Problem 137
Question
Determine whether each statement makes sense or does not make sense, and explain your reasoning. If the statement makes sense, graph the new function on the domain \([0,24]\). If the statement does not make sense, correct the function in the statement and graph the corrected function on the domain \([0,24]\) I decided to keep the house \(5^{\circ}\) warmer than before, so I reprogrammed the thermostat to \(y=f(t)+5\).
Step-by-Step Solution
Verified Answer
The statement 'I decided to keep the house \(5^{\circ}\) warmer than before, so I reprogrammed the thermostat to \(y=f(t)+5\)' makes sense. The function \(y = f(t) + 5\) represents a thermostat setting that is 5 degrees warmer than the original setting represented by \(f(t)\). To graph the function, shift the graph of \(f(t)\) upwards by 5 units in the domain \([0,24]\).
1Step 1: Statement Analysis
Look at the statement 'I decided to keep the house \(5^{\circ}\) warmer than before, so I reprogrammed the thermostat to \(y=f(t)+5\). This represents a vertical shift of the function \(f(t)\) upwards by 5 units. This is because the value of \(f(t)\) at every point is increased by 5. So, this statement makes sense and is logically correct because if you want the house to be warmer by \(5^{\circ}\), you would increase the thermostat setting by \(5^{\circ}\).
2Step 2: Graphing the New Function
The domain given in the problem is \([0,24]\). To graph the function \(y = f(t) + 5\), take the graph of \(f(t)\) and shift it upwards by 5 units. The appearance of the function will not change, it will just be 5 units higher than before on the y-axis.
Key Concepts
Vertical ShiftThermostat ProgrammingTemperature Adjustment
Vertical Shift
A vertical shift in functions is a fundamental concept in mathematics, especially when studying function transformations. By vertically shifting a function, you adjust every point of the function in the vertical direction. In simple terms, adding or subtracting a constant from the function shifts it up or down along the y-axis.
For instance, if you have a function \(f(t)\), and you transform it to \(y=f(t)+5\), each output is increased by 5. This is why it is called a vertical shift. The graph of the new function will look identical to the original, except it will be positioned higher or lower on the y-axis.
Key points about vertical shift transformations:
For instance, if you have a function \(f(t)\), and you transform it to \(y=f(t)+5\), each output is increased by 5. This is why it is called a vertical shift. The graph of the new function will look identical to the original, except it will be positioned higher or lower on the y-axis.
Key points about vertical shift transformations:
- Adding a constant shifts the function upwards.
- Subtracting a constant shifts the function downwards.
- The shape of the graph remains unchanged.
- The domain of the function is unaffected.
Thermostat Programming
Thermostat programming offers a seamless way to control the environment within our homes by adjusting temperature settings according to our preferences. When you program a thermostat, you're essentially telling it to keep indoor temperature levels consistent with a desired schedule.
If you adjust a thermostat using a function like \(y=f(t)+5\), you effectively set a target temperature that's consistently 5 degrees warmer than the previous settings over time. This programming can provide both comfort and energy efficiency. Here are some basics of thermostat programming:
If you adjust a thermostat using a function like \(y=f(t)+5\), you effectively set a target temperature that's consistently 5 degrees warmer than the previous settings over time. This programming can provide both comfort and energy efficiency. Here are some basics of thermostat programming:
- It allows for setting schedules that match your daily routine.
- You can have different settings for weekends or weekdays.
- Programming helps save energy by reducing heating or cooling when it is not needed.
- Modern thermostats often come with features like remote access, letting you adjust settings from your smartphone.
Temperature Adjustment
Temperature adjustment is a practical application of the mathematical concept of function transformations, particularly vertical shifts. When you adjust a thermostat for a warmer or cooler setting, you're essentially changing the function guiding the temperature.
The purpose behind adjusting temperatures can vary, but it is usually driven by the need for comfort or to conserve energy. Using our earlier example, if you decide to make your home 5 degrees warmer, it directly translates to adjusting your temperature function to \(y=f(t)+5\). This represents a very straightforward transformation:
The purpose behind adjusting temperatures can vary, but it is usually driven by the need for comfort or to conserve energy. Using our earlier example, if you decide to make your home 5 degrees warmer, it directly translates to adjusting your temperature function to \(y=f(t)+5\). This represents a very straightforward transformation:
- It maintains a consistent increase or decrease in desired setting.
- Ensures comfort by preventing drastic fluctuations in temperature.
- Allows for better energy management through precise control.
Other exercises in this chapter
Problem 135
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View solution Problem 136
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View solution Problem 140
Determine whether each statement makes sense or does not make sense, and explain your reasoning. If the statement makes sense, graph the new function on the dom
View solution Problem 141
Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(f(x)=|x|\) and \(g(
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