Problem 73 - Algebra and Trigonometry

Question

Sales Growth The annual sales of a certain company can be modeled by the function \(f(t)=4+0.01 t^{2}\) , where \(t\) represents years since 1990 and \(f(t)\) is measured in millions of dollars. (a) What shifting and shrinking operations must be performed on the function \(y=t^{2}\) to obtain the function \(y=f(t) ?\) (b) Suppose you want \(t\) to represent years since 2000 instead of \(1990 .\) What transformation would you have to apply to the function \(y=f(t)\) to accomplish this? Write the new function \(y=g(t)\) that results from this transformation.

Step-by-Step Solution

Answer

(a) Vertically shrink by 0.01 and shift up by 4. (b) New function: \(g(t') = 5 + 0.01t'^2 + 0.2t'\).

Step 1: Understand the Original Function

The original function is given by \(f(t) = 4 + 0.01 t^2\). This function models the annual sales in millions of dollars, where \(t\) represents years since 1990.

Step 2: Identify Shifts and Shrinks

For part (a), compare the function \(y = t^2\) with \(y = f(t) = 4 + 0.01 t^2\). To go from \(y = t^2\) to \(y = 0.01 t^2\), a vertical shrinking of factor 0.01 is applied. Additionally, the function is vertically shifted upwards by 4 units due to the constant addition of 4.

Step 3: Set New Year Representation

For part (b), since \(t\) should represent years since 2000 instead of 1990, we denote the new \(t\) as years since 1990 subtracted by 10, i.e., \(t' = t - 10\).

Step 4: Apply the Transformation

Substitute \(t = t' + 10\) into the original function: \(f(t') = 4 + 0.01(t' + 10)^2\).

Step 5: Expand the New Function

Expand \(f(t') = 4 + 0.01(t' + 10)^2\) to get the new function: \(f(t') = 4 + 0.01(t'^2 + 20t' + 100)\). Simplify this to get \(f(t') = 5 + 0.01t'^2 + 0.2t'\).

Step 6: Write the New Function

The new function \(y = g(t')\), representing sales in millions for years since 2000, is \(g(t') = 5 + 0.01t'^2 + 0.2t'\).

Key Concepts

Vertical Shift

In the function transformation process, a vertical shift happens when we add or subtract a constant to the function's output. This operation moves the graph of the function up or down on the coordinate plane.
For the function given by the exercise, starting with the basic quadratic function, \(y = t^2\):- Adding a constant value shifts the graph upwards.- Subtraction would shift it downwards, but that's not the case here.Consider the transformation from \(y = t^2\) to \(y = f(t) = 4 + 0.01t^2\).
The term \(+4\) in the function causes a **vertical shift** upwards by 4 units. This is because any output value of \(t^2\) will now be augmented by 4, shifting the entire graph of the function higher without altering its shape. Thus, all values of \(f(t)\) will be higher than their corresponding \(t^2\) values by exactly 4 units.

Vertical Shrink

Vertical shrink involves scaling down the height of the graph by multiplying the function's output by a constant factor between 0 and 1. This compression makes the graph look "flatter."For the given function, we transform from the standard quadratic \(y = t^2\) to \(y = 0.01 t^2\). Here's how:

1. The factor \(0.01\) affects the size of \(t^2\), effectively shrinking the graph vertically by 1% of its original height.
2. Instead of climbing sharply, the function's curve appears much closer to the x-axis.

This change, known as a **vertical shrink**, compresses the entire upward growth, flattening its appearance in a controlled manner without affecting the horizontal position of any point on the graph. As a result, the progression of sales over time as indicated by the function's output becomes less steep.

Time Transformation

Time transformation in function transformation refers to changing the variable representing time to a different reference point or scale. This concept is handy when needing to represent data or functions based on alternative temporal starting points.In part (b) of the exercise, the goal is to change the reference of \(t\) from "years since 1990" to "years since 2000." To achieve this, we can manipulate the timing variable:

• Express the new variable, \(t'\), as \(t = t' + 10\). Here, \(t'\) represents years since 2000.
• Substitute \(t = t' + 10\) into the original function to adjust the timing: \[f(t') = 4 + 0.01(t' + 10)^2\]
• Expanding yields: \[f(t') = 5 + 0.01 t'^2 + 0.2 t'\]
• Now, \(g(t') = 5 + 0.01 t'^2 + 0.2 t'\) is the new function.

This process effectively moves the timeline forward by 10 years. It modifies how we interpret each function point without needing an entirely new equation.