A parabola is a two-dimensional curve, symmetrical with respect to an axis and having an arcuate shape. Each point on the parabola is equidistant from a fixed point (the focus) and a straight line (the directrix). To draw a parabola, you need to find its vertex and many x and y coordinates on either side of the vertex in order to draw the path to follow. If you want to know how to draw a parabola, start with Step 1.
Steps
Part 1 of 2: Drawing a Parable
Step 1. Distinguish the parts of the parable
You may have been given some information before starting, and knowing the terminology will help you avoid unnecessary steps. Here are the parts of the parable you need to know:
- Fire. A fixed point within the parable that is used for its formal definition.
- Director. A fixed straight line. The parabola is the locus of points that are equidistant from a fixed point called the focus and from the directrix.
- The axis of symmetry. The axis of symmetry is a vertical line that crosses the vertex of the parabola. On each side of the symmetry axis, the parabola is reflected.
- The summit. The point where the axis of symmetry crosses the parabola is called the vertex. If the parabola opens upwards, then the vertex is the minimum point; if it is facing down, the vertex is the maximum point.
Step 2. Know the equation of the parabola
The equation of the parabola is y = ax2+ bx + c. It can also be written in the form y = a (x - h) 2 + k, but, in our example, we will focus on the first.
- If a in the equation is positive, then the parabola is facing upwards, like a "U", and has a minimum point. If a is negative, then it faces down and has a maximum point. If you have trouble remembering this point, think of it this way: an equation with a positive a is happy; an equation with a negative is sad.
- Suppose you have the following equation: y = 2x2 -1. This parable will look like a "U" since a is equal to 2, therefore positive.
- If your equation has a y squared instead of an x squared, then it will open to the side, right or left, like a "C" or "C" facing left. For example, the parabola y2 = x + 3 opens to the right, like a "C".
Step 3. Find the axis of symmetry
Remember that the axis of symmetry is the line that passes through the vertex of the parabola. It corresponds with the x coordinate of the vertex, which is the point where the axis of symmetry meets the parabola. To find the axis of symmetry, use this formula: x = -b / 2a
- In the example, you can see that a = 2, b = 0 and c = 1. Now, you can calculate the axis of symmetry by replacing the points: x = -0 / (2 x 2) = 0.
- Your axis of symmetry is x = 0.
Step 4. Find the vertex
Once you have the symmetry axis, you can substitute the x value to find the corresponding y coordinate. These two coordinates identify the vertex of the parabola. In this case, you should substitute 0 into 2x2 -1 to get the y coordinate. y = 2 x 02 -1 = 0 -1 = -1. Your vertex is (0, -1), which is the point where the parabola meets the y axis.
The vertex values are also known as the (h, k) coordinates. Your h is 0 and your k is -1. If the equation of 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 have to do any mathematical calculations to find it: just interpret the graph correctly
Step 5. Create a table with x values
In this step, you need to create a table where you enter the x values in the first column. This table will contain the coordinates you will need to draw the parabola.
- The average value of x should be the symmetry axis.
- You should include 2 values above and below the mean value of x in the table, for symmetry reasons.
- In your example, enter the value of the symmetry axis, x = 0, in the center of the table.
Step 6. Calculate the y coordinate values
Substitute each value of x in the equation of the parabola and calculate the values of y. Enter the calculated values of y into 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
Step 7. Enter the calculated y values in the table
Now that you have found at least 5 coordinate pairs of the parabola, you are practically ready to draw it. Based on your work, you now possess the following points: (-2, 7), (-1, 1), (0, -1), (1, 1), (2, 7). Now, you can return to the idea that the parabola is reflected with respect to its axis of symmetry. This means that the y coordinates of the points that are reflections of each other will be the same. The y coordinates for the x coordinates of -2 and 2 are both 7, the y coordinates for the x coordinates of -1 and 1 are both 1, and so on.
Step 8. Draw the points of the table on the graph
Each row of the table forms points (x, y) on the coordinate plane. Draw all points in the table on the coordinate plane.
- The x axis goes from left to right; the y axis from bottom to top.
- The positive numbers of the y are located above the point (0, 0) and the negative numbers of the y axis are located below the point (0, 0).
- The positive numbers of the x axis are to the right of (0, 0) and the negative ones to the left of the point (0, 0).
Step 9. Connect the dots
To draw the parabola, connect the points found in the previous step. The graph in your example will look like a U. Make sure you connect the points using a curved line, instead of connecting them with straight segments. This will allow you to accurately represent the appearance of the parable. You can also draw arrows pointing up or down at the ends of the parabola, depending on which direction it is facing. This indicates that the parabola graph will continue outside the graph.
Part 2 of 2: Moving the Graph of the Parabola
If you want to know a shortcut to move the parabola without having to calculate the vertex and different points on it, then you need to understand how to read the equation of a parabola and move it up, down, right or left. Start with the basic parabola: y = x2. This has a vertex (0, 0) and is facing upwards. Some points on it are for example (-1, 1), (1, 1), (-2, 4), (2, 4), and so on. You can understand how to move the parabola depending on the equation you have.
Step 1. Move the parabola graph upwards
Take the equation y = x2 +1. All you have to do is move the original parabola up one unit, so the vertex is now (0, 1) instead of (0, 0). It will always have exactly the same shape as the original parabola, but each y coordinate will be higher than one unit. So instead of (-1, 1) and (1, 1), you would have (-1, 2) and (1, 2), and so on.
Step 2. Move the parabola graph down
Take the equation y = x2 -1. All you have to do is move the original parabola down one unit, so that the vertex is now (0, -1) instead of (0, 0). It will always have exactly the same shape as the original parabola, but each y coordinate will be one unit lower. So instead of (-1, 1) and (1, 1), you would have (-1, 0) and (1, 0), and so on.
Step 3. Move the parabola graph to the left
Take the equation y = (x + 1)2. All you have to do is move the original parabola to the left by one unit, so that the vertex is now (-1, 0) instead of (0, 0). It will always have exactly the same shape as the original parabola, but each x coordinate will be more to the left of a unit. So instead of (-1, 1) and (1, 1), you would have (-2, 1) and (0, 1), and so on.
Step 4. Move the parabola graph to the right
Take the equation y = (x - 1)2. All you have to do is move the original parabola to the right by one unit, so that the vertex is now (1, 0) instead of (0, 0). It will always have exactly the same shape as the original parabola, but each x coordinate will be more to the right of a unit. So instead of (-1, 1) and (1, 1), you would have (0, 1) and (2, 1), and so on.
