# 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

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 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

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

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

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