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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
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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
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Find the vertex, focus, directrix and axis of symmetry of the
parabola whose equation is x2 = 24y
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Step 1
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(x)2 = 24(y)

Separate the x and y variable

Step 3
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(x – 0)2 = (4)( )(y-0)

Add a factor of 4 and divide 24 by another 4

Step 5
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Identify the value of the variables
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The focus is (h, k + c), and it is (0, 0 + 6), then (0, 6)
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The axis of symmetry is the line x = h, so it is x = 0 or the y-axis
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Find the vertex, focus, directrix and axis of symmetry of the
parabola whose equation is y = x2+4x – 68
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Step 1
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Put the y variable alone on the right
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We need to complete the square of x2 + 4x, so get rid of
-68 by adding 68 on both sides
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x2+4x = y + 68
Step 5
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So,
x2+4x + 4= y + 68 + 4

Step 6
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(x + 2)2 = y + 72
Step 8
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But what about 4c? (y+72) actually has 1 right
before it as a factor
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To find 4c, divide 1 by 4
and that’s ¼
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(x + 2)2 = 1(y + 72)
(x + 2)2 = 4(1/4)(y + 72)
(x – h)2 = 4c(y-k)

Step 9
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4|Page

EQUATIONS AND GRAPHS

Answer:

07/31/2021

The vertex is (h,k) and it is the origin (-2,-72)
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75)
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25
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EXAMPLE 3
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Note that the general equation is
the value of y in terms of x
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(2x+1)2 = 2(y – 2)

Step 2
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4x2 + 4x + 1 = 2y – 4

Step 4
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4x2 + 4x + 5= 2y

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