co-vertices
(h, k ± 𝑐) are the coordinates of the foci (plural of focus)
Where c2 = a2 – b2
GRAPH
y
Vertex (h, k + a)
Focus
(h, k + c)
Major axis
k
Co-Vertex
Co-Vertex
(h - b, k)
(h + b, k)
(h, k)
Minor axis
(h, k - c)
Focus
x
Vertex (h, k - a)
h
7|Page
EQUATIONS AND GRAPHS
07/31/2021
EXAMPLE 4
...
Step 2
...
(
)
=1
=1
Note that the value of a here is greater than b
...
The length of the major axis is 2a, so 2(4) = 8 units
...
The coordinates of the vertices are (h+ 𝑎, k) which
is (0+4, 0) (4,0), and (h− 𝑎, k) which is (0-4, 0)
(-4, 0)
...
The coordinates of the foci are (h + 𝑐, k) which is
(0+√7, 0) (√𝟕, 0), and (h − 𝑐, k) which is
(0-√7, 0) (-√𝟕, 0)
...
Its equation may have similarities to the
equation of the ellipse except for the sign
...
Where
(
)
-
(
)
=1
(h, k) is the center of the hyperbola, and a > b
( ℎ ± 𝑎, k) are the coordinates of the vertices
(h, 𝑘 ± 𝑏) are the coordinates of the co-vertices
2a is the length of the transverse axis
2b is the length of the conjugate axis
(ℎ ± 𝑐, k) are the coordinates of the foci (plural of focus)
y - k = ± (𝑥 − ℎ) are the equations of the asymptotes
Where c2 = a2 + b2
GRAPH
Co-vertex
Transverse Axis
Conjugate axis
Vertex
Vertex
Focus
Focus
Center
asymptote
Co-vertex
asymptote
9|Page
EQUATIONS AND GRAPHS
07/31/2021
Standard Equation: (Transverse Axis is vertical)
b
...
Find the center, vertices, co-vertices, the length of transverse
axis and conjugate axis, foci, and asymptotes of the hyperbola whose
equation is
(
)
-
(
Step 1
...
Step 3
...
(The transverse axis is horizontal)
(
)
(
)
-
(
)
(
)
=1
to
(
)
-
(
)
=1
=1
Determine the values of the variables
...
A
...
1
...
y = x2 + 2x + 1
B
...
3
...
(
)
(
)
+
+
(
)
(
)
=1
=1
C
...
5
...
(
)
(
)
−
−
(
(
)
)
=1
(The transverse axis is horizontal)
=1
(The transverse axis is vertical)
12 | P a g e
EQUATIONS AND GRAPHS
07/31/2021
PRACTICE TEST 5
...
1
...
(x-5)2 = 4(y+1)
as
(x-h)2 = 4c(y-k)
Step 2
...
(x – h)2 = 4c(y-k)
Answer:
The vertex is (h,k) and it is (5,-1)
...
The directrix is the line y = k – c, so it is y = -1 – 1, then y = -2
...
2
...
y = x2 + 2x + 1
Step 2
...
(x + 1)2 = y
Step 4
...
The focus is (h, k + c), and it is (-1, 0 + ¼ ), then (-1, 0
...
25
...
B
...
Step 1
...
(
)
(
)
+
+
(
)
(
)
= 1 as
(
)
+
(
)
=1
=1
Note that the value of a here is greater than b
...
- The length of the major axis is 2a, so 2(10) = 20
units
...
- The coordinates of the vertices are (h+a, k)
which is (-5+10, -7) (5,-7), and (h-a, k) which
is (-5-10, -7) (-15, -7)
...
- The coordinates of the foci are (h+c, k), which is
(-5+4√𝟔, -7) and (h-c, k) which is (-5-4√𝟔, -7)
...
Step 1
...
(
)
(
)
+
+
(
)
(
)
= 1 as
(
)
+
(
)
=1
=1
Note that the value of a here is greater than b
...
-
The length of the major axis is 2a, so 2(7) = 14 units
...
- The coordinates of the vertices are (h, k+a)
which is (2, -3+7) (2,4), and (h, k – a) which
is (2, -3–7) (2, -10)
...
14 | P a g e
EQUATIONS AND GRAPHS
07/31/2021
- The coordinates of the foci are (h, k+c), which is
(2, -3+√𝟑𝟑) and (h, k – c) which is (2, -3–√𝟑𝟑)
C
...
Step 1
...
(
)
h=4
)
-
= 1 to
(
)
k = -6
c2 = a 2 + b2
=
(
)
-
(
)
=1
=1
a=5
b=3
25 + 9
=
c2 = 34
c = √𝟑𝟒
Answer:
The center is (h, k), and it is (4, -6)
The vertices are ( ℎ ± 𝑎, k), so these are (4 + 5, -6)
(9, -6) and (4-5, -6) (-1, -6)
The co-vertices are (h, 𝑘 ± 𝑏), so these are (4, -6+3)
(4, -3), and (4, -6-3) (4, -9)
The length of transverse axis is 2a, so it is 2(5) = 10 units
The length of the conjugate axis is 2b, so it is 2(3) = 6 units
The foci are (ℎ ± 𝑐, k), so these are (4+√𝟑𝟒, -6), and
(4-√𝟑𝟒, -6)
...
Step 1
...
(
07/31/2021
)
−
(
)
=1
h = -1
k=8
c2 = a 2 + b2
=
a=9
81 + 121
b = 11
=
c2 = 202
c = √𝟐𝟎𝟐
Answer:
The center is (h, k), and it is (-1, 8)
The vertices are (h, 𝑘 ± 𝑎), so these are (-1, 8+9)
(-1, 17) and (-1, 8-9) (-1, -1)
The co-vertices are (ℎ ± 𝑏, k), so these are (-1+11, 8)
(10, 8), and (-1-11, 8) (-12, 8)
The length of transverse axis is 2a, so it is 2(9) = 18 units
The length of the conjugate axis is 2b, so it is 2(11) = 22 units
The foci are (h, 𝑘 ± 𝑐), so these are (-1, 8+√𝟐𝟎𝟐), and
(-1, 8-√𝟐𝟎𝟐)
Title: Equations and Graphs - Differential Calculus
Description: In this study note about the equations and graphs, you will learn the parts of the conic sections such as parabolas, ellipses, and hyperbolas. Given this sections, you will also learn their own equations (standard and general) given such properties and variables. You will also learn on how to identify their graph on the rectangular coordinate system. In addition, you will improve your learnings because there have example problems in this study note and there is also a practice test on the last part so that you'll try to challenge your knowledge and increase your learning! Happy studying!
Description: In this study note about the equations and graphs, you will learn the parts of the conic sections such as parabolas, ellipses, and hyperbolas. Given this sections, you will also learn their own equations (standard and general) given such properties and variables. You will also learn on how to identify their graph on the rectangular coordinate system. In addition, you will improve your learnings because there have example problems in this study note and there is also a practice test on the last part so that you'll try to challenge your knowledge and increase your learning! Happy studying!