Problem 40 - Precalculus : Building Concepts and Connections

Question

Use the verbal description to find an algebraic expression for the function. The graph of the function \(f(t)\) is formed by translating the graph of \(h(t)=t^{2} 2\) units to the right and 6 units upward.

Step-by-Step Solution

Answer

The function \(f(t)\) is given by \(f(t) = (t-2)^2 + 6\).

Step 1: Recognize Translations

A translation is a transformation that moves every point of a figure the same distance in the same direction. In this case, the graph of the function is moving 2 units to the right and 6 units upwards. This information helps us understand the changes made to the function h(t) = \(t^2\).

Step 2: Translate Horizontally

In a function, a horizontal shift or translation involves replacing the variable t with \(t-2\) (because it is a right shift by 2 units). This gives us the function \(h(t) = (t-2)^2\).

Step 3: Translate Vertically

A vertical shift involves adding a constant to the function. Because the graph is shifted 6 units upward, this translates to adding 6 to the function obtained in the previous step. This gives us \(h(t) = (t-2)^2 + 6\).

Step 4: Identify the New Function

The new function obtained after both translations, \(f(t)\), is therefore \(f(t) = (t-2)^2 + 6\).

Key Concepts

Algebraic Expression

When we talk about an algebraic expression, we refer to a mathematical phrase that can include numbers, variables, and operation symbols. These expressions are used to describe relationships and to represent values in a succinct manner.

For instance, the expression t² is algebraic and denotes the square of a variable t. When we adjust this expression to account for changes like shifts and transformations, we modify it while maintaining its algebraic nature. In our example, we translated t² by applying shifts, which led to a new algebraic expression to describe the transformed function.

Horizontal Shift

A horizontal shift occurs when a function moves left or right along the x-axis. In the function transformation, this can be visualized by shifting the graph of a function without changing its shape.

To apply a horizontal shift to a function, such as h(t) = t², we replace the input variable with \(t - h\) if the shift is to the right by h units, or with \(t + h\) if the shift is to the left by h units. In the given exercise, since the shift is 2 units to the right, we substitute t with \(t - 2\) resulting in the expression \(h(t) = (t-2)^2\).

Vertical Shift

A vertical shift translates a graph up or down along the y-axis. It is akin to lifting or lowering the whole graph without distorting its outline.

To carry out a vertical shift, we add or subtract a constant value k from the entire function. If we're moving the graph up by k units, we add k; if moving it down, we subtract k. In our exercise, where the function h(t) after the horizontal shift was \( (t-2)^2 \) and needed to be moved 6 units upward, we added 6 to get the new vertically shifted function \(h(t) = (t-2)^2 + 6\).

Function Transformation

The term function transformation encompasses any change to a graph of a function that alters its position, shape, or size. This can include moves (shifts), stretches, compressions, and reflections.

In the context of our problem, the transformations are purely translational: moving the graph without altering its shape. Combining the horizontal shift to the right and the vertical shift upwards resulted in the final transformed function \(f(t) = (t-2)^2 + 6\). Function transformations are essential for understanding how the graphical representation of a function relates to its algebraic expression and provides a powerful tool for visualizing complex changes in a simple way.