Reflecting parabolas

If you reflect parabolas you have to remember the following:

$x$ axis reflection multiply whole equation by $-1$
$y$ axis reflection replace $x$ with $-x$

Example 1

Reflect the following parabola through the $x$ axis and then the $y$ axis.

$x^2-2x+4$

First plot $x^2-2x+4$

Try different values of $x$

$x=3$	$y=3^2-2times3+4$	$=9-6+4$	$=7$
$x=2$	$y=2^2-2times2+4$	$=4-4+4$	$=4$
$x=1$	$y=1^2-2times1+4$	$=1-2+4$	$=3$
$x=0$	$y=0^2-2times0+4$	$=0-0+4$	$=4$
$x=-1$	$y=(-1)^2-2times(-1)+4$	$=1+2+4$	$=7$

Reflect this parabola through the x an y axis

Reflecting the parabola through $x$ axis, remember $x$ axis reflection multiply the whole equation by $-1$ .

$x^2-2x+4$ becomes

$-1(x^2)-1(-2x)+\-1times4$

$-1(x^2)+2x-4$

Now plot this against the original graph.

Try different values of $x$

$x=3$	$y=-1(3^2)+2times3+4$	$=-9+6-4$	$=-7$
$x=2$	$y=-1(2^2)+2times2+4$	$=-4+4-4$	$=-4$
$x=1$	$y=-1(1^2)+2times1+4$	$=-1+2-4$	$=-3$
$x=0$	$y=0^2+2times0+4$	$=0+0-4$	$=-4$
$x=-1$	$y=-1(-1^2)+2times(-1)+4$	$=-1-2-4$	$=-7$

Now reflect this parabola through the x axis

Now the second part of the question.

Reflecting the parabola through the $y$ axis.

Remember:

$y$ axis reflection replace $x$ with $-x$ .

$x^2-2x+4$ becomes

$(-x)^2-2times(-x)+4$

$(-x)^2+2x+4$

Now plot this against the original graph.

Try different values of $x$

$x=2$	$y=(-2)^2+2times2+4$	$=4+4+4$	$=12$
$x=1$	$y=(-1)^2+2times1+4$	$=1+2+4$	$=7$
$x=0$	$y=0^2+2times0+4$	$=0+0+4$	$=4$
$x=-1$	$y=(-1times(-1))^2+2times(-1)+4$	$=1-2+4$	$=3$
$x=-2$	$y=(-1times(-2))^2+2times(-2)+4$	$=4-4+4$	$=4$
$x=-3$	$y=(-1times(-3))^2+2times(-3)+4$	$=9-6+4$	$=7$

Now reflect through y

Example 2

Reflect the following parabola through the $x$ axis and then the $y$ axis.

$y=x^2-6x+4$

First plot $x^2-6x+4$

Try different values of $x$

$x=6$	$y=6^2-6times6+4$	$=36-36+4$	$=4$
$x=5$	$y=5^2-6times5+4$	$=25-30+4$	$=-1$
$x=4$	$y=4^2-6times4+4$	$=16-24+4$	$=-4$
$x=3$	$y=3^2-6times3+4$	$=9-18+4$	$=-5$
$x=2$	$y=2^2-6times2+4$	$=4-12+4$	$=-4$
$x=1$	$y=1^2-6times1+4$	$=1-6+4$	$=-1$
$x=0$	$y=0^2-6times0+4$	$=0-0+4$	$=4$

Reflect this parabola through the x an y axis

Reflecting the parabola through the $x$ axis remember $x$ axis reflection multiply whole equation by $-1$ .

$x^2-6x+4$

$-1(x^2)-1(-6x)+\-1times4$

$-1(x^2)+6x-4$

Now try and plot this against the original graph.

Try different values of $x$

$x=6$	$y=-1(6^2)+6times6-4$	$=-36+36-4$	$=-4$
$x=5$	$y=-1(5^2)+6times5-4$	$=-25+30-4$	$=1$
$x=4$	$y=-1(4^2)+6times4-4$	$=-16+24-4$	$=4$
$x=3$	$y=-1(3^2)+6times3-4$	$=-9+18-4$	$=5$
$x=2$	$y=-1(2^2)+6times2-4$	$=-4+12-4$	$=4$
$x=1$	$y=-1(1^2)+6times1-4$	$=-1+6-4$	$=1$
$x=0$	$y=0^2+6times0-4$	$=-0+0-4$	$=-4$

Reflect the x axis first

Reflecting the parabola through the $y$ axis.

Remember

$y$ axis reflection replace $x$ with $-x$ .

$x^2-6x+4$ becomes

$(-x)^2 -6(-x)+4$

$(-x)^2+6x+4$

Now plot this against the original graph.

Try different values of $x$

$x=2$	$y=(-2^2)+6times2+4$	$=4+12+4$	$=20$
$x=1$	$y=(-1^2)+6times1+4$	$=1+6+4$	$=11$
$x=0$	$y=0^2+6times0+4$	$=0+0+4$	$=4$
$x=-1$	$y=(-1times-1)^2+6times-1+4$	$=1-6+4$	$=-1$
$x=-2$	$y=(-1times-2)^2+6times-2+4$	$=4-12+4$	$=-4$
$x=-3$	$y=(-1times-3)^2+6times-3+4$	$=9-18+4$	$=-5$
$x=-4$	$y=(-1times-4)^2+6times-4+4$	$=16-24+4$	$=-4$
$x=-5$	$y=(-1times-5)^2+6times-5+4$	$=25-30+4$	$=-1$
$x=-6$	$y=(-1times-6)^2+6times-6+4$	$=36-36+4$	$=4$

Now reflect through the y axis