Graphing a Parabola in a Cartesian Coordinate System
What Is a Parabola?
A parabola is an open plane curve that is created by the junction of a right circular cone with a plane parallel to its side. The set of points in a parabola are equidistant from a fixed line. A parabola is a graphical illustration of a quadratic equation or seconddegree equation. Some of the examples representing a parabola are the projectile motion of a body that follows a parabolic curve path, suspension bridges in the shape of a parabola, reflecting telescopes, and antennae. The general forms of a parabola are:
Cy^{2} + Dx +Ey + F = 0
where C ≠ 0 and D ≠ 0
Ax^{2} + Dx + Ey + F = 0
where A ≠ 0 and D ≠ 0
Different Forms of Parabolic Equations
The general formula Cy2 + Dx +Ey + F = 0 is a parabolic equation whose vertex is at (h, k) and the curve opens either to the left or right. The two reduced and specific forms of this general formula are:
(y  k)^{2} = 4a (x  h)
(y  k)^{2} =  4a (x  h)
On the other hand, the general formula Ax2 + Dx + Ey + F = 0 is a parabolic equation whose vertex is at (h, k) and the curve opens either upward or downward. The two reduced and specific forms of this general formula are:
(x  h)^{2} = 4a (y  k)
(x  h)^{2} =  4a (y  k)
If the vertex of the parabola is at (0, 0), these general equations have reduced standard forms.
y^{2} = 4ax
y^{2} =  4ax
x^{2} = 4ay
x^{2} =  4ay
Properties of a Parabola
A parabola has six properties.
1. The vertex of a parabola is at the middle of the curve. It can either be at the origin (0, 0) or any other location (h, k) in the Cartesian plane.
2. The concavity of a parabola is the orientation of the parabolic curve. The curve may open either upward or downward, or to the left or right.
3. The focus lies on the axis of symmetry of a parabolic curve. It is a distance 'a' units from the vertex of the parabola.
4. The axis of symmetry is the imaginary line containing the vertex, focus, and the midpoint of the directrix. It is the imaginary line that separates the parabola into two equal sections mirroring each other.
Equation in Standard Form
 Vertex
 Concavity
 Focus
 Axis of Symmetry


y^2 = 4ax
 (0, 0)
 right
 (a , 0)
 y = 0

y^2 = 4ax
 (0, 0)
 left
 (a, 0)
 y = 0

(y  k)^2 = 4a (x  h)
 (h, k)
 right
 (h + a, k)
 y = k

(y  k)^2 = 4a (x  h)
 (h, k)
 left
 (h  a, k)
 y = k

x^2 = 4ay
 (0, 0)
 upward
 (0, a)
 x = 0

x^2 = 4ay
 (0, 0)
 downward
 (0, a)
 x = 0

(x  h)^2 = 4a (y  k)
 (h, k)
 upward
 (h, k + a)
 x = h

(x  h)^2 = 4a (y  k)
 (h, k)
 downward
 (h, k  a)
 x = h

5. The directrix of a parabola is the line that is parallel to both axes. The distance of the directrix from the vertex is 'a' units from the vertex and '2a' units from the focus.
6. Latus rectum is a segment passing through the parabolic curve's focus. The two ends of this segment lie on the parabolic curve (±a, ±2a).
Equation in Standard Form
 Directrix
 Ends of Latus Rectum


y^2 = 4ax
 x = a
 (a, 2a) and (a, 2a)

y^2 = 4ax
 x = a
 (a, 2a) and ( a, 2a)

(y  k)^2 = 4a (x  h)
 x = h  a
 (h + a, k + 2a) and (h +a, k  2a)

(y  k)^2 = 4a (x  h)
 x = h + a
 (h  a, k + 2a) and (h  a, k  2a)

x^2 = 4ay
 y = a
 (2a, a) and (2a, a)

x^2 = 4ay
 y = a
 (2a, a) and (2a, a)

(x  h)^2 = 4a (y  k)
 y = k  a
 (h  2a, k + a) and (h + 2a, k + a)

(x  h)^2 = 4a (y  k)
 y = k + a
 (h  2a, k  a) and (h + 2a, k  a)

Different Graphs of a Parabola
The focus of a parabola is n units away from the vertex and is directly on the right side or left side if it opens to the right or left. On the other hand, the focus of a parabola is directly above or below the vertex if it opens upward or downward. If the parabola opens to the right or left, the axis of symmetry is either the xaxis or parallel to the xaxis. If the parabola opens upward or downward, the axis of symmetry is either the yaxis or parallel to yaxis. Here are the graphs of all equations of a parabola.
Guide in Graphing Any Parabola
1. Identify the concavity of the parabolic equation. Refer for the directions of the opening of the curve to the given table above. It could be opening to the left or right, or upward or downward.
2. Locate the vertex of the parabola. The vertex can either be (0, 0) or (h, k).
3. Locate the focus of the parabola.
4. Identify the coordinate of the latus rectum.
5. Locate the directrix of the parabolic curve. The location of the directrix is the same distance of the focus from the vertex but in the opposite direction.
6. Graph the parabola by drawing a curve joining the vertex and the coordinates of the latus rectum. Then to finish it, label all the significant points of the parabola.
Problem 1: A Parabola Opening to the Right
Given the parabolic equation, y^{2} = 12x, determine the following properties and graph the parabola.
a. Concavity (direction in which the graph opens)
b. Vertex
c. Focus
d. Latus rectum coordinates
e. The line of symmetry
f. Directrix
Solution
The equation y^{2} = 12x is in the reduced form y^{2} = 4ax where a = 3.
a. The concavity of the parabolic curve is opening to the right since the equation is in the form y^{2} = 4ax.
b. The vertex of the parabola with a form y^{2} = 4ax is at (0, 0).
c. The focus of a parabola in the form y^{2} = 4ax is at (a, 0). Since 4a is equal to 12, the value of a is 3. Therefore, the focus of the parabolic curve with equation y^{2} = 12x is at (3, 0). Count 3 units to the right.
d. The latus rectum coordinates of the equation y^{2} = 4ax is at (a, 2a) and (a, 2a). Since the segment contains the focus and is parallel to the yaxis, we add or subtract 2a from the yaxis. Therefore, the latus rectum coordinates are (3, 6) and (3, 6).
e. Since the parabola's vertex is at (0, 0) and is opening to the right, the line of symmetry is y = 0.
f. Since the value of a = 3 and the graph of the parabola opens to the right, the directrix is at x = 3.
Problem 2: A Parabola Opening to the Left
Given the parabolic equation, y^{2} =  8x, determine the following properties and graph the parabola.
a. Concavity (direction in which the graph opens)
b. Vertex
c. Focus
d. Latus rectum coordinates
e. The line of symmetry
f. Directrix
Solution
The equation y^{2} =  8x is in the reduced form y^{2} =  4ax where a = 2.
a. The concavity of the parabolic curve is opening to the left since the equation is in the form y^{2} =  4ax.
b. The vertex of the parabola with a form y^{2} =  4ax is at (0, 0).
c. The focus of a parabola in the form y^{2} =  4ax is at (a, 0). Since 4a is equal to 8, the value of a is 2. Therefore, the focus of the parabolic curve with equation y^{2} =  8x is at (2, 0). Count 2 units to the left.
d. The latus rectum coordinates of the equation y^{2} =  4ax is at (a, 2a) and (a, 2a). Since the segment contains the focus and is parallel to the yaxis, we add or subtract 2a from the yaxis. Therefore, the latus rectum coordinates are (2, 4) and (2, 4).
e. Since the parabola's vertex is at (0, 0) and is opening to the left, the line of symmetry is y = 0.
f. Since the value of a = 2 and the graph of the parabola opens to the left, the directrix is at x = 2.
Problem 3: A Parabola Opening Upward
Given the parabolic equation x^{2} = 16y, determine the following properties and graph the parabola.
a. Concavity (direction in which the graph opens)
b. Vertex
c. Focus
d. Latus rectum coordinates
e. The line of symmetry
f. Directrix
Solution
The equation x^{2} = 16y is in the reduced form x^{2} = 4ay where a = 4.
a. The concavity of the parabolic curve is opening upward since the equation is in the form x^{2} = 4ay.
b. The vertex of the parabola with a form x^{2} = 4ay is at (0, 0).
c. The focus of a parabola in the form x^{2} = 4ay is at (0, a). Since 4a is equal to 16, the value of a is 4. Therefore, the focus of the parabolic curve with equation x^{2} = 4ay is at (0, 4). Count 4 units upward.
d. The latus rectum coordinates of the equation x^{2} = 4ay is at (2a, a) and (2a, a). Since the segment contains the focus and is parallel to the xaxis, we add or subtract a from the xaxis. Therefore, the latus rectum coordinates are (16, 4) and (16, 4).
e. Since the parabola's vertex is at (0, 0) and is opening upward, the line of symmetry is x = 0.
f. Since the value of a = 4 and the graph of the parabola opens upward, the directrix is at y = 4.
Problem 4: A Parabola Opening Downward
Given the parabolic equation (x  3)^{2} =  12(y + 2), determine the following properties and graph the parabola.
a. Concavity (direction in which the graph opens)
b. Vertex
c. Focus
d. Latus rectum coordinates
e. The line of symmetry
f. Directrix
Solution
The equation (x  3)^{2} =  12(y + 2) is in the reduced form (x  h)^{2} =  4a (y  k) where a = 3.
a. The concavity of the parabolic curve is opening downward since the equation is in the form (x  h)^{2} =  4a (y  k).
b. The vertex of the parabola with a form (x  h)^{2} =  4a (y  k) is at (h, k). Therefore, the vertex is at (3, 2).
c. The focus of a parabola in the form (x  h)^{2} =  4a (y  k) is at (h, ka). Since 4a is equal to 12, the value of a is 3. Therefore, the focus of the parabolic curve with equation (x  h)^{2} =  4a (y  k) is at (3, 5). Count 5 units downward.
d. The latus rectum coordinates of the equation (x  h)^{2} =  4a (y  k) is at (h  2a, k  a) and (h + 2a, k  a) Therefore, the latus rectum coordinates are (3, 5) and (9, 5).
e. Since the parabola's vertex is at (3, 2) and is opening downward, the line of symmetry is x = 3.
f. Since the value of a = 3 and the graph of the parabola opens downward, the directrix is at y = 1.
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