Vertex at Origin :
y2 = 4ax (opens right, a > 0)
y2 = -4ax (opens right, a > 0)
x2 = 4ay (opens up, a > 0)
x2 = -4ay (opens down, a > 0)
Vertex at (h, k) :
(y - k)2 = 4a(x - h) (opens right, a > 0)
(y - k)2 = -4a(x - h) (opens right, a > 0)
(x - h)2 = 4a(y - k) (opens up, a > 0)
(x - h)2 = -4a(y - k) (opens down, a > 0)
Vertex at Origin :
y = ax2 (opens up, a > 0)
y = -ax2 (opens down, a > 0)
x = ay2 (opens right, a > 0)
x = -ay2 (opens left, a > 0)
Vertex at (h, k) :
y = a(x - h)2 + k (opens up, a > 0)
y = -a(x - h)2 + k (opens down, a > 0)
x = a(y - k)2 + h (opens right, a > 0)
y = -a(y - k)2 + h (opens left, a > 0)
y = a(x - p)(x - q) (opens up, a > 0)
y = -a(x - p)(x - q) (opens down, a > 0)
x = a(y - p)(y - q) (opens right, a > 0)
x = -a(y - p)(y - q) (opens left, a > 0)
y = ax2 + bx + c (opens up, a > 0)
y = -ax2 + bx + c (opens down, a > 0)
x = ay2 + by + c (opens right, a > 0)
x = -ay2 + by + c (opens left, a > 0)
Problem 1 :
Find the standard form equation of the parabola that has vertex at origin and focus at (0, 1).
Solution :
Plot the vertex (0, 0) and focus (0, 1) on the xy-plane.
The parabola is open up with vertex at origin.
Standard form equation of a parabola that opens up with vertex at origin :
x2 = 4ay
Distance between the vertex and focus is 1 unit.
That is, a = 1.
x2 = 4(1)y
x2 = 4y
Problem 2 :
Find the standard form equation of the parabola that has vertex at (2, -1) and focus at (-1, -1).
Solution :
Plot the vertex (0, 0) and focus (0, 1) on the xy-plane.
The parabola is open to the left with vertex at (2, -1).
Standard equation of a parabola that opens left with vertex at (h, k) :
(y - k)2 = -4a(x - h)
Vertex (h, k) = (2, -1).
(y + 1)2 = -4a(x - 2)
Distance between the vertex and focus is 3 units.
That is, a = 3.
(y + 1)2 = -4(3)(x - 2)
(y + 1)2 = -12(x - 2)
Problem 3 :
Find the vertex form equation of the parabola :
Opens left or right, Vertex (0,0), Passes through (-16, 2)
Solution :
Vertex form equation of a parabola that opens left or right with vertex at origin :
x = ay2
It passes through (-16, 2). Substitute (x, y) = (-16, 2).
-16 = a(2)2
-16 = a(4)
Divide each side by 4.
-4 = a
Vertex form equation of the parabola :
x = -4y2
Problem 4 :
Find the vertex form equation of the parabola :
Opens left or right, Vertex (-1, -2), Passes through (11, 0)
Solution :
Vertex form equation of a parabola that opens left or right with vertex at (h, k) :
x = a(y - k)2 + h
Vertex (h, k) = (-1, -2).
x = a(y + 2)2 - 1
It passes through (11, 0). Substitute (x, y) = (11, 0).
11 = a(0 + 2)2 - 1
11 = a(2)2 - 1
11 = 4a - 1
Add 1 to each side.
12 = 4a
Divide each side by 4.
3 = a
Vertex form equation of the parabola :
x = 3(y + 2)2 - 1
Problem 5 :
Write the intercept form equation of the parabola shown below.
Solution :
Intercept form equation of the above parabola :
y = a(x - p)(x - q)
Because x-intercepts are (-1, 0) and (2, 0),
x = -1 -----> x + 1 = 0
x = 2 -----> x - 2 = 0
Then,
y = a(x + 1)(x - 2)
It passes through (0, -4). Substitute (x, y) = (0, -4).
-4 = a(0 + 1)(0 - 2)
-4 = a(1)(-2)
-4 = -2a
Divide each side by -2.
2 = a
Intercept form equation of the parabola :
y = 2(x + 1)(x - 2)
Problem 6 :
Find the equation of the parabola in general form :
Opens up or down, Vertex (3, 1), Passes through (1, 9)
Solution :
First, find the equation of the parabola in vertex form, then convert it to general form.
Vertex form equation of a parabola that opens up or down with vertex at (h, k) :
y = a(x - h)2 + k
Vertex (h, k) = (3, 1).
y = a(x - 3)2 + 1
It passes through (1, 9). Substitute (x, y) = (1, 9).
9 = a(1 - 3)2 + 1
9 = a(-2)2 + 1
9 = 4a + 1
Subtract 1 from each side.
8 = 4a
Divide each side by 4.
2 = a
Vertex form equation of the parabola :
y = 2(x - 3)2 + 1
Convert the above vertex form equation to general form.
y = 2[x2 - 2(x)(3) + 32] + 1
y = 2(x2 - 6x + 9) + 1
Distribute.
y = 2x2 - 12x + 18 + 1
y = 2x2 - 12x + 19
Problem 7 :
Write the following general form equation of the parabola in vertex form.
y = 3x2 - 18x + 29
Solution :
y = 3x2 - 18x + 29
y = 3(x2 - 6x) + 29
y = 3[x2 - 2(x)(3) + 32 - 32] + 29
y = 3[(x - 3)2 - 9] + 29
y = 3(x - 3)2 - 27 + 29
y = 3(x - 3)2 + 2
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