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This essential trains for: SAT-I, SAT-II, GMAT, AMC-10, AMC-12.

Translation up/down (in the y-direction)

If a is a positive number, i.e.:

and f(x) is a function, by adding a to each y-value, the graph of the function is shifted up by a units.

By subtracting a from each y-value, the graph of the function is shifted down by a units.

Examples:

Function | Translation | Translated Function |

up by 3/2 units | ||

down by 1.5 units |

Translation left/right (in the x-direction)

If a is a positive number, i.e.:

and f(x) is a function, by adding a to each x-value, the graph of the function is shifted left by a units.

By subtracting a from each x-value, the graph of the function is shifted right by a units.

This may seem completely counter-intuitive compared to the up/down translation, but think what happens if each x-value is increased by some amount - we obtain the y-values 'sooner', i.e. for lower x-values. A similar reasoning applies to the 'left shift'.

Examples:

Function | Translation | Translated Function |

right by π units | ||

left by 1 unit |

Horizontal Compression/Expansion

If κ is a positive, supra-unitary (i.e. greater than 1) number, i.e.:

multiplying a number x by κ results in a number larger than x in absolute value:

and dividing a number x by κ results in a number smaller than x in absolute value:

This means that the effects of the two transformations are to bring the graph points farther/closer to the origin along the x-axis, thus expanding/compressing the graph.

Multiply all x-values by a constant greater than 1 to compress the graph of the function in the x-direction with respect to the fixed point f(0):

Divide all x-values by a constant greater than 1 to expand the graph of the function in the x-direction with respect to the fixed point f(0):

Examples:

Function | Transformation | Transformed Function |

x-compression by a factor of 2 | ||

x-expansion by a factor of 3 | ||

x-expansion by a factor of 2 |

Vertical Compression/Expansion

If κ is a positive, supra-unitary (i.e. greater than 1) number, i.e.:

multiplying a number x by κ results in a number larger than x in absolute value:

and dividing a number x by κ results in a number smaller than x in absolute value:

This means that the effects of the two transformations are to bring the graph points farther/closer to the origin along the y-axis, thus expanding/compressing the graph.

Multiply all y-values by a constant greater than 1 to compress the graph of the function in the y-direction with respect to the fixed point f(x)=0:

Divide all y-values by a constant greater than 1 to expand the graph of the function in the y-direction with respect to the fixed point f(x)=0:

Examples:

Function | Transformation | Transformed Function |

y-compression by a factor of 2 | ||

y-expansion by a factor of 3 | ||

y-expansion by a factor of 2 |

Reflection

Reflecting a graph means to obtain a graph symmetric to the original one with respect to a given line.

Simple cases of reflection are: reflection over the x-axis, over the y-axis and over the line y=x (first bisector of the plane of coordinates).

Reflection over the x-axis is done by taking the opposite of each y-value of the graph. Note that the x-intercepts are fixed points under this transformation.

The reflected function is:

Reflection over the y-axis is done by replacing each y-value with the value obtained for -x.

Note that the y-intercepts are fixed points under this transformation.

The reflected function is:

Reflection over the first bisector of the plane is more tricky.

Reflecting a function over the line y=x means we interchange x and y. Therefore, the two functions are inverses of one another. However, some functions __do not have an inverse__ - the reflection results in a graph that does not satisfy __the vertical line test__ and, therefore, is not the graph of a function.

The equation of the reflected function is found by computing the inverse of the given function. Note that this is not always possible without implementing restrictions.

Example 1: Well-behaved (i.e. bijective) function that has an inverse:

Solve for x:

and interchange x with y to obtain the inverse:

Example 2: Reflecting a function that does not produce another function:

If we try to 'invert' this function we end up with an ambiguous definition (it is not well defined which value to choose: the positive, or the negative one):