9435
/www/nrich/html/content/id/9435/
1
2012-09-18T13:00:44
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<mdo:image src="function.png" style="float: right;"></mdo:image><br></br>
<br></br>
<br></br>
The curve $y=x^{2}-6x+11$ is rotated through $180^\circ$ about the origin.<br></br>
<br></br>
What is the equation of the new curve? <br></br>
<br></br>
<br></br>
<br></br>
<br></br>
<br></br>
<br></br>
<br></br>
<br></br>
<br></br>
<br></br>
<br></br>
If you liked this problem, <a href="/773">here is an NRICH task</a> which challenges you to use similar mathematical ideas.<br></br>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<br></br>
$x^{2} - 6x + 11 = (x - 3)^{2} + 2$.<br></br>
<br></br>
When the curve is rotated $180^\circ$ about the origin, the equation of the new curve will be<br></br>
$y = -(x + 3)^{2} - 2$<br></br>
$= -x^{2} - 6x - 9 - 2$<br></br>
$= -x^{2} - 6x - 11$.<br></br>
<br></br>
<br></br>
Note: the image of point $(a,b)$ under a $180^\circ$ rotation about the origin is the point $(-a,-b)$. An alternative method, therefore, is to replace $x$ and $y$ in the original equation by $-x$ and $-y$ respectively.<br></br>
<br></br>
<br></br>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">Graphs of Functions - Stage 4 Short Problem, UKMT 1997-2006 p 73 Q18</mdoxml>
2
3
0
0
0
1
0
Weekly Problem 48 - 2012
The curve $y=x^2−6x+11$ is rotated through $180^\circ$ about the origin. What is the equation of the new curve?
Secondary Mapping Document
Graphs of functions
Secondary Mapping Document
DisplayCabinet