How to Graph a Parabola

Step-by-Step Guide

1. 1

   Step 1: Understand the parts of a parabola.

   You may be given certain information prior to beginning, and knowing the terminology will help you avoid any unnecessary steps.
   
   Here are the parts of the parabola that you'll need to know:
   The focus.
   
   A fixed point on the interior of the parabola that is used for the formal definition of the curve.
   
   The direct x.
   
   A fixed straight line.
   
   The parabola is the locus of points where any given point is of equal distance from the focus and the directrix.
   
   The axis of symmetry.
   
   The axis of symmetry is a vertical line that passes through the turning point of the parabola.
   
   Each side of the axis of symmetry is a mirror image.
   
   The vertex.
   
   The point where the axis of symmetry crosses the parabola is called the vertex of the parabola.
   
   If the parabola opens upward, then the vertex is a minimum point; if it opens downward, then the vertex is a maximum point.
2. 2

   Step 2: Know the equation of a parabola.

   The equation of a parabola is y = ax2+ bx + c.
   
   It can also be written in the form y = a(x – h)2 + k, but we will focus on the first form of the equation in this example.
   
   If the a in the equation is positive, then the parabola opens upward, like a "U"
   
   and has a minimum point.
   
   If the a is negative, then it opens downward and has a minimum point.
   
   If you have trouble remembering this, think of it this way: an equation with a positive a value looks like a smile; an equation with a negative a value looks like a frown.Let’s say you have the following equation: y = 2x2
   -1.
   
   This parabola will be shaped like a "U" because the a value, 2, is positive.
   
   If your equation has a squared y coordinate instead of a squared x-coordinate, then it will open up sideways, to the right or a left, like a "C" or a "C" facing the left.
   
   For example, the parabola x2 = y + 3 opens to the right, like a "C". , Remember that the axis of symmetry is the vertical line that passes through the turning point of the parabola.
   
   It is the same as the x coordinate of the vertex, which is the point where the axis of symmetry crosses the parabola.
   
   To find the axis of symmetry, use this formula: x =
   -b/2aUsing the example, you can see that a = 2, b = 0, and c =
   1.
   
   Now, you can calculate the axis of symmetry by plugging in the numbers: x =
   -0/(2 x 2) =
   0.
   
   Your axis of symmetry is x =
   0. , Once you have your axis of symmetry, you can plug that value in for x to get the y coordinate.
   
   These two coordinates will give you the vertex of the parabola.
   
   In this case, you should plug 0 in to 2x2
   -1 to get the y coordinate. y = 2 x 02
   -1 = 0
   -1 =
   -1.
   
   Your vertex is (0,-1), which is the point at which the parabola crosses the y axis.The points of the vertex are also known as the points (h, k).
   
   Your h is 0 and your k is
   -1.
   
   If the equation for the parabola is written in the form y = a(x – h)2 + k, then your vertex is simply the point (h, k), and you don't need to do any math to find it beyond interpreting the graph correctly. , In this step, you need to create a table where you put the values of x in the first column.
   
   This table will give you the coordinates you need to graph your parabola.
   
   The middle value of x should be the axis of symmetry.
   
   You should include 2 values above and below the middle value for x in the table, for the sake of symmetry.
   
   In your example, put the value of the axis of symmetry, x = 0, in the middle of the table. , Substitute each value of x in the equation of the parabola and calculate the corresponding values of y.
   
   Insert the calculated values of y in the table.
   
   In your example, the equation of the parabola is calculated as follows:
   For x =
   -2, y is calculated as: y = 2 x (-2)2
   - 1 = 8
   - 1 = 7 For x =
   -1, y is calculated as: y = 2 x (-1)2
   - 1 = 2
   - 1 = 1 For x = 0, y is calculated as: y = 2 x (0)2
   - 1 = 0
   - 1 =
   -1 For x = 1, y is calculated as: y = 2 x (1)2
   - 1 = 2
   - 1 = 1 For x = 2, y is calculated as: y = 2 x (2)2
   - 1 = 8
   - 1 = 7 , Now that you've found at least 5 coordinate pairs for the parabola, you're almost ready to graph it.
   
   Based on your work, you now have the following points: (-2, 7), (-1, 1), (0,
   -1), (1, 1), (2, 7).
   
   Now, you can think back to the idea that the parabola is reflected over the axis of symmetry.
   
   This means that the y coordinates of points that are direct reflections of each other across the axis of symmetry will be the same.
   
   The y-coordinates for the x-coordinates
   -2 and 2 are both 7, the y-coordinates for the x-coordinates
   -1 and 1 are both 1, and so on. , Each row of the table forms a coordinate (x, y) on the coordinate plane.
   
   Draw all dots with the coordinates given in the table on the coordinate plane.
   
   The x-axis goes left and right; the y-axis goes up and down.
   
   The positive numbers on the y-axis are above the point (0, 0) and the negative numbers on the y-axis are below the point (0, 0).
   
   The positive numbers on the x-axis are to the right of the point (0, 0) and the negative numbers on the x-axis are to the left of the point (0, 0). , To graph the parabola, connect the dots given in the previous step.
   
   The graph of your example will look like a U.
   
   Make sure that you connect the dots using a curved line, instead of connecting them to look like line segments.
   
   This will create the most accurate image of the parabola.
   
   You can also draw arrows pointing upward or downward at each end of the parabola, depending on which direction it is facing.
   
   This will indicate that the parabola's graph will continue to move outside of the coordinate plane., Take the equation y = x2 +1.
   
   All you have to do is shift this original parabola up 1 unit, so that the vertex is now (0, 1) instead of (0, 0).
   
   It will still have the same exact shape as the original parabola, but every y-coordinate will be shifted up 1 unit.
   
   So, instead of (-1, 1) and (1, 1), you have (-1, 2) and (1, 2), and so on. , Take the equation y = x2
   -1.
   
   All you have to do is shift this original parabola down 1 unit, so that the vertex is now (0,
   -1) instead of (0, 0).
   
   It will still have the same exact shape as the original parabola, but every y-coordinate will be shifted down 1 unit.
   
   So, instead of (-1, 1) and (1, 1), you have (-1, 0) and (1, 0), and so on. , Take the equation y = (x + 1)2.
   
   All you have to do is shift this original parabola to the left 1 unit, so that the vertex is now (-1, 0) instead of (0, 0).
   
   It will still have the same exact shape as the original parabola, but every x-coordinate will be shifted to the left 1 unit.
   
   So, instead of (-1, 1) and (1, 1), you have (-2, 1) and (0, 1), and so on. , Take the equation y = (x
   - 1)2.
   
   All you have to do is shift this original parabola to the left 1 unit, so that the vertex is now (1, 0) instead of (0, 0).
   
   It will still have the same exact shape as the original parabola, but every x-coordinate will be shifted to right left 1 unit.
   
   So, instead of (-1, 1) and (1, 1), you have (0, 1) and (2, 1), and so on.
3. 3

   Step 3: Find the axis of symmetry.

4. 4

   Step 4: Find the vertex.

5. 5

   Step 5: Set up a table with values of x.

6. 6

   Step 6: Calculate the values of the y-coordinates.

7. 7

   Step 7: Insert the calculated values of y in the table.

8. 8

   Step 8: Plot the points from the table on the coordinate plane.

9. 9

   Step 9: Connect the dots.

10. 10

    Step 10: Shift the graph of a parabola up.

11. 11

    Step 11: Shift the graph of a parabola down.

12. 12

    Step 12: Shift the graph of a parabola to the left.

13. 13

    Step 13: Shift the graph of a parabola to the right.

Detailed Guide

About the Author