{"id":6255,"date":"2011-06-30T12:51:52","date_gmt":"2011-06-30T04:51:52","guid":{"rendered":"http:\/\/www.intmath.com\/blog\/?p=6255"},"modified":"2022-04-27T10:17:52","modified_gmt":"2022-04-27T02:17:52","slug":"how-to-reflect-a-graph-through-the-x-axis-y-axis-or-origin","status":"publish","type":"post","link":"https:\/\/www.intmath.com\/blog\/mathematics\/how-to-reflect-a-graph-through-the-x-axis-y-axis-or-origin-6255","title":{"rendered":"How to reflect a graph through the x-axis, y-axis or Origin?"},"content":{"rendered":"

This mail came in from reader Stuart recently:<\/p>\n

\n

Can you explain the principles of a graph involving y<\/em> = \u2212f<\/em>(x<\/em>)<\/span> being a reflection of the graph y<\/em> = f<\/em>(x<\/em>)<\/span> in the x<\/em>-axis and the graph of y<\/em> = f<\/em>(\u2212x<\/em>)<\/span> a reflection of the graph y<\/em> = f<\/em>(x<\/em>) <\/span> in the y-<\/em>axis?<\/p>\n

Thanks<\/p>\n<\/blockquote>\n

My reply<\/h2>\n

Hello Stuart<\/p>\n

Let's see what this means via an example.<\/p>\n

Let f<\/em>(x<\/em>) = 3x<\/em> + 2<\/span><\/p>\n

If you are not sure what it looks like, you can graph it using this graphing facility<\/a>.<\/p>\n

You'll see it is a straight line, slope 3 (which is positive, i.e. going uphill as we go left to right) and y<\/em>-intercept 2.<\/p>\n

Now let's consider \u2212f<\/em>(x<\/em>)<\/span>.<\/p>\n

This gives us<\/p>\n

\u2212f<\/em>(x<\/em>) =<\/em> \u22123x<\/em> \u2212 2<\/span><\/p>\n

Our new line has negative slope (it goes down as you scan from left to right) and goes through \u22122 on the y<\/em>-axis.<\/p>\n

When you graph the 2 lines on the same axes, it looks like this:<\/p>\n

\"graph<\/p>\n

Note that if you reflect the blue graph (y<\/em> = 3x<\/em> + 2)<\/span> in the x<\/em>-axis, you get the green graph (y<\/em> = \u22123x<\/em> \u2212 2)<\/span> (as shown by the red arrows).<\/p>\n

What we've done is to take every y<\/em>-value and turn them upside down (this is the effect of the minus<\/strong> out the front).<\/p>\n

Now for f<\/em>(\u2212x<\/em>)<\/h2>\n

Similarly, let's do f<\/em>(\u2212x<\/em>)<\/span>.<\/p>\n

Since f<\/em>(x<\/em>) = 3x<\/em> + 2<\/span>, then<\/p>\n

f<\/em>(\u2212x<\/em>) = \u22123x<\/em> + 2<\/span> (replace every \"x<\/em>\" with a \"\u2212x<\/em>\").<\/p>\n

Now, graphing those on the same axes, we have:<\/p>\n

\"graph<\/p>\n

Note that the effect of the \"minus\" in f<\/em>(\u2212x<\/em>)<\/span> is to reflect the blue original line (y<\/em> = 3x<\/em> + 2)<\/span> in the y<\/em>-axis, and we get the green line, which is (y<\/em> = \u22123x<\/em> + 2)<\/span>. The green line also goes through 2 on the y<\/em>-axis.<\/p>\n

\"24x7<\/a><\/p>\n

Further Example<\/h2>\n

Here's an example using a cubic graph.<\/p>\n

Blue graph:<\/strong> f<\/em>(x<\/em>) = x<\/em>3<\/sup> \u2212 3x<\/em>2<\/sup> + x<\/em> \u2212 2<\/span><\/p>\n

Reflection in x-axis (green):<\/strong> \u2212f<\/em>(x<\/em>) = \u2212x<\/em>3<\/sup> + 3x<\/em>2<\/sup> \u2212 x<\/em> + 2<\/span><\/p>\n

\"graph<\/p>\n

Now to reflect in the y-axis.<\/p>\n

\"graph<\/p>\n

Blue graph:<\/strong> f<\/em>(x<\/em>) = x<\/em>3<\/sup> \u2212 3x<\/em>2<\/sup> + x<\/em> \u2212 2<\/span><\/p>\n

Reflection in y-axis (green): <\/strong>f<\/em>(\u2212x<\/em>) = \u2212x<\/em>3<\/sup> \u2212 3x<\/em>2<\/sup> \u2212 x<\/em> \u2212 2<\/span><\/p>\n

Even and Odd Functions<\/h2>\n

We really should mention even and odd functions<\/strong> before leaving this topic.<\/p>\n

For each of my examples above, the reflections in either the x<\/em>- or y<\/em>-axis produced a graph that was different<\/strong>. But sometimes, the reflection is the same as the original graph. We say the reflection \"maps on to\" the original.<\/p>\n

Even Functions<\/strong><\/p>\n

An even <\/strong>function has the property f<\/em>(\u2212x<\/em>) = f<\/em>(x<\/em>)<\/span>. That is, if we reflect an even function in the y<\/em>-axis, it will look exactly like the original.<\/p>\n

An example of an even function is f<\/em>(x<\/em>) = x<\/em>4<\/sup> \u2212 29x<\/em>2<\/sup> + 100<\/span><\/p>\n

\"graph<\/p>\n

The above even function is equivalent to:<\/p>\n

f<\/em>(x<\/em>) = (x + <\/em>5)(x <\/em> + 2)(x <\/em>\u2212 2)(x \u2212<\/em> 5)<\/span><\/p>\n

Note if we reflect the graph in the y<\/em>-axis, we get the same graph (or we could say it \"maps onto\" itself).<\/p>\n

Odd Functions<\/strong><\/p>\n

An odd <\/strong>function has the property f<\/em>(\u2212x<\/em>) = \u2212f<\/em>(x<\/em>)<\/span>.<\/p>\n

This time, if we reflect our function in both <\/strong>the x<\/em>-axis and y<\/em>-axis, and if it looks exactly like the original, then we have an odd function.<\/p>\n

This kind of symmetry is called origin symmetry<\/strong>. An odd function either passes through the origin (0, 0)<\/span> or is reflected through the origin.<\/p>\n

An example of an odd function is f<\/em>(x<\/em>) = x<\/em>3<\/sup> \u2212 9x<\/em><\/span><\/p>\n

\"graph<\/p>\n

The above odd function is equivalent to:<\/p>\n

f<\/em>(x<\/em>) = x<\/em>(x + <\/em>3)(x <\/em>\u2212 3)<\/span><\/p>\n

Note if we reflect the graph in the x<\/em>-axis, then the y<\/em>-axis, we get the same graph.<\/p>\n

More examples of Even and Odd functions<\/h2>\n

There some more examples on this page: Even and Odd Functions<\/a><\/p>\n

Knowing about even and odd functions is very helpful when studying Fourier Series<\/a>.<\/p>\n

I hope that all makes sense, Stuart.<\/p>\n

 <\/p>\n

\"24x7<\/a><\/p>\n

 <\/p>\n

See the 8 Comments<\/a> below.<\/p>\n","protected":false},"excerpt":{"rendered":"

\"What<\/a>A reader asks how to graph f(-x) and -f(x), and what does it mean?<\/p>\n","protected":false},"author":5,"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\/6255"}],"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\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/comments?post=6255"}],"version-history":[{"count":3,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/6255\/revisions"}],"predecessor-version":[{"id":13071,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/6255\/revisions\/13071"}],"wp:attachment":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/media?parent=6255"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/categories?post=6255"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/tags?post=6255"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}