Interpret the graph of a linear model

Horizontal t is time in minutes; vertical V is volume in litres. The segment joins (0,120) to (15,0). How does the volume change each minute?

A tank loses 8 litres each minute at a constant rate. Predict its signed volume change over two minutes

Horizontal t is minutes; vertical V is litres. Change the rate and starting volume. A marks two minutes and B four minutes. Compare their volumes. The segment ends when the tank is empty

Which calculation gives the signed volume change over two minutes?

Read values in their stated units

The graph supplies a relationship, not just a picture. Read the scale and the meaning of each axis

1. 1Starting value Read the output where the input is zero
2. 2Signed rate Divide the output change by the input change, preserving their order \(\displaystyle b=\frac{y_2-y_1}{x_2-x_1}\)
3. 3Scaled axis A reading on a thousands axis must be multiplied by one thousand to give base units \(\displaystyle 6\text{ thousand dollars}=\$6000\)
4. 4Input units and domain Convert an input to the horizontal-axis unit and check that it lies in the supported interval \(\displaystyle 12\text{ months}=1\text{ year}\)

Supported ages

\(\displaystyle 0\le t\le8\text{ years}\)

Horizontal t is years; vertical V is thousands of dollars. A=(0,36), B=(8,12). Find the signed change in value each year, in dollars

Find the signed slope in graph units, then convert thousands of dollars to dollars

The van model has a signed yearly value change of −$3000. What is the magnitude of its annual loss?

Requested age

\(\displaystyle 75\text{ months}\)

Supported ages

\(\displaystyle 0\le t\le8\text{ years}\)

Horizontal t is years; vertical V is thousands of dollars. A=(0,36), B=(8,12). Predict the value at an age of seventy-five months, in dollars

Convert months to years before using the model, then convert the output from thousands of dollars

Check the scale and both conversions

Use the two displayed endpoint readings for the model, and keep graph units distinct from dollars

1. 1Rate The signed slope is in thousands of dollars per year \(\displaystyle \begin{aligned}b&=\frac{12-36}{8-0}=-3\\-3\times1000&=-3000\end{aligned}\)
2. 2Convert the age Divide months by twelve to obtain years \(\displaystyle 75\div12=6.25\text{ years}\)
3. 3Predict in graph units The starting value is thirty-six graph units \(\displaystyle V=36-3(6.25)=17.25\)
4. 4Convert the response Each graph unit represents one thousand dollars \(\displaystyle 17.25\times1000=17250\)

Model or data

\(\displaystyle \begin{gathered}V=36-3t\\0\le t\le8\end{gathered}\)

Horizontal t is years; vertical V is thousands of dollars. At what age does the van model predict a value of $18,000? Give the age in months

Model or data

\(\displaystyle V=36-3t\)

The supported van model covers ages zero to eight years. Does substituting nine years justify quoting the resulting value as a supported prediction?

Horizontal t is years; vertical V is thousands of dollars. A=(0,36), B=(8,12). Where is the first error in this reading of the model?

Model or data

\(\displaystyle V=36-3t\quad\text{(thousands)}\)

At age four years, the van sells for $25,000. How many dollars higher is this observation than the model prediction?

One observed sale is $1000 above the model prediction. Which conclusion is supported?