# How to draw y^2 = x - 2?

By Murray Bourne, 09 Apr 2009

Nuaja, a subscriber to the IntMath Newsletter, wrote recently:

How do I know how the graph should look like: For example:

y^{2}=x- 2?

The first thing I recognize in that equation is the *y*^{2} term, which tells me it will be a **parabola**. (It won't be a circle, ellipse or hyperbola because there is an *x* term, but no *x*^{2} term. See Conic Sections.)

Let's start with the most basic parabola y = x^{2} and build up to the required answer.

## Example 1: *y* = *x*^{2}

You could draw up a table and calculate the *y*-values for a set of *x*-values, like this:

x |
-4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|---|---|---|---|

y |
16 | 9 | 4 | 1 | 0 | 1 | 4 | 9 | 16 |

This gives us a series of points (-4,16), (-3,9), (-2,4) up to (4,16).

You then join these dots with a smooth curve and get something like the following.

Notice that the vertex of the parabola (the "pointy" end) is at the origin, (0, 0).

Now for all the curves that I draw below, I'm not going to draw up a table. It becomes tedious, and it can lead to incorrect graphs. It is better to be able to **recognize** the graph type (from the equation) and then know how to sketch it in the right place and with the right orientation.

I will consider the effect of small changes to the equation and then sketch my curve.

All of the following graphs have the **same size and shape** as the above curve. I am just moving that curve around to show you how it works.

## Example 2: *y* = *x*^{2} − 2

The only difference with the first graph that I drew (*y* = *x*^{2}) and this one (*y* = *x*^{2} − 2) is the "minus 2". The "minus 2" means that all the *y*-values for the graph need to be moved down by 2 units.

So we just take our first curve and move it down 2 units. Our new curve's vertex is at −2 on the *y*-axis.

Next, we see how to move the curve up (rather than down).

## Example 3: *y* = *x*^{2} + 3

The "plus 3" means we need to add 3 to all the *y*-values that we got for the basic curve *y* = *x*^{2}. The resulting curve is 3 units higher than *y* = *x*^{2}. Note that the vertex of the curve is at (0, 3) on the *y*-axis.

Next we see how to move a curve left and right.

## Example 4: *y* = (*x* − 1)^{2}

Note the brackets in this example - they make a big difference!

If we think about *y* = (*x* − 1)^{2} for a while, we realize the *y*-value will always be positive, except at *x* = 1 (where *y* will equal 0).

Before sketching, I will check another (easy) point to make sure I have the curve in the right place. Putting x = 0 is usually easy, so I substitute and get

*y* = (0 − 1)^{2}

= 1

So the curve passes through (0, 1).

Here is the graph of *y* = (*x* − 1)^{2}.

## Example 5: *y* = (*x* + 2)^{2}

With similar reasoning to the last example, I know that my curve is going to be completely above the *x*-axis, except at *x* = −2.

The "plus 2" in brackets has the effect of moving our parabola 2 units to the left.

## Rotating the Parabola

The original question from Anuja asked how to draw y^{2} = x − 4.

In this case, we don't have a simple *y* with an x^{2} term like all of the above examples. Now we have a situation where the parabola is rotated.

Let's go through the steps, starting with a basic rotated parabola.

## Example 6: *y*^{2} = *x*

The curve *y*^{2} = *x* represents a parabola rotated 90° to the right.

We actually have 2 functions,

*y* = √*x* (the top half of the parabola); and

*y* = −√*x* (the bottom half of the parabola)

Here is the curve *y*^{2} = *x*. It passes through (0, 0) and also (4,2) and (4,−2).

[Notice that we get 2 values of *y* for each value of *x* larger than 0. This is not a **function**, it is called a **relation**.]

## Example 7: (*y* + 1)^{2} = *x*

If we think about the equation (*y* + 1)^{2} = *x* for a while, we can see that *x* will be positive for all values of *y* (since any value squared will be positive) except *y* = −1 (at which point *x* = 0).

In the equation (*y* + 1)^{2} = *x*, the "plus 1" in brackets has the effect of moving our rotated parabola down one unit.

## Example 8: (*y* − 3)^{2} = *x*

Using similar reasoning to the above example, the "minus 3" in brackets has the effect of moving the rotated parabola up 3 units.

Finally we are ready to answer the question posed by Nuaja.

## Example 9: *y*^{2} = *x* − 2

You can hopefully imagine what is going to happen now. We have a *y*^{2} term , so it means it will be a rotated parabola.

When *x* = 2, *y* = 0. The value of *x* cannot be less than 2, otherwise when we try to evaluate *y* we would be trying to find the square root of a negative number. Since out numbers are all real numbers, *x* must be greater than or equal to 2.

The "minus 2" term has the effect of shifting our parabola 2 units to the right.

I hope you can see now that if the equation was *y*^{2} = *x* + 2 (with a "plus"), then we would need to shift our rotated parabola to the left by 2 units.

So Nuaja, I hope that answers your question.

Like all things, the best way how to learn graph sketching is through practice. Also, be observant and note the effect of **plus**, **minus** and **brackets** in each example.

See the 72 Comments below.