1.4: Linear Functions

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Stalactite Growth (Linear Model)

The length of a stalactite is modeled by the formula:
$L(t)=17.75+\left(\frac{1}{250}\right) t$, where $\mathrm{L}(\mathrm{t})$ represents the length (in inches) and t is the time in years since measurement.

[a)] Enter $L(t)=17.75+\left(\frac{1}{250}\right) t$ into Desmos.

[b)] Identify the vertical intercept and explain what it represents in context.

[c)] Use the graph to estimate the stalactite's length after 500 years. Compare with the algebraic calculation.

Slope from Two Points

[a)] Plot the points $(3,5)$ and $(9,23)$ in Desmos.

[b)] Use Desmos to draw the line through the points and confirm the slope visually.

[c)] Describe how the steepness of the line matches your calculation.

Car Depreciation

A new Toyota RAV4 costs $\$ 23,500$. The car's value depreciates linearly to $\$ 18,823$ in three years.

[a)] Write the equation for the car's value.

[b)] Graph the function in Desmos.

[c)] Use the graph to estimate the car's value after 5 years.

[d)] Compare this with your algebraic calculation.

Advertising and Units Sold

A company sells 300 units with no advertising. For each additional \$5000 spent, 20 more units are sold.

[a)] Write a linear equation for units sold.

[b)] Enter your equation into Desmos.

[c)] Use the graph to estimate sales at \$40,000 spent.

[d)] Zoom out and describe how the slope represents rate of change in sales.

Graphing Nonlinear Function}

Consider $y=-x\left(\dfrac{x-1000}{900}\right)$.

[a)] Enter the function into Desmos.

[b)] Adjust the viewing window to $-500 \leq \mathrm{x} \leq 500$ and $-500 \leq \mathrm{y} \leq 500$.

[c)] Compare how the graph looks zoomed in vs. zoomed out. Explain why it appears linear at first and nonlinear at a larger scale.

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