This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

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Transformations are a way for us to take elementary functions and move them around throughout the cartesian plane. There are three distinct categories of transformations. Namely:

- translations, which move the function up, down, left or right
- reflections, which reflect the function over a given axis
- stretches, which stretch or compress the function by a factor of a given number.

Transformations can be applied to many aspects of real life including architecture, aviation, engineering, and even simple day-to-day tasks like looking in a mirror or turning on a faucet.

Vertical translations are what causes a function to move up and down.

Given an elementary function \(y=f(x)\), simply adding a number c to the function will cause it to move c units UP. This looks like \(y=f(x)+c\).

To move a function DOWN c units, we subtract the number c from the function as follows: \(y=f(x)−c\).

**Example:**

Consider the elementary function \(y=x^2\). Look at what happens to the graph when we add or subtract 3 from the function:

The graph of \(y=x^2\):

The graph of \(y=x^2+3\) moves the function 3 units up:

The graph of \(y=x^2-3\) moves the function 3 units down:

We can translate a function horizontally (left and right) by adding or subtracting a number c to the x-coordinate inside the function.

Adding c to the x-coordinate \(y=f(x+c)\) will result in the function moving LEFT c units.

Subtracting c from the x-coordinate \(y=f(x-c)\) will move the functions RIGHT c units.

**Example:**

Consider the elementary function \(y=\sqrt{x}\). Look at what happens when we add or subtract 2 to the x-coordinate within the function:

The graph of \(y=\sqrt{x}\):

The graph of \(y=\sqrt{x+2}\) moves the function 2 units to the left:

The graph of \(y=\sqrt{x-2}\) moves the function 2 units to the right:

Reflections over the x-axis are represented by negating the function as follows:

\(y=−f(x)\)

Reflections over the y-axis are represented by negating the x-coordinate within the function as follows:

\(y=f(−x)\)

**Example:**

Consider the elementary function \(y=2^x\). Observe what happens when we negate the function or the x-coordinate:

The graph of \(y=2^x\):

The graph of \(y=-2^x\) results in a reflection over the X-AXIS:

The graph of \(y=2^{-x}\) results in a reflection over the Y-AXIS:

We express vertical stretches and compressions by multiplying the whole function by a number. If the number is greater than 1, we have a VERTICAL STRETCH but if it's less than 1, we have a VERTICAL COMPRESSION.

\(y=cf(x)\) represents a vertical stretch by a factor of c

\(y=\frac{1}{c}f(x)\) represents a vertical compression by a factor of c

**Example: **

Consider again the elementary function \(y=\sqrt{x}\). Observe what happens when we multiply the function by \(4\) and \(\frac{1}{4}\)

The graph of \(y=\sqrt{x}\):

The graph of \(y=4\sqrt{x}\) vertically stretches the function by a factor of 4:

The graph of \(y=\frac{1}{4}\sqrt{x}\) vertically compresses the function by a factor of 4:

For horizontal stretches and compressions, we multiply the x-coordinate within the function by a number. If the number is greater than 1, it's a HORIZONTAL COMPRESSION and if the number is less than 1, it's a HORIZONTAL STRETCH.

\(y=f(cx)\) represents a horizontal compression by a factor of c and

\(y=f(\frac{x}{c})\) represents a horizontal stretch by a factor of c.

**Example:**

Consider the elementary function \(y=x^3\). Observe what happens to the graphs when we multiply the x-coordinate by \(3\) and \(\frac{1}{3}\).

The graph of \(y=x^3\):

The graph of \(y=(3x)^3\) horizontally compresses the function by a factor of 3:

The graph of \(y=(\frac{x}{3})^3\) horizontally stretches the function by a factor of 3:

Transformation | Description |
---|---|

\(y=f(x)+c\) | Moves the function c units up |

\(y=f(x)-c\) | Moves the function c units down |

\(y=f(x+c)\) | Moves the function c units left |

\(y=f(x-c)\) | Moves the function c units right |

\(y=-f(x)\) | Reflects the function over the x-axis |

\(y=f(-x)\) | Reflects the function over the y-axis |

\(y=cf(x)\) | Stretches the function vertically by a factor of c |

\(y=\frac{1}{c}f(x)\) | Compresses the function vertically by a factor of c |

\(y=f(cx)\) | Compresses the function horizontally by a factor of c |

\(y=f(\frac{x}{c})\) | Stretches the function horizontally by a factor of c |

Graph the following functions. You can check your answer by graphing the function on a graphing software such as Desmos or Geogebra:

\(y=5(logx+3)−2\)

\(y=−3\sqrt{2x+1}-8\)

\(y=−5x^3+4\)

- Last Updated: Jul 17, 2024 3:24 PM
- URL: https://libraryguides.centennialcollege.ca/mathhelp
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