1.) y² = -36x
y² = -4ax
(y - 0)² = -(4)(9)(x - 0)
y² = -36x
4a = -36
a = -9
h = 0 ; k = 0 ; a = -9
Vertex (h , k)
Vertex (0 , 0)
Focus (a, 0)
Focus (-9, 0)
Directrix: x = -a
Directrix: x = -(-9)
Directrix: x = 9
Axis of symmetry: y = 0
2.) 5x² = 100y
x² = 4ay
(x - 0)² = 4(5)(y - 0)
x² = 20y
4a = 20
a = 5
h = 0 ; k = 0 ; a = 5
Focus (0, a)
Focus (0, 5)
Directrix: y = -a
Directrix: y = -5
Axis of symmetry: x = 0
3.) y² + 4x - 14y = -53
y² - 14y + 7² = - 4x - 53 + 7²
y² - 14y + 7² = -4x - 4
(y - 7)² = -(4)(1)(x + 1)
y² = 4x
4a = 4
a = 1
h = -1 ; k = 7 ; a = 1
Vertex (-1 , 7)
Focus(-a , 0)
X = x + 1 ; Y = y - 7
x = X - 1 ; y = Y + 7
x = -1 - 1 = - 2 ; y = 0 + 7 = 7
Focus (-2, 7)
Directrix: x = a
X = x + 1
x = X - 1
x = 1 - 1 = 0
Directrix: x = 0
Y = y - 7
y = Y + 7
y = 0 + 7
Axis of symmetry: y = 7
4.) y² - 2x + 2y - 1 = 0
y² + 2y + 1² = 2x + 1 + 1²
(y + 1)² = 2x + 2
(y + 1)² = 2(x + 1)
y² = 4ax
y² = 2x
4a = 2
a = 1/2
h = -1 ; k = -1 ; a = 1/2
Vertex (-1 , -1)
Focus(a , 0)
X = x + 1 ; Y = y + 1
x = X - 1 ; y = Y - 1
x = 1/2 - 1 = - 1/2 ; y = 0 - 1 = -1
Focus (-1/2, -1)
Directrix: X = -a
x = -1/2 - 1 = -3/2
Directrix: x = -3/2
Y = y + 1
y = Y - 1
y = 0 - 1
Axis of symmetry: y = -1
5.) 2x² - 12x + 28y = 38
2x² - 12x = -28y + 38
2(x² - 6x + 3²) = - 28y + 38 + 2(3²)
2(x² - 6x + 3²) = -28y + 38 + 18
2(x² - 6x + 3²) = -28y + 56
2(x² - 6x + 3²) = -28(y - 2)
(x² - 6x + 3²) = -14(y - 2)
(x - 3)² = -14(y - 2)
x² = -4ay
4a = -14
a = -14/4
a = -7/2
h = 3 ; k = 2 ; a = -7/2
Vertex (3 , 2)
Focus(0 , a)
X = x - 3 ; Y = y - 2
x = X + 3 ; y = Y + 2
x = 0 + 3 = 3 ; y = -7/2 + 2 = -3/2
Focus (3, -3/2)
Directrix: Y = a
Y = y - 2
y = Y + 2
y = 7/2 + 2 = 11/2
Directrix: y = 11/2
Axis of symmetry: x = 3
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Answers & Comments
Determine the vertex, focus, directrix and axis of symmetry of the parabola with the given equation. Sketch the graph and include these pts and lines
Problem 1:
1.) y² = -36x
y² = -4ax
(y - 0)² = -(4)(9)(x - 0)
y² = -36x
4a = -36
a = -9
h = 0 ; k = 0 ; a = -9
Vertex (h , k)
Vertex (0 , 0)
Focus (a, 0)
Focus (-9, 0)
Directrix: x = -a
Directrix: x = -(-9)
Directrix: x = 9
Axis of symmetry: y = 0
Problem 2:
2.) 5x² = 100y
x² = 4ay
(x - 0)² = 4(5)(y - 0)
x² = 20y
4a = 20
a = 5
h = 0 ; k = 0 ; a = 5
Vertex (h , k)
Vertex (0 , 0)
Focus (0, a)
Focus (0, 5)
Directrix: y = -a
Directrix: y = -5
Axis of symmetry: x = 0
Problem 3:
3.) y² + 4x - 14y = -53
y² - 14y + 7² = - 4x - 53 + 7²
y² - 14y + 7² = -4x - 4
y² = -4ax
(y - 7)² = -(4)(1)(x + 1)
y² = 4x
4a = 4
a = 1
h = -1 ; k = 7 ; a = 1
Vertex (h , k)
Vertex (-1 , 7)
Focus(-a , 0)
X = x + 1 ; Y = y - 7
x = X - 1 ; y = Y + 7
x = -1 - 1 = - 2 ; y = 0 + 7 = 7
Focus (-2, 7)
Directrix: x = a
X = x + 1
x = X - 1
x = 1 - 1 = 0
Directrix: x = 0
Y = y - 7
y = Y + 7
y = 0 + 7
Axis of symmetry: y = 7
Problem 4:
4.) y² - 2x + 2y - 1 = 0
y² + 2y + 1² = 2x + 1 + 1²
(y + 1)² = 2x + 2
(y + 1)² = 2(x + 1)
y² = 4ax
y² = 2x
4a = 2
a = 1/2
h = -1 ; k = -1 ; a = 1/2
Vertex (h , k)
Vertex (-1 , -1)
Focus(a , 0)
X = x + 1 ; Y = y + 1
x = X - 1 ; y = Y - 1
x = 1/2 - 1 = - 1/2 ; y = 0 - 1 = -1
Focus (-1/2, -1)
Directrix: X = -a
X = x + 1
x = X - 1
x = -1/2 - 1 = -3/2
Directrix: x = -3/2
Y = y + 1
y = Y - 1
y = 0 - 1
Axis of symmetry: y = -1
Problem 5:
5.) 2x² - 12x + 28y = 38
2x² - 12x = -28y + 38
2(x² - 6x + 3²) = - 28y + 38 + 2(3²)
2(x² - 6x + 3²) = -28y + 38 + 18
2(x² - 6x + 3²) = -28y + 56
2(x² - 6x + 3²) = -28(y - 2)
(x² - 6x + 3²) = -14(y - 2)
(x - 3)² = -14(y - 2)
x² = -4ay
4a = -14
a = -14/4
a = -7/2
h = 3 ; k = 2 ; a = -7/2
Vertex (h , k)
Vertex (3 , 2)
Focus(0 , a)
X = x - 3 ; Y = y - 2
x = X + 3 ; y = Y + 2
x = 0 + 3 = 3 ; y = -7/2 + 2 = -3/2
Focus (3, -3/2)
Directrix: Y = a
Y = y - 2
y = Y + 2
y = 7/2 + 2 = 11/2
Directrix: y = 11/2
Axis of symmetry: x = 3
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