Graphing a Parabola in a Cartesian Coordinate System

Updated on June 30, 2018
JR Cuevas profile image

JR has a Bachelor of Science in Civil Engineering and specializes in Structural Engineering. He loves to write anything about education.

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 second-degree 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:

Cy2 + Dx +Ey + F = 0

where C ≠ 0 and D ≠ 0

Ax2 + 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.

y2 = 4ax

y2 = - 4ax

x2 = 4ay

x2 = - 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
Table 1: Standard Equations of a Parabola

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 x-axis or parallel to the x-axis. If the parabola opens upward or downward, the axis of symmetry is either the y-axis or parallel to y-axis. Here are the graphs of all equations of a parabola.

Graph of Different Equations of a Parabola
Graph of Different Equations of a Parabola | Source
Graph of Different Forms of Parabola
Graph of Different Forms of Parabola | Source

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, y2 = 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 y2 = 12x is in the reduced form y2 = 4ax where a = 3.

a. The concavity of the parabolic curve is opening to the right since the equation is in the form y2 = 4ax.

b. The vertex of the parabola with a form y2 = 4ax is at (0, 0).

c. The focus of a parabola in the form y2 = 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 y2 = 12x is at (3, 0). Count 3 units to the right.

d. The latus rectum coordinates of the equation y2 = 4ax is at (a, 2a) and (a, -2a). Since the segment contains the focus and is parallel to the y-axis, we add or subtract 2a from the y-axis. 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.

Graph of a Parabola Opening to the Right in Cartesian Coordinate System
Graph of a Parabola Opening to the Right in Cartesian Coordinate System | Source

Problem 2: A Parabola Opening to the Left

Given the parabolic equation, y2 = - 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 y2 = - 8x is in the reduced form y2 = - 4ax where a = 2.

a. The concavity of the parabolic curve is opening to the left since the equation is in the form y2 = - 4ax.

b. The vertex of the parabola with a form y2 = - 4ax is at (0, 0).

c. The focus of a parabola in the form y2 = - 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 y2 = - 8x is at (-2, 0). Count 2 units to the left.

d. The latus rectum coordinates of the equation y2 = - 4ax is at (-a, 2a) and (-a, -2a). Since the segment contains the focus and is parallel to the y-axis, we add or subtract 2a from the y-axis. 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.

Graph of a Parabola Opening to the Left in Cartesian Coordinate System
Graph of a Parabola Opening to the Left in Cartesian Coordinate System | Source

Problem 3: A Parabola Opening Upward

Given the parabolic equation x2 = 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 x2 = 16y is in the reduced form x2 = 4ay where a = 4.

a. The concavity of the parabolic curve is opening upward since the equation is in the form x2 = 4ay.

b. The vertex of the parabola with a form x2 = 4ay is at (0, 0).

c. The focus of a parabola in the form x2 = 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 x2 = 4ay is at (0, 4). Count 4 units upward.

d. The latus rectum coordinates of the equation x2 = 4ay is at (-2a, a) and (2a, a). Since the segment contains the focus and is parallel to the x-axis, we add or subtract a from the x-axis. 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.

Graph of a Parabola Opening Upward in Cartesian Coordinate System
Graph of a Parabola Opening Upward in Cartesian Coordinate System | Source

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, k-a). 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.

Graph of a Parabola Opening Downward in Cartesian Coordinate System
Graph of a Parabola Opening Downward in Cartesian Coordinate System | Source

Did you learn from the examples?

See results

Questions & Answers

    © 2018 Ray

    Comments

      0 of 8192 characters used
      Post Comment

      • JR Cuevas profile imageAUTHOR

        Ray 

        4 months ago from Philippines

        I hope you and your boy are enjoying. Graphing parabolas are easy given you are knowledgeable of the different properties and familiar with the equations.The tables provided would be a great help. Thank you again, Sir Eric!

      • Ericdierker profile image

        Eric Dierker 

        4 months ago from Spring Valley, CA. U.S.A.

        Very interesting. I am just going to read it and not try to pass a test. Thanks

        We are are working on the concepts of graphs.

      working

      This website uses cookies

      As a user in the EEA, your approval is needed on a few things. To provide a better website experience, owlcation.com uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

      For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at: https://owlcation.com/privacy-policy#gdpr

      Show Details
      Necessary
      HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
      LoginThis is necessary to sign in to the HubPages Service.
      Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
      AkismetThis is used to detect comment spam. (Privacy Policy)
      HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
      HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
      Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
      CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
      Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the googleapis.com or gstatic.com domains, for performance and efficiency reasons. (Privacy Policy)
      Features
      Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
      Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
      Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
      Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
      Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
      VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
      PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
      Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
      MavenThis supports the Maven widget and search functionality. (Privacy Policy)
      Marketing
      Google AdSenseThis is an ad network. (Privacy Policy)
      Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
      Index ExchangeThis is an ad network. (Privacy Policy)
      SovrnThis is an ad network. (Privacy Policy)
      Facebook AdsThis is an ad network. (Privacy Policy)
      Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
      AppNexusThis is an ad network. (Privacy Policy)
      OpenxThis is an ad network. (Privacy Policy)
      Rubicon ProjectThis is an ad network. (Privacy Policy)
      TripleLiftThis is an ad network. (Privacy Policy)
      Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
      Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
      Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
      Statistics
      Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
      ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
      Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)