Question

The function \(H=f(t)\) gives the temperature, \(H^{\circ} \mathrm{F},\) of an object \(t\) minutes after it is taken out of the refrigerator and left to sit in a room. Write a new function in terms of \(f(t)\) for the temperature if: (a) The object is taken out of the refrigerator 5 minutes later. (Give a reasonable domain for your function.) (b) Both the refrigerator and the room are \(10^{\circ} \mathrm{F}\) colder.

Short answer

Question: Create a new function for each case: (a) if the object is taken out of the refrigerator 5 minutes later, and (b) if both the refrigerator and room are 10 degrees Fahrenheit colder. Answer: (a) For an object taken out of the refrigerator 5 minutes later: \(H = f(t - 5)\), where the domain is \(5 \leq t \leq n\); (b) If both the refrigerator and the room are 10 degrees Fahrenheit colder: \(H = f(t) - 10\).

Step-by-step solution

Shift the function horizontally

When the object is taken out 5 minutes later, we need to shift the function horizontally to account for this delay. To do this, we need to replace t with (t-5) within the function, resulting in a new function: \(H = f(t - 5)\).

Determine a reasonable domain

Since the object is taken out of the refrigerator 5 minutes later, the new domain of the function will start from t=5. The end point of the domain would depend on how long we want to observe the temperature change of the object. In this case, since it is not specified, we can say the end point is t=n, where n is the arbitrary number of total minutes being considered. Therefore, the domain of the new function would be \(5 \leq t \leq n\).

The final answer for case (a)

For an object taken out of the refrigerator 5 minutes later, the new function is \(H = f(t - 5)\), where the domain of the function is \(5 \leq t \leq n\). ## Case (b): 10 degrees Fahrenheit colder ##

Subtract 10 from the function

For this scenario, both the refrigerator and the room are 10 degrees Fahrenheit colder. To account for this change in temperature, we simply need to subtract 10 from the original function, meaning our new function would be: \(H = f(t) - 10\).

The final answer for case (b)

If both the refrigerator and the room are 10 degrees Fahrenheit colder, the new function is \(H = f(t) - 10\).

Key concepts

Horizontal Shift

Imagine you are graphing a function that represents the temperature over time. If the object is left outside the refrigerator 5 minutes later than expected, we have to adjust our function to reflect this delay. To achieve this, we perform a horizontal shift. How? We replace the variable in our function, usually written as

• \( f(t) \) with \( f(t - 5) \)

This means the curve of the function moves 5 units to the right on the time axis, representing that everything is starting 5 minutes later.

This kind of transformation doesn’t change the shape or direction of the graph. It merely shifts the entire graph horizontally. This is crucial when creating accurate real-world data models like our temperature example.

Vertical Shift

A vertical shift involves moving the graph up or down along the vertical axis. In thermal-related functions, this transformation is used when there is a uniform temperature change in the environment. Let's consider our scenario where both the refrigerator and room are 10 degrees colder.

Here, we reflect this new condition by simply subtracting the value from the output of the original function.

• \( H = f(t) - 10 \)

This adjustment means that every temperature value predicted by the function is decreased by 10 degrees.

Vertical shifts don’t impact the domain, but they ensure the function accurately reflects changes in the actual temperature conditions.

Domain Adjustment

When functions undergo transformations, it can often affect the domain. The domain is the set of "input" values (or the times in this context) for which the function is defined.

If the object is taken out later by 5 minutes, the earliest time we examine, the start of our domain, moves from time zero to time five as represented by:

• \(5 \leq t \leq n \)

This helps us establish a realistic model, as we are now measuring from 5 minutes onwards.

The end of the domain, noted as \(t = n\), is usually defined by the duration of interest or observation, and can be adjusted based on specific requirements or the context of data collection.

Temperature Function

The concept of a temperature function is a practical application of function transformation in daily life. It allows us to predict how the temperature of an object changes over time when exposed to different conditions.

The initial function \( H = f(t) \) can be considered a baseline, reflecting the standard or expected temperatures over time. Adjustments through horizontal or vertical shifts allow us to modify this function to align with real-world events.

Temperature functions are particularly vital in fields like meteorology, cooking, and science. They help in predicting temperature changes effectively, ensuring actions like safety checks or timing adjustments are based on accurate data. This makes mastering these transformations beneficial for effectively applying mathematical models to practical scenarios.