Learn how to do reflections functions in transformations of graphs. Practice reflection of graph in y-axis.

- We define modulus function

\(f(x)=|x|=\begin {cases}{x\,\,\,\, if\,\,\, x \geq0\\-x\,\,\,\,\,if\,\,\,\,\,x<0}\end {cases}\)

The idea of the function is to return the absolute value of the input quantity \(x\)

\(\therefore \;|5|=5\) and \(|-6|=-(-6)=6\)

- The above is the graph of \(f(x)=|x|\) v/s \(x.\)It resembles \(y=x\) line for \(x\geq 0\) and \(y=-x\) line for \(x<0\).
- The negative modulus function is defined as

\(f(x)=-|x|=\begin {cases}{-x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, if\,\,\, x \geq0\\-(-x)=x\,\,\,\,\,if\,\,\,\,\,x<0}\end {cases}\)

The above is the graph of \(f(x)=-|x|\) v/s \(x\).

- To obtain \(-f(x)\) from \(f(x)\) we find the value of the function at particular \(x\) and then negate the value obtained while in \(f(-x)\), we negate the value of \(x\) first and then find the value of the function at that negated value.
- Sketch of \(-f(x)\) is obtained by taking reflection of graph of \(f(x)\) about \(x\) axis.
- Sketch of \(f(-x)\) is obtained by taking reflection of graph of \(f(x)\) about \(y\) axis.

A 11

B –21

C 20

D –15

Suppose we know the graph of \(f(x)\) then graph of following function can be constructed using some appropriate transformation on \('f'\).

S.No. | Function | Transformation on \(f(x)\) |
---|---|---|

1 | \(y=c\,f(x)\;\;(c>0)\) | Stretch vertically by a factor of 'c'. |

2 | \(y=c\,f(x)\;\;(0<c<1)\) | Shrink vertically by a factor of 'c'. |

3 | \(y=f(c\,x)\;\;(c>1)\) | Shrink horizontally by a factor of 'c'. |

4 | \(y=f(c\,x)\;\;(0<c<1)\) | Stretch horizontally by a factor of 'c'. |

5 | \(y=-f(x)\) | Take reflection about \(x\)-axis. |

6 | \(y=f(-x)\) | Take reflection about \(y\)-axis. |

e.g.

If we know the graph of a certain function say \(f(x)\) then we can sketch the graphs of many functions which are related to \('f'\) by making some appropriate changes in the graph of \('f'\).

The following tables show the function on the left and the transformation or change required to be done to the graph of \('f'\) obtain the graph of this function.

S.No. | Function | Transformation on \(f(x)\) |
---|---|---|

1 | \(y=f(x)+c\;\;(c>0)\) | Shift the graph 'c' units upwards |

2 | \(y=f(x)-c\;\;(c>0)\) | Shift the graph 'c' units downwards |

3 | \(y=f(x-c)\;\;(c>0)\) | Shift the graph 'c' units to right |

4 | \(y=f(x+c)\;\;(c>0)\) | Shift the graph 'c' units to left. |

e.g.

We have shifted the graph of\( f(x)\) by 1 units in the upward direction.

- To sketch the graph of \(f(-x)\) when graph of \(f(x)\) is known or given, we take the reflection of entire graph in y-axis (y-axis acts as a mirror).

- To sketch three graph of |f(x)| from the graph of 'f', we retain the part of the graph which is above x-axis as it is (|f(x)| = f(x) if \(f(x) \geq 0\)) and part below x-axis is reflected about x-axis.

e.g.

- To sketch the graph of \(-f(x)\) when graph of \(f(x)\) is known or given, we take the reflection of the entire graph in x-axis. (x-axis acts as a mirror)