Introduction

A parabola is defined as a curve in which any given point lies at an equidistant from the focus and the directrix. Representing or plotting a parabola on a graph is termed as a Parabola graph. There is a stepwise series of points that help to determine and thereafter plot the points on the graph. We will study these in the topic in detail.

Parts of a Parabola - It contains the following parts -

Focus It is a fixed point inside a parabola which is used in general, the formal definition of the given curve.

The Directrix - It is a fixed straight line. The series of points that forms a parabola is located at equal distance from focus and directrix.

The Axis of symmetry - It is also a straight line that passes through the vertex and is situated at an equal distance from corresponding points on the arms of a parabola.

The vertex - It is a point where the axis of symmetry crosses the parabola.

Equation of Parabola - y=ax²+bx+c is the general equation of a parabola. In this, if a is positive, the parabola moves upwards, and if it is negative, parabola moves downwards.

Finding the Axis of Symmetry - The formula used to find the axis of symmetry is x= -b/2a.

Finding the Vertex - Refers to finding out the value of y coordinate.

Set up a Table - Create a table with x in column number 1. Axis of symmetry is the middle value of x. Setting up the table is the most important part of making a parabola graph.

Calculate the Values of Y - After putting in the value of x in each equation, you can derive the value of y. Let us take some examples to understand it clearly -

If x = -2, y = (2) (-2)2 - 1 = 8 - 1 = 7

If x = -1, y = (2) (-1)2 - 1 = 2 - 1 = 1

If x = 0, y = (2) (0)2 - 1 = 0 - 1 = -1

If x = 1, y = (2) (1)2 - 1 = 2 - 1 = 1

If x = 2, y = (2) (2)2 - 1 = 8 - 1 = 7

Insert the derived values of y in the table. Now, because we have found the values, we are all set to graph it. 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.

Now, we are ready to plot the points. Each row consists of pairs (x, y).

The last and important step is to graph a parabola. All we have to do is connect the points that were plotted in the previous step.

Note - We can draw an arrow at both the ends of the parabola, which points away from the vertex. These arrows indicate that the parabola continues indefinitely.

The one main thing required to learn how to shift the parabola is to actually read the equation. Let’s take an example. In the equation y=x². It has a vertex at the points (0,0) and tends to open upwards. The points on it are (-1, 1), (1, 1), (-2, 4), and (2, 4).

Shifting a Parabola Upwards - Let’s make an equation y=x²+1. This tends to shift the parabola in upwards direction with 1 unit. The vertex becomes ( 0,1). Though the shape will stay the same but y will shift towards upwards direction with 1 unit.

Shifting the Parabola Downwards - Let’s take the example of equation y=x²-1. As we are shifting the parabola downwards to 1 unit, the vertex now becomes (0,1). The shape of the parabola will stay the same, but y will shift downward with 1 unit.

Shifting the Parabola to the Left - Let’s take an example of the equation y=(x+1)². This is shifting the parabola towards left, and the vertex now becomes (-1,0). The shape of the parabola stays the same, but x shifts one unit towards the left.

Shifting the Parabola to the Right - let us take the equation y=(x-1)². The vertex becomes (1,0). The shape remains the same, but x shifts towards the right with 1 unit.

The parabolic graph is a smooth, U shaped curve that depends on the sign that its coefficient carries on whether it will open upwards or downwards. The better and the intercepted are the two points that make it very easy to graph a parabola. Another thing that is used in the process of graphing a parabola is the axis of symmetry. The points that are present on either side of this carry the same value. If the graph has x-intercept a, you can also stick to other points in the graph.

FAQ (Frequently Asked Questions)

1. How can I make sure that I Draw a Perfect Parabola?

Here is a simple trick to it that will make sure that the parabola you draw is accurate. You only need to draw a smooth line through the points you have made the graph out of. Just keep the U shape of the parabola in mind, and don’t forget to put the arrows at the ends.

2. How can I Draw a Graph for x = - 4x^{2} + 32?

All you need to do is follow the steps of making a graph. Make a table having two columns. Select the values for x. Then calculate the value of y with the equation f(x). Then plot the value of the pair on the grid and connect the points. You are done.

3. How can we Derive the Formula for a Parabola?

There is no such method of finding a formula for a given parabola. The equation will be provided along if the parabola has been mentioned.

4. If in Case, the Value of the Axis of Symmetry is Anything Else Apart From 0, How can we Make the Table?

First of all, this is a wrong notation that for you to make a table, the value of the axis of symmetry should be zero. The things that are actually required to make a table is the value of out of which you can very easily derive the value of y. Plot the values of the pair on the graph, comment the points, and you are ready with a parabolic graph.