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` |
