﻿ Parabola Worksheet | Problems & Solutions

# Parabola Worksheet

Parabola Worksheet
• Page 1
1.
For the equation of a parabola $x$2 = 20$y$, 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.]

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

#### 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.

3.
Find the vertex of the parabola $y$ = 6$x$2 - 24$x$ + 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.]

4.
Find the vertex and axis of symmetry of the parabola $y$2 - 4$x$ + 4$y$ + 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.]

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

#### 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

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

#### 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

7.
Given the parabola with focus (0, 3) and directrix $y$ = - 3. Determine its equation and which direction it opens.
 a. $x$2 = 3$y$, downwards b. $x$2 = 12$y$, downwards c. $x$2 = 12$y$, upwards d. $x$2 = 3$y$, 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.

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.

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.

10.
What is the focus and directrix for the parabola $x$ = $y$2 - 14$y$ + 3?
 a. (- $\frac{183}{4}$, 7), 4$x$ = 185 b. (7, $\frac{183}{4}$), 4$x$ = - 185 c. (- $\frac{183}{4}$, - 7), 4$x$ = - 185 d. (- $\frac{183}{4}$, 7), 4$x$ + 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.