Math 30



Math 30 Transformations

Function [pic]

Replacement Transformation Image Point

[pic] if [pic] [pic]

[pic] if [pic] [pic]

[pic]

Replacement Transformation Image Point

[pic] if [pic] [pic]

[pic] if [pic] [pic]

[pic]

Replacement Transformation Image Point

[pic] vertical stretch about the x-axis [pic]

by a factor of [pic]

Replacement Transformation Image Point

[pic] horizontal stretch about the [pic]

y-axis by a factor of [pic]

Replacement Transformation Image Point

[pic] reflection in the x-axis [pic]

(vertically)

Replacement Transformation Image Point

[pic] reflection in the y-axis [pic]

(horizontally)

Replacement Transformation Image Point

[pic] reflection in the line [pic] [pic]

Transformations

[pic] [pic]

Function has moved h units left and k units up.

Image point [pic]

[pic] [pic]

Function has been stretched vertically by a factor of 2

Image point [pic]

[pic] [pic]

Function has been stretched vertically by a factor of [pic], reflected about the x-axis, stretched horizontally by a factor of 3, horizontally translated 3 units left and vertically translated 2 units down.

Image point [pic]

[pic] Transformation Replacement Image Point

[pic] [pic] [pic]

[pic] [pic] [pic]

[pic] [pic] [pic]

[pic] [pic] [pic]

[pic] [pic] [pic]

[pic] [pic] [pic]

[pic] [pic] [pic]

Domain and Range

Recall the definitions for domain and range:

Domain: the set of x-values represented by the graph or equation of the function.

Range: the set of y-values represented by the graph or the equation of the function

The following table shows how transformations may affect the domain and range of a function.

|Transformation |Image |Domain |Range |

|[pic] |[pic] |No |Yes |

|[pic] |[pic] |Yes |No |

|[pic] |[pic] |No |Yes |

|[pic] |[pic] |Yes |No |

|[pic] |[pic] |No |Yes |

|[pic] |[pic] |Yes |No |

The graph has a domain of [pic]

and range of [pic]. State the new

domain and range of the function

[pic].

The function is stretched vertically

by a factor of [pic] and translated horizontally 2 units left.

So, [pic]

New domain :[pic], new range: [pic]

Intercepts

Recall the definition of the x-intercept and y-intercept.

x-intercept: the x-coordinate where the graph of a line or a function intersects the x-axis. [pic]

y-intercept: the y-coordinate where the graph of a line or a function intersects the y-axis. [pic]

To determine the intercepts algebraically:

x-intercept: set [pic] y-intercept: set [pic]

Graphs of Transformations

[pic] The graph of [pic]is shown below:

Key Image Points:

[pic]

Sketch the graph of :

[pic] [pic]

translated horizontally or shifted vertically stretched by a factor of

2 units left and translated 2 and reflected about the y-

vertically or shifted 3 units axis (i.e. horizontally)

down

Image points changes:

[pic] [pic]

Given the graph, determine the equation of [pic].

[pic] the graph of [pic], where [pic]is a quadratic function is shown below. The y-intercept of [pic] is [pic]and the maximum point of [pic]. What is the equation of [pic]?

[pic]

Solution:

The reciprocal of [pic]is 9, so the y-intercept will be 9. The parabola will have a minimum point at [pic]which is the vertex. This illustrates that there is a horizontal shift of 2 and a vertical shift of 1; therefore the equation of this function is [pic] or [pic]. To find the value of a, substitute [pic] the new y-intercept into the equation. The value of [pic], so the final equation is [pic]

-----------------------

Vertical Translations [pic]

Horizontal Translations [pic]

Vertical Stretches [pic]

Horizontal Stretches [pic]

Vertical Reflections [pic]

Horizontal Reflections [pic]

Inverse Functions [pic]

Remember to factor out b and move the parameters associated with y back to the left side.

Remember the ordering of transformations:

1. Stretches/Reflections

2. Translations

For stretches, the replacement and the factor are reciprocals of each other.

Remember the notation for inverse [pic]

[pic]

[pic]

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