# How to Trace a Parable: 13 Steps (with Pictures)

A parabola is a two-dimensional, symmetrical curve, shaped like an arc. Any point in a parabola is equidistant from a fixed point (focus) and a fixed straight line (guideline). To trace a parabola, you need to find its vertex, as well as several x and y coordinates on each side of the vertex, in order to mark the path it takes. If you want to know how to plot a parable, see Step 1 to get started.

## Steps

### Part 1 of 2: Tracing a Parable

#### Step 1. Understand the parts of the parable

You may have certain information before you begin, and knowing the terminology will help you avoid unnecessary steps. These are the parts of the parable you will need to know:

• The focus. A fixed point inside the parabola, used for the formal definition of the curve.
• The guideline. A fixed straight line. The parabola is the geometric locus where any given point is at the same distance from the focus and the guideline.
• 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 symmetry axis is a reflection of the other.
• The apex. The point where the axis of symmetry intersects the parabola is called the vertex of the parabola. If the concavity of the parabola is upwards, the vertex is a minimum point; if it is down, the vertex is a maximum point.

#### Step 2. Know the equation of the parable

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 let's focus on the first form of the equation in this example.

• If the a in the equation is positive, then the parabola has an upward concavity, "U" shape, and a minimum point. If a is negative, then the parabola has downward concavity and a maximum point. If you have trouble remembering this, think of it this way: an equation with a positive a looks like a smile; an equation with a negative a looks like a frown.
• Let's say you have the following equation: y = 2x2 -1. This parabola will be "U" shaped because the value of a, 2, is positive.
• If your equation has a square y coordinate instead of an x, then the concavity will be on either side, right or left, like a "C" or an inverted "C". For example, the parable x2 = y + 3 is concave to the right side, like a "C".

#### Step 3. Find the axis of symmetry

Remember that the axis of symmetry is the vertical line through the turning point of the parabola. It's the same as the x-coordinate of the vertex, which is the point at which the symmetry axis intersects the parabola. To find the symmetra axis, use this formula: x = -b/2a

• Using the example, you can see that a = 2, b = 0, and c = 1. Now you can calculate the axis of symmetry by substituting the numbers: x = -0/(2 x 2) = 0.
• Its symmetry axis is x = 0.

#### Step 4. Find the vertex

Once you have your axis of symmetry, you can substitute the value of x and find the coordinate of y. These two coordinates will give the vertex of the parabola. In that case, you should replace 0 in place of 2x2 -1 to get to the y coordinate. y = 2 x 02 -1 = 0 -1 = -1. Its vertex is (0, -1), which is the point where the parabola intersects the y-axis.

### Vertex points are also known as (h, k) points. Your h is 0 and your k is -1. If the parabola equation is written in the form y = a(x – h)2 + k, its vertex is simply the point (h, k), and you don't need to do any more calculations to find it other than interpret the graph

#### Step 5. Build a table with x values

In this step, you need to create a table where you will put the x values ​​in the first column. This table will give you the coordinates you need to plot your parabola.

• The central value of x must be the axis of symmetry.
• You must include two values ​​above and below the center value of x in the table for symmetry reasons.
• For the example, place the symmetry axis value, x = 0, in the middle of the table.

#### Step 6. Calculate the y coordinate values

Substitute each value of x into the equation of the parabola and calculate the corresponding values ​​of y. Enter the calculated values ​​for y into the table. In the example, the equation for the parabola is calculated as follows:

• For x = -2, y is calculated by: y = 2 x (-2)2 - 1 = 8 - 1 = 7
• For x = -1, y is calculated by: y = 2 x (-1)2 - 1 = 2 - 1 = 1
• For x = 0, y is calculated by: y = 2 x (0)2 - 1 = 0 - 1 = -1
• For x = 1, y is calculated by: y = 2 x (1)2 - 1 = 2 - 1 = 1
• For x = 2, y is calculated by: y = 2 x (2)2 - 1 = 8 - 1 = 7

#### Step 7. Enter the calculated values ​​of y into the table

Now that you've found at least 5 pairs of coordinates for the parabola, you're almost ready to plot it. Based on your work, you now have the following points: (-2, 7), (-1, 1), (0, -1), (1, 1), (2, 7). Now you can get back to the idea that each side of the parabola's axis of symmetry is a reflection of the other. The y coordinates for the x -2 and 2 coordinates are both 7, the y coordinates for x -1 and 1 are both 1 and so on.

#### Step 8. Mark the points on the table in the coordinate plane

Each row in the table forms a coordinate (x, y) in the coordinate plane. Mark all points with the coordinates given in the table in the coordinate plane.

• The c axis goes left and right; the y axis goes up and down.
• Positive numbers on the y axis are above the point (0, 0) and negative numbers below.
• Positive numbers on the x-axis are to the right of the point (0, 0) and negative numbers to the left.

#### Step 9. Connect the dots

To trace the parabola, connect the points marked in the previous step. The example graph will look like a U. Be sure to connect the points by making a curve rather than a straight line. This will create the most accurate image of the parable. You can also draw arrows pointing up or down at each end of the parabola, depending on its direction. This will indicate that the parabola graph continues beyond the coordinate plane.

### Part 2 of 2: Shifting the Graphic of a Parable

If you want a quick way to shift a parabola without having to find the vertex and make multiple points, then you have to understand how to read a parabola equation and learn how to shift it up, down, left or right. Start with the basic parable: y = x2. This one has the vertex (0, 0) and the concavity upwards. Some points of it include (-1, 1), (1, 1), (-2, 4), (2, 4), and so on. You can learn to shift the parabola based on the equation you're working with.

#### Step 1. Move the parabola graph up

Take the equation y = x2 +1. All you have to do is shift the original parabola up 1 unit so that the vertex is (0, 1) instead of (0, 0). It will still have the same shape as the original parabola, but all the y coordinates will be increased by 1 unit. So instead of (-1, 1) and (1, 1), you get (-1, 2) and (1, 2), and so on.

#### Step 2. Shift the parabola graph downwards

Take the equation y = x2 -1. All you have to do is shift the original parabola down 1 unit so that the vertex is (0, -1) instead of (0, 0). It will still have the same shape as the original parabola, but all the y coordinates will be decreased by 1 unit. So instead of (-1, 1) and (1, 1), you get (-1, 0) and (1, 0), and so on.

#### Step 3. Shift the parabola graphic to the left

Take the equation y = (x + 1)2. All you have to do is shift the original parabola 1 unit to the left so that the vertex is (-1, 0) instead of (0, 0). It will still have the same shape as the original parabola, but all the x coordinates will be decreased by 1 unit. So instead of (-1, 1) and (1, 1), you get (-2, 1) and (0, 1), and so on.

#### Step 4. Shift the parabola graphic to the right

Take the equation y = (x - 1)2. All you have to do is shift the original parabola 1 unit to the right so that the vertex is (1, 0) instead of (0, 0). It will still have the same shape as the original parabola, but all the x coordinates will be increased by 1 unit. So instead of (-1, 1) and (1, 1), you get (0, 1) and (2, 1), and so on.