Question

Michele Lake and Coral Lake, located near the Columbia Ice Fields, are the only two lakes in Alberta in which rare golden trout live. Suppose the graph represents the number of golden trout in Michelle Lake in the years since 1970. Let the function \(f(t)\) represent the number of fish in Michelle Lake since \(1970 .\) Describe an event or a situation for the fish population that would result in the following transformations of the graph. Then, use function notation to represent the transformation. a) a vertical translation of 2 units up. b) a horizontal translation of 3 units to the right.

Short answer

a) \( f(t) + 2 \); b) \( f(t-3) \).

Step-by-step solution

Understanding the Vertical Translation

A vertical translation shifts the graph up or down. For a vertical translation of 2 units up, every point on the graph moves 2 units higher. This can be thought of as an increase in the overall fish population by a constant number.

Representing the Vertical Translation in Function Notation

To express this transformation mathematically, add 2 to the function: Given function: oIf the transformation is a vertical translation of 2 units up, the new function is: \[ f(t) = f(t) + 2 \]

Understanding the Horizontal Translation

A horizontal translation shifts the graph left or right. For a horizontal translation of 3 units to the right, every point on the graph moves 3 units to the right. This means the same number of fish appears 3 years later.

Representing the Horizontal Translation in Function Notation

To express this transformation mathematically, subtract 3 from the variable inside the function: Given function: oIf the transformation is a horizontal translation of 3 units to the right, the new function is: \[ f(t) = f(t-3) \]

Key concepts

Vertical Translation

A vertical translation involves shifting a graph up or down. This affects the y-values (output) of the function. If the number of golden trout in Michele Lake increases steadily by a constant number, this is a vertical translation up. For instance, suppose we have a function, \( f(t) \), representing the fish population, and this population experiences a vertical increase. To represent an upward shift of 2 units in our function, we adjust every output value by adding 2. Hence, the new function is expressed as \( f(t) + 2 \).
In essence, each point on the graph moves 2 units higher. Unlike vertical stretches or compressions which alter the shape of the graph, vertical translations simply shift it up or down, keeping the shape intact. Understanding vertical translations is crucial because they demonstrate how consistent changes affect an entire system proportionally.

Horizontal Translation

Horizontal translation shifts a graph left or right, altering the x-values (inputs) of the function. This modification doesn't change the shape of the graph, but shifts it along the x-axis. If the number of golden trout appears 3 years later, this results in a horizontal translation to the right.
Suppose \( f(t) \) is the function representing the fish population in Michele Lake. To achieve a horizontal translation of 3 units to the right, we replace \( t \) with \( (t - 3) \). Thus, the new function becomes \( f(t - 3) \).
This means every point on the graph now shifts 3 units to the right, representing events occurring 3 years later. It's vital to grasp horizontal translations as they showcase how time-frame changes in data interpretation affect the function.

Function Notation

Function notation is a streamlined way to represent and understand relationships between variables. It uses the format \( f(x) \), where \( f \) is the function name, and \( x \) represents the input.
For example, \( f(t) \) shows that \( f \) is a function depending on the variable \( t \). When graph transformations occur, function notation helps succinctly represent these changes. A vertical translation of 2 units up would be noted as \( f(t) \) shifting to \( f(t) + 2 \).
Similarly, a horizontal translation of 3 units to the right modifies the input, becoming \( f(t - 3) \). This notation simplifies complex transformations, allowing easier communication and understanding of mathematical manipulations.

Graph Transformations

Graph transformations include various methods of altering the position, shape, or orientation of a graph. The primary types include translations (both vertical and horizontal), stretches, compressions, and reflections.

Vertical and horizontal translations are pivotal because they deal with shifting the graph without altering its shape. As seen with the golden trout example:
\ul \ - Vertical Translation: Each point on the graph moves 2 units up, represented by \( f(t) + 2 \).
\ - Horizontal Translation: Every point shifts 3 units to the right, represented by \( f(t - 3) \).
These transformations are crucial for understanding how data changes over time (horizontal) or due to consistent adjustments (vertical). Recognizing these changes assists in comprehensively analyzing function behavior and graph movements.