Lesson 4.2 : Modeling with Linear Functions

New Concept

This lesson bridges algebra and reality by teaching you to build linear models from descriptions and data. You'll translate situations with constant change into functions like $f(x) = mx+b$ to analyze relationships and predict future outcomes.

What’s next

You've got the foundation. Soon, we'll walk through interactive examples and then you'll solve challenge problems using diagrams and real-world data sets.

Property

Identify changing quantities and define descriptive variables. Find the initial value ($b$) and the constant rate of change ($m$). The rate of change is constant, so we can start with the linear model $M(t) = mt + b$. Then we can substitute the intercept and slope provided.

Examples

• A new plant is 5 cm tall and grows 2 cm each week. The height $H$ after $t$ weeks is modeled by $H(t) = 2t + 5$. After 4 weeks, the height will be $H(4) = 2(4) + 5 = 13$ cm.

• You have 200 dollars and spend 10 dollars each day. The money you have left, $M$, after $d$ days is $M(d) = -10d + 200$. You will run out of money when $0 = -10d + 200$, so $d=20$ days.

Property

Some real-world problems provide the vertical axis intercept, which is the constant or initial value. Once the vertical axis intercept is known, the $x$-intercept (horizontal axis intercept) can be calculated. The $y$-intercept is the initial amount of a debt, or an initial value. The rate of change, or slope, is also given.
We can then use the slope-intercept form and the given information to develop a linear model: $f(x) = mx + b$.

Examples

• A gym membership costs 30 dollars upfront and 20 dollars per month. The total cost $C$ for $x$ months is $C(x) = 20x + 30$. The initial cost is the $y$-intercept, 30 dollars.

• A phone's battery is at 90% and loses 5% every hour. The battery percentage $P$ after $t$ hours is $P(t) = -5t + 90$. It will be empty when $0 = -5t + 90$, which means $t=18$ hours.

Property

Given a word problem that includes two pairs of input and output values, use the linear function to solve a problem.

1. Identify the input and output values.
2. Convert the data to two coordinate pairs.
3. Find the slope: $m = \frac{\operatorname{change\ in\ output}}{\operatorname{change\ in\ input}} = \frac{y_2 - y_1}{x_2 - x_1}$.
4. Write the linear model using the slope and one of the points to find the $y$-intercept $b$.
5. Use the model to make a prediction.

Examples

• A tree was 10 feet tall in 2015 and 16 feet tall in 2018. The slope is $m = \frac{16-10}{2018-2015} = \frac{6}{3} = 2$ feet per year. Let $t$ be years since 2015. The model is $H(t) = 2t + 10$.

• A subscription service had 500 users in January and 900 users in May. Let $t=0$ for January. The points are $(0, 500)$ and $(4, 900)$. The slope is $m = \frac{900-500}{4-0} = 100$ users/month. The model is $U(t) = 100t + 500$.

Property

It is useful for many real-world applications to draw a picture to gain a sense of how the variables representing the input and output may be used to answer a question. If a right triangle is sketched, the Pythagorean Theorem relates the sides.
Using the Pythagorean Theorem, we get: $D(t)^2 = A(t)^2 + E(t)^2$, where $A(t)$ and $E(t)$ are the distances traveled along each leg of the triangle and $D(t)$ is the direct distance between the endpoints.

Examples

• Two ships leave the same port at noon. Ship A sails west at 15 knots, and Ship B sails south at 20 knots. The distance between them after $t$ hours is $D(t) = \sqrt{(15t)^2 + (20t)^2} = \sqrt{225t^2 + 400t^2} = \sqrt{625t^2} = 25t$ nautical miles.

• A bird flies north at 12 mph and the wind blows it east at 5 mph. Its speed relative to the ground is $\sqrt{12^2 + 5^2} = \sqrt{144+25} = \sqrt{169} = 13$ mph. The distance from the start after $t$ hours is $D(t) = 13t$.

Property

Given a situation that represents a system of linear equations, write the system of equations and identify the solution.

1. Identify the input and output of each linear model.
2. Identify the slope and $y$-intercept of each linear model.
3. Find the solution by setting the two linear functions equal to one another and solving for $x$, or find the point of intersection on a graph. This point is where the models have equal value.

Examples

• Company A rents a car for 30 dollars a day plus 25 cents per mile. Company B charges 50 dollars a day plus 15 cents per mile. To find when they cost the same, set $30 + 0.25d = 50 + 0.15d$. Solving gives $0.10d = 20$, so $d=200$ miles.

• Phone Plan A is 40 dollars per month. Plan B is 20 dollars per month plus 5 dollars per GB of data. They cost the same when $40 = 20 + 5g$, which means $g=4$ GB. For more than 4 GB, Plan A is cheaper.