Translations of graphs
Introduction to Translations of Graphs
- Translations of graphs involve shifting the graph of a function to a different location in the Cartesian plane without changing its shape or orientation.
- A translation can occur along the x-axis, the y-axis, or both simultaneously.
- In mathematical notation, a translation can be described as (x, y) → (x + a, y + b). This indicates all points (x, y) on the original graph are moved ‘a’ units horizontally and ‘b’ units vertically to reach their new position in the translated graph.
Horizontal Translations
- A horizontal translation moves the graph to the left or the right along the x-axis.
- A translation of
(x, y) → (x + a, y)involves moving the graph a units to the left if ‘a’ is negative, or a units to the right if ‘a’ is positive.
Vertical Translations
- A vertical translation involves moving the graph up or down along the y-axis.
- A translation of
(x, y) → (x, y + b)means the graph moves b units down if ‘b’ is negative, or b units up if ‘b’ is positive.
Combining Translations
- A graph can be translated both horizontally and vertically at the same time.
- This is denoted as
(x, y) → (x + a, y + b), meaning the graph is moved a units horizontally (left for negative ‘a’, right for positive ‘a’) and b units vertically (down for negative ‘b’, up for positive ‘b’).
Examples of Translations
- For instance, consider the function f(x) = x². A translation
(x, y) → (x + 2, y - 3)would result in the new function f(x) = (x - 2)² - 3. - The function f(x) = x³ when translated through
(x, y) → (x - 1, y + 2)would result in the new function f(x) = (x + 1)³ + 2.
Conclusion
- Understanding how to perform and interpret graph translations is vital in algebra. It allows the exploration of how alterations to function rules can affect the graph of the function.
- Clear comprehension of the effects of positive and negative translations on the coordinates of a graph enables the prediction of how function graphs will behave under these transformations.
- Regular practice with different function graphs and translations helps reinforce learnt concepts and aids in efficient problem-solving.