Summary
Highlights
The video introduces translations as the first topic in the Math 30-1 transformations unit, focusing on horizontal and vertical movements. It begins by reviewing quadratic functions from Math 20, specifically how the vertex form (e.g., y = (x - 4)^2 + 2) indicates a translation of 4 units to the right and 2 units up from the parent function y = x^2. The key takeaway is that every coordinate on the graph undergoes the exact same translation, maintaining the shape of the function.
A translation moves a graph up, down, left, or right without distorting its shape. Every coordinate on the graph is moved by the same amount in the same direction. The video elaborates on this, emphasizing that the function's overall form remains identical, only its position changes on the coordinate plane.
Vertical translations are introduced using the notation f(x) + K, where K determines the vertical shift. A positive K value means the graph moves up, and a negative K value means it moves down. If K is on the left side of the equation (e.g., y - K = f(x)), the sign is flipped to determine the direction of the vertical shift. Only the y-coordinate of every point changes, while the x-coordinate remains the same.
Horizontal translations use the notation f(x - H), where H dictates the horizontal movement. Unlike vertical translations, the horizontal shift is 'opposite' to the sign; f(x - H) means H units to the right, and f(x + H) means H units to the left. Only the x-coordinate of every point changes during a horizontal translation.
When both horizontal and vertical translations are applied, the general form becomes f(x - H) + K. The video demonstrates how to describe these combined transformations in words and how to apply them to specific coordinates using mapping (x, y) → (x + H, y + K), remembering that H is subtracted from x in the function notation but added to x in the coordinate mapping for a rightward shift. An example with an absolute value function is used to illustrate these concepts.
The discussion shifts to how translations affect the domain (all possible x-values) and range (all possible y-values) of a function. It reviews traditional set notation for domain and range from earlier math classes. The video then introduces interval notation as a new, often simpler way to express domain and range, using round brackets for excluded values (open circles or infinity) and square brackets for included values (closed circles).
An example demonstrates how to find the new domain and range of a transformed graph without necessarily sketching it. By knowing the original domain and range and the translation amounts (e.g., 2 units left and 5 units down), one can directly adjust the x-values for the domain and the y-values for the range. The video emphasizes that transformations applied to the function directly correspond to operations on the domain and range values.
The final examples involve identifying the transformations that have occurred between an original (dotted line) and a transformed (solid line) graph. From these identified translations, the corresponding new equation in the form y - K = f(x - H) can be written. The impact on domain and range is also briefly revisited, reinforcing the direct relationship between transformations and changes to the domain and range values.