True False
regular
There are many relationships between measurements that exhibit proportional changes.
Proportional Changes
Speed
Suppose a car is traveling on the highway at a constant speed of \(60 \, mph\).
\(\blacktriangleright \) Whenever the distance measurement changes by \(60 \, miles\), the time measurement changes
by \(1 \, hour\).
\(\blacktriangleright \) Whenever the time measurement changes by \(1 \, hour\), the distance measurement changes
by \(60 \, miles\).
- When the time changes by \(2\) hours, the distance changes by \(\answer {120}\) miles.
- When the time changes by \(5\) hours, the distance changes by \(\answer {300}\) miles.
- When the time changes by \(0.5\) hours, the distance changes by \(\answer {30}\) miles.
We have a constant conversion factor of \(\frac {60 \, miles}{1 \, hour}\) for converting time changes into distance
changes.
- When the distance changes by \(2\) miles, the time changes by \(\answer {\frac {1}{30}}\) hours.
- When the distance changes by \(5\) miles, the time changes by \(\answer {\frac {1}{12}}\) hours.
- When the distance changes by \(0.5\) miles, the time changes by \(\answer {\frac {1}{120}}\) hours.
We have a constant conversion factor of \(\frac {1 \, hour}{60 \, mile}\) for converting distance changes into time
changes.
\(\blacktriangleright \) One viewpoint is that, in this situation, \(60 \, miles = 1 hour\). When one change occurs, the other must occur as well.
\(\Delta \)
Mathematics has a shorthand symbol for “change”. It is an uppercase Greek delta: \(\Delta \)
\(\blacktriangleright \) In the previous example, it might be more appropriate to say \(\Delta 60 \, miles = \Delta 1 hour\) . It is the change in the measurements that is proportional.
Temperature
Temperature can be measured in degrees Fahrenheit or degrees Celsius and these measurements change proportionally.
\(\blacktriangleright \) Whenever the Fahrenheit measurement changes by \(9^{\circ }\), the Celsius measurement
changes by \(5^{\circ }\).
\(\blacktriangleright \) Whenever the Celsius measurement changes by \(5^{\circ }\), the Fahrenheit measurement
changes by \(9^{\circ }\).
Water freezes at \(0^{\circ }\) C and \(32^{\circ }\) F. From here, if the Celsius measurement changes by \(100^{\circ }\), then the Fahrenheit measurement changes by \(180^{\circ }\). \(18\) each of \(5\)’s and \(9\)’s. We get water boiling at \(100^{\circ }\) celsius and \(212^{\circ }\) degrees.
In this situation, \(\frac {\Delta 5^{\circ }C}{\Delta 9^{\circ }F}\) or just \(\frac {5^{\circ }C}{9^{\circ }F}\) is the conversion factor from Fahrenheit change to Celsius change, and \(\frac {9^{\circ }F}{5^{\circ }C}\) is the conversion factor from Celsius to Fahrenheit.
Rates
Those conversions are examples of rates.
Rates
A rate is a ratio between two quantities with different units.
We usually represent rates with factions, although we also use phrases involving “per” or equalities.
For us, rates measure how fast one quantity changes compared to the change in another quantity.
Speed
Suppose a car is traveling on the highway at a constant speed of \(60 \, mph\).
\(\blacktriangleright \) Whenever the distance measurement changes by \(60 \, miles\), the time measurement changes by \(1 \, hour\).
\(\blacktriangleright \) Whenever the time measurement changes by \(1 \, hour\), the distance measurement changes by \(60 \, miles\).
If we compare these related measurement changes in a rate, then we get
\[ \frac {\Delta Distance}{\Delta Time} = \frac {60 \, miles}{1 \, hour} = 60 \,miles \, per \, hour \]
This example gives us the equation for linear motion: \(Distance = Rate \times Time\), which we could represent here with a function.
\[ D(t) = R \cdot t = \frac {60 \, miles}{1 \, hour} \cdot t \]
Temperature
Temperature can be measured in degrees Fahrenheit or degrees Celsius and these measurements have a function relationship. This relationship has a special property.
\(\blacktriangleright \) Whenever the Fahrenheit measurement changes by \(9^{\circ }\), the Celsius measurement
changes by \(5^{\circ }\).
\(\blacktriangleright \) Whenever the Celsius measurement changes by \(5^{\circ }\), the Fahrenheit measurement
changes by \(9^{\circ }\).
Water freezes at \(0^{\circ }\) C and \(32^{\circ }\) F. From here, if the Celsius measurement changes by \(100^{\circ }\), then the Fahrenheit measurement changes by \(180^{\circ }\). \(20\) each of \(5\)’s and \(9\)’s. We get water boiling at \(100^{\circ }\) celsius and \(212^{\circ }\) degrees.
If we compare these related measurement changes in a rate, then we get
\[ \frac {\Delta Degrees}{\Delta Celsius} = \frac {9^{\circ }F}{5^{\circ }C} \, \text { or } \, \frac {\Delta Celsius}{\Delta Degrees} = \frac {5^{\circ }C}{9^{\circ }F} \]
This second example gives us the conversion between Fahrenheit change and Celsius change, which we could express with a function. We just have to remember that \(0^{\circ }C = 32^{\circ }F\).
\[ F(C) = 32 + \frac {9}{5} \cdot C \]
The formula converts each change of \(5\) in \(C\) into a change of \(9\) for \(F\).
Constant Rate of Change
A rate is a comparison of how measurements CHANGE. A rate is not a
comparison of the measurement values, but a comparison of how they change.
Two measurements share a constant rate of change of \(\tfrac {A}{B}\) if whenever measurement #1 changes by \(A\), then measurement #2 changes by \(B\) and vice versa, and this rate is not dependent on the amount of each measurement present. The rate is the same throughout the situation. It is a constant.
Linear Functions
Linear functions are those functions where the domain and range share a constant rate of change.
Each linear function has its own constant rate of change.
Suppose \(L\) is a linear function. Let \(a\) and \(b\) be numbers in the domain of \(L\). Then \(L(a)\) and \(L(b)\) are the corresponding range values.
Since \(L\) is a linear function, we know that \(\frac {L(b) - L(a)}{b - a} = constant\). And, this works for ANY two domain numbers.
Otherwise, it is not a linear function.
Constant Rate of Change
Suppose \(f\) is a function, which contains the pairs \((3, 7)\), \((5, 17)\), and \((6, 23)\).
The rate-of-change from \(3\) to \(5\) is \(\answer {5}\).
The rate-of-change from \(5\) to \(6\) is \(\answer {6}\).
Somewhere in history, \(m\) became a popular choice for the constant rate of change of a linear function.
\[ \frac {L(b) - L(a)}{b - a} = m \]
No matter which two numbers you select from the domain of \(L\), the rate of change always turns out to be \(m\). Each linear function has its own \(m\) - its own constant rate of change.
Mileage vs. Weight
How does the weight of a car affect its gas mileage?
\(392\) makes and models of automobiles manufactured between 1970 and 1982 were considered.
- How many gallons of gasoline were used to travel \(100\) miles? (gallons)
- What is the weight of the car? (\(1000\) pounds)
These pairings are plotted below.
We can see from the graph that this is not the graph of a function. It is a data
plot.
Our goal is to make predictions about other cars from this data. Therefore, we would like a model for this data. That is, we would like a function that approximates the data relationship. The easiest model is a linear model.
A linear model would be a linear function whose graph does the best job a line can do of pretending to be the dots. The line should pass through the “middle of the dots”.
In the DESMOS window above, change the values of “\(m\)” and “\(b\)” to find such a line.
When you have your line, turn on the model found by DESMOS by clicking on the circle next to \(y_1 \sim a_1 x + a_2\).
Source: regressit.com
DESMOS found the linear model \(y_1 = 0.450808 x_1 + 0.820606\)
The units of \(x_1\) are \(1000 \, lbs\).
The units of \(y_1\) are \(gallons\).
What are the units of \(0.450808\)?
\(gallons\) \(pounds\) \(\frac {gallon}{pound}\) \(\frac {pound}{gallon}\)
DESMOS found the linear model \(y_1 = 0.450808 x_1 + 0.820606\)
The units of \(x_1\) are \(1000 \, lbs\).
The units of \(y_1\) are \(gallons\).
Which is the correct statement?
For every \(1000 \, pounds\) added to a car, an extra \(0.450808 \, gallons\) of gas is needed to travel \(100 \, miles\). Every added
gallon of gas can carry \(4508.08 \, pounds\) an additional \(100 \, miles\).
ooooo-=-=-=-ooOoo-=-=-=-ooooo
more examples can be found by following this link
More Examples of Linear Functions