squareCircleZ

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

How do I know how the graph should look like: For example: y² = x – 2?

The first thing I recognize in that equation is the y² 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 x² term. See Conic Sections.)

Let’s start with the most basic parabola y = x² and build up to the required answer.

Example 1: y = x²

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 = x² − 2

The only difference with the first graph that I drew (y = x²) and this one (y = x² − 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 = x² + 3

The “plus 3″ means we need to add 3 to all the y-values that we got for the basic curve y = x². The resulting curve is 3 units higher than y = x². 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 y² = x − 4.

In this case, we don’t have a simple y with an x² 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: y² = x

The curve y² = 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 y² = 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: y² = x − 2

You can hopefully imagine what is going to happen now. We have a y² 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 y² = 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.

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