{"id":2301,"date":"2009-04-09T17:37:34","date_gmt":"2009-04-09T09:37:34","guid":{"rendered":"http:\/\/www.intmath.com\/blog\/?p=2301"},"modified":"2019-11-18T20:10:23","modified_gmt":"2019-11-18T12:10:23","slug":"how-to-draw-y2-x-2","status":"publish","type":"post","link":"https:\/\/www.intmath.com\/blog\/mathematics\/how-to-draw-y2-x-2-2301","title":{"rendered":"How to draw y^2 = x - 2?"},"content":{"rendered":"

Nuaja, a subscriber to the IntMath Newsletter, wrote recently:<\/p>\n

\n

How do I know how the graph should look like: For example: y<\/em>2<\/sup> = x<\/em> - 2?<\/span><\/p>\n<\/blockquote>\n

The first thing I recognize in that equation is the y<\/em>2<\/sup> term, which tells me it will be a parabola<\/strong>. (It won't be a circle, ellipse or hyperbola because there is an x<\/em> term, but no x<\/em>2<\/sup> term. See Conic Sections<\/a>.)<\/p>\n

Let's start with the most basic parabola y = x2<\/sup> and build up to the required answer.<\/p>\n

Example 1: y<\/i> = x<\/i>2<\/sup><\/span><\/h2>\n

You could draw up a table and calculate the y<\/em>-values for a set of x<\/em>-values, like this:<\/p>\n\n\n\n\n
x<\/em><\/th>\n-4<\/td>\n-3<\/td>\n-2<\/td>\n-1<\/td>\n0<\/td>\n1<\/td>\n2<\/td>\n3<\/td>\n4<\/td>\n<\/tr>\n
y<\/em><\/th>\n16<\/td>\n9<\/td>\n4<\/td>\n1<\/td>\n0<\/td>\n1<\/td>\n4<\/td>\n9<\/td>\n16<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

This gives us a series of points (-4,16), (-3,9), (-2,4) up to (4,16).<\/p>\n

You then join these dots with a smooth curve and get something like the following.<\/p>\n

\"y<\/p>\n

Notice that the vertex of the parabola (the \"pointy\" end) is at the origin, (0, 0).<\/p>\n

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<\/strong> the graph type (from the equation) and then know how to sketch it in the right place and with the right orientation.<\/p>\n

I will consider the effect of small changes to the equation and then sketch my curve.<\/p>\n

All of the following graphs have the same size and shape<\/strong> as the above curve. I am just moving that curve around to show you how it works.<\/p>\n

Example 2: y<\/em> = x<\/em>2<\/sup> − 2<\/h2>\n

The only difference with the first graph that I drew (y<\/em> = x<\/em>2<\/sup>) and this one (y<\/em> = x<\/em>2<\/sup> − 2) is the \"minus 2\". The \"minus 2\" means that all the y<\/em>-values for the graph need to be moved down by 2 units.<\/p>\n

So we just take our first curve and move it down 2 units. Our new curve's vertex is at −2 on the y<\/em>-axis.<\/p>\n

\"parabola-2\"<\/p>\n

Next, we see how to move the curve up (rather than down).<\/p>\n

Example 3: y<\/em> = x<\/em>2<\/sup> + 3<\/h2>\n

The \"plus 3\" means we need to add 3 to all the y<\/em>-values that we got for the basic curve y<\/em> = x<\/em>2<\/sup>. The resulting curve is 3 units higher than y<\/em> = x<\/em>2<\/sup>. Note that the vertex of the curve is at (0, 3) on the y<\/em>-axis.<\/p>\n

\"parabola-3\"<\/p>\n

Next we see how to move a curve left and right.<\/p>\n

Example 4: y<\/em> = (x<\/em> − 1)2<\/sup><\/h2>\n

Note the brackets in this example - they make a big difference!<\/p>\n

If we think about y<\/em> = (x<\/em> − 1)2<\/sup> for a while, we realize the y<\/em>-value will always be positive, except at x<\/em> = 1 (where y<\/em> will equal 0).<\/p>\n

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<\/p>\n

y<\/em> = (0 − 1)2<\/sup><\/p>\n

= 1<\/p>\n

So the curve passes through (0, 1).<\/p>\n

Here is the graph of y<\/em> = (x<\/em> − 1)2<\/sup>.<\/p>\n

\"parabola-4\"<\/p>\n

Example 5: y<\/em> = (x<\/em> + 2)2<\/sup><\/h2>\n

With similar reasoning to the last example, I know that my curve is going to be completely above the x<\/em>-axis, except at x<\/em> = −2.<\/p>\n

The \"plus 2\" in brackets has the effect of moving our parabola 2 units to the left.<\/p>\n

\"parabola-5\"<\/p>\n

Rotating the Parabola<\/h2>\n

The original question from Anuja asked how to draw y2<\/sup> = x − 4.<\/p>\n

In this case, we don't have a simple y<\/em> with an x2<\/sup> term like all of the above examples. Now we have a situation where the parabola is rotated.<\/p>\n

Let's go through the steps, starting with a basic rotated parabola.<\/p>\n

Example 6: y<\/em>2<\/sup> = x<\/em><\/h2>\n

The curve y<\/em>2<\/sup> = x<\/em> represents a parabola rotated 90\u00b0 to the right.<\/p>\n

We actually have 2 functions,<\/p>\n

y<\/em> = √x<\/em> (the top half of the parabola); and<\/p>\n

y<\/em> = −√x<\/em> (the bottom half of the parabola)<\/p>\n

Here is the curve y<\/em>2<\/sup> = x<\/em>. It passes through (0, 0) and also (4,2) and (4,−2).<\/p>\n

\"parabola-6\"<\/p>\n

[Notice that we get 2 values of y<\/em> for each value of x<\/em> larger than 0. This is not a function<\/strong>, it is called a relation<\/strong>.]<\/p>\n

Example 7: (y<\/em> + 1)2<\/sup> = x<\/em><\/h2>\n

If we think about the equation (y<\/em> + 1)2<\/sup> = x<\/em> for a while, we can see that x<\/em> will be positive for all values of y<\/em> (since any value squared will be positive) except y<\/em> = −1 (at which point x<\/em> = 0).<\/p>\n

In the equation (y<\/em> + 1)2<\/sup> = x<\/em>, the \"plus 1\" in brackets has the effect of moving our rotated parabola down one unit.<\/p>\n

\"parabola-7\"<\/p>\n

Example 8: (y<\/em> − 3)2<\/sup> = x<\/em><\/h2>\n

Using similar reasoning to the above example, the \"minus 3\" in brackets has the effect of moving the rotated parabola up 3 units.<\/p>\n

\"parabola-8\"<\/p>\n

Finally we are ready to answer the question posed by Nuaja.<\/p>\n

Example 9: y<\/em>2<\/sup> = x<\/em> − 2<\/h2>\n

You can hopefully imagine what is going to happen now. We have a y<\/em>2<\/sup> term , so it means it will be a rotated parabola.<\/p>\n

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

The \"minus 2\" term has the effect of shifting our parabola 2 units to the right.<\/p>\n

\"parabola-9\"<\/p>\n

I hope you can see now that if the equation was y<\/em>2<\/sup> = x<\/em> + 2 (with a \"plus\"), then we would need to shift our rotated parabola to the left by 2 units.<\/p>\n

So Nuaja, I hope that answers your question.<\/p>\n

Like all things, the best way how to learn graph sketching is through practice. Also, be observant and note the effect of plus<\/strong>, minus<\/strong> and brackets<\/strong> in each example.<\/p>\n

See the 72 Comments<\/a> below.<\/p>\n","protected":false},"excerpt":{"rendered":"

How do we draw parabolas that are moved around the x-y axes, and rotated?<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_mo_disable_npp":""},"categories":[4],"tags":[134],"_links":{"self":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/2301"}],"collection":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/comments?post=2301"}],"version-history":[{"count":2,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/2301\/revisions"}],"predecessor-version":[{"id":12272,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/2301\/revisions\/12272"}],"wp:attachment":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/media?parent=2301"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/categories?post=2301"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/tags?post=2301"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}