Parabolas
The standard equation of a parabola looks like y = x2. It can also look like y = ax2 ± 𝑏𝑥 ± 𝑐 or
Y = a(x – h)2 + k.
The value of a makes the parabola narrow or wide. As the value of a increases then the parabola gets narrower.
The value of a also tells whether the parabola opens up or down. If a > 0 then it open up and if a<0 then it open down.
The value of b or h moves it right or left.
The value of c or k moves it up or down.
The center of a parabola is at (-b/2a, y(-b/2a)) or at (h,k).
The axis of symmetry is at x = -b/2a or at x = h.
The x-intercepts are found by setting y=0 and then factoring, using quadratic, or using the square root method.
To graph a parabola find the vertex and axis of symmetry. Then find at least two points on each side of the axis of
symmetry and do a best sketch.
Y = x2 + 4x + 7
Vertex x =- b/2a = -4/2(1) = -2 to find y plug in -2 for x
y = (-2)2 + 4(-2) + 7 = 3
Vertex is at (-2,3) axis of symmetry is x = -2
Or
Y = x2 + 4x +4 + 3 or y = (x + 2)2 + 3
Vertex is at (-2,3) axis of symmetry is x = -2
Intercepts are at
X
y
−4±√4 2 −4(1)(7)
2(1)
-4
7
=
−4±√−12
=
2
-3
4
-2 ± 𝑖√3 not real numbers so no x-intercepts
-2
3
-1
4
0
7
Y = 2x2 -8x + 5
Vertex x = -b/2a = -(-8)/2(2) = 2 plug in 2 for x to find y y = 2(2)2 -8(2) + 5 = -3
Vertex is at (2,-3) axis of symmetry is x = 2
Y = 2(x2 – 4x + 4) – 3
y = 2(x – 2)2 – 3
Vertex is at (2, -3) axis of symmetry is x = 2
Intercepts are at
X
y
−(−8)±√(−8)2 −4(2)(5)
2(2)
0
5
=
8 ±√24 8 ±4.9
= 4 =
4
1
-1
3.225 or .775
2
-3
3
-1
4
5
Y = -2x2 + 6
Vertex x = -b/2a = 0/2(-2) = 0
plug in 0 for x y = -2(0)2 + 6 = 6
Vertex is at (0, 6) axis of symmetry is x = 0
Intercepts - 2x2 + 6 = 0
X
Y
-2
-2
-2x2 = -6
x2 = 3 x = ±√3
-1
4
0
6
x = + 1.73 or = 1.73
1
4
2
-2
The difference in x = ay2 ± 𝑏𝑦 ± 𝑐 or x = a(y – k)2 + h is that these open right and left. Vertex is at (x(-b/2a),-b/2a) or
(h,k). Axis of symmetry is a y = -b/2a or y = k.
Y-intercepts are found when x = 0 by factoring, quadratic, or square root method.
To graph a parabola find the vertex and axis of symmetry. Then find at least two points on each side of the axis of
symmetry and do a best sketch.
X = 2y2 + 5
Vertex y = -b/2a = 0/2(2) = 0 plug in y = 0 and find x x = 2(0)2 + 5 = 5
Vertex is at (5,0) axis of symmetry is at Y = 0
5
Intercepts are 0 = 2y2 +5 2y2 = -5 y2 = -5/2 y = ±√− 2 not a real number so no y intercepts
Y
x
-2
13
-1
7
0
5
1
7
X = y2 + 4y + 3
Vertex y = -b/2a = -4/2(1) = -2 plug in y = -2 and find x x = (-2)2 + 4(-2) + 3 = -1
Vertex is at (-1, -2) axis of symmetry is y = -2
Or x = y2 + 4y + 4 -1 x = (y + 2)2 – 1
Vertex is at (-1,-2) axis of symmetry is y = -2
Intercepts 0 = y2 + 4y + 3
0 = (y + 3)(y + 1) y = -3 or -1
2
13
X = y2 -7y + 10
Vertex y = -b/2a = -(-7)/2(1) = 7/2 plug in 7/2 for y
x = (7/2)2 –7(7/2) + 10 = 49/4 -49/2 + 10 =
49/4 -98/4 + 40/4 = -9/4
Vertex is at (-9/4, 7/2) axis of symmetry is y = 7/2
Or x = y2 -7y + 49/4 -9/4
x = (y -7/2)2 – 9/4
Vertex is at (-9/4,7/2) axis of symmetry is y = 7/2
Y intercepts 0 = y2 -7y + 10 0 = (y – 5)(y – 2) y = 5 or 2
y
x
2
0
3
-2
7/2
-9/4
4
-2
5
0