Change

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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 a capital Greek Delta: $\Delta$

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.

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 $\answer {6}$ .

$f$ is a linear function.

True False

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.

This browser does not support embedded elements.

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 $100 \, miles$ .

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more examples can be found by following this link
More Examples of Linear Functions