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Parabola Worksheet

Parabola Worksheet
  • Page 1
 1.  
For the equation of a parabola x2 = 20y, find the axis of symmetry and the equation of directrix.
a.
y - axis, x = - 5
b.
x = 0, y = 5
c.
x - axis, y = 5
d.
y - axis, y = - 5


Solution:

The axis of symmetry of parabola x2 = 20y is y - axis.

4c = 20
[Compare with x2 = 4cy.]

c = 5

Directrix is y = - 5
[Directrix is y + c = 0.]


Correct answer : (4)
 2.  
Find the equation of the parabola with focus at (- 7, 0), x - axis as the axis of symmetry and vertex at origin.
a.
y2 = - 7x
b.
y2 = - 28x
c.
y2 = 28x
d.
x2 = 28y


Solution:

The equation of a parabola with vertex at the origin, focus (a, 0), and x - axis as the axis of symmetry is y2 = 4ax.

(a, 0) = (- 7, 0), So a = - 7

y2 = 4(- 7)x
[Replace a with - 7.]

y2 = - 28x is the equation of the parabola.


Correct answer : (2)
 3.  
Find the vertex of the parabola y = 6x2 - 24x + 27.
a.
(3, 2)
b.
(5, 4)
c.
(2, 3)
d.
(2, 4)


Solution:

y = 6(x2 - 4x) + 27
[Factor 6x2 - 24x.]

y = 6(x2 - 4x + 4) + 27 - 24
[Complete the square.]

y = 6(x - 2)2 + 3

So, the vertex is (h, k) = (2, 3).
[Compare with y = a(x - h)2 + k.]


Correct answer : (3)
 4.  
Find the vertex and axis of symmetry of the parabola y2 - 4x + 4y + 20 = 0.
a.
(- 4, - 2), x = - 2
b.
(4, - 2), y = - 2
c.
(- 4, 4), x = - 4
d.
(4, 2), y = - 2


Solution:

y2 - 4x + 4y + 20 = 0

4x = y2 + 4y + 20
[Solve for x in terms of y.]

x = 1 / 4y2 + y + 5

x = 1 / 4(y2 + 4y + 4) + 5 - 1
[Complete the square.]

x = 1 / 4(y + 2)2 + 4

h = 4, k = - 2
[Compare with x = a(y - k)2 + h.]

So, the vertex is (4, - 2) and the axis of symmetry is y = - 2.
[vertex: (h, k), axis of symmetry: y = k.]


Correct answer : (2)
 5.  
For the equation y = - 8x 2 + 112x - 392, find the focus and directrix.
a.
Focus: (7, 1 32), directrix: x = - 1 32
b.
Focus: (7, - 1 32), directrix: y = 1 32
c.
Focus (7, 1 32), directrix y = - 1 32
d.
Focus: (1 32, 7), directrix: x = 1 32


Solution:

y = - 8x 2 + 112x - 392

y = - 8(x 2 - 14x + 49)
[Factor.]

y = - 8(x - 7) 2 + 0
[Complete the square.]

h = 7, k = 0 and a = - 8
[Compare with y = a(x - h)2 + k.]

Focus is (h, (k + 14a)) = (7, 0 + 14(-8)) = (7, - 1 / 32)

Directrix is y = k - 14a = 0 - 1 / 4×-8 = 1 / 32, So, y = 1 / 32


Correct answer : (2)
 6.  
What is the standard form of the equation of a parabola with focus (9, 0), directrix x = - 9?
a.
y2 = 4x
b.
y2 = - 36x
c.
y2 = 36x
d.
y = 36x2


Solution:

The equation of a parabola with focus (a, 0) and directrix x = - a is y2 = 4ax.

y2 = 4(9) x
[Replace a with 9.]

y2 = 36x


Correct answer : (3)
 7.  
Given the parabola with focus (0, 3) and directrix y = - 3. Determine its equation and which direction it opens.
a.
x2 = 3y, downwards
b.
x2 = 12y, downwards
c.
x2 = 12y, upwards
d.
x2 = 3y, upwards


Solution:

A parabola with focus (0, c) and directrix y = - c, equation is given by x2 = 4cy.

Since c = 3 > 0, it open upwards.

x2 = 4(3)y = 12y
[Replace c with 3.]

So, the equation of parabola is x2 = 12y and it open upwards.


Correct answer : (3)
 8.  
If Bill threw a ball straight up, then what shape would the graph of height of the ball verses time be?
a.
ellipse
b.
hyperbola
c.
circular
d.
parabola


Solution:

The path of the ball is a parabola.


Correct answer : (4)
 9.  
Name the chord drawn through the focus and perpendicular to the axis of the parabola.
a.
focal Chord
b.
latus rectum
c.
focal beam
d.
asymptote


Solution:

The chord drawn through the focus and perpendicular to the axis of the parabola is a latus rectum.


Correct answer : (2)
 10.  
What is the focus and directrix for the parabola x = y2 - 14y + 3?
a.
(- 183 4, 7), 4x = 185
b.
(7, 183 4), 4x = - 185
c.
(- 183 4, - 7), 4x = - 185
d.
(- 183 4, 7), 4x + 185 = 0


Solution:

x = y2 - 14y + 3

x = (y2 - 14y + 49) + 3 - 49
[Complete the square.]

x = (y - 7)2 - 46

(h, k) = (- 46, 7), a = 1
[Compare with x = a(y - k)2 + h.)

h + 14a = - 46 + 14(1) = - 183 / 4
So, focus is (- 183 / 4, 7).
[Focus: (h + 14a, k).]

x = h - 14a = - 46 - 1 / 4 = - 185 / 4
[Directrix: x = h - 14a.]

x = - 185 / 4, so 4x + 185 = 0

Therefore, focus is (- 183 / 4, 7) and directrix is 4x + 185 = 0.


Correct answer : (4)

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