2.3Modeling with Linear Functions
Build linear models from two data points or a verbal description, interpret slope and y-intercept in context, and use the model to make predictions via interpolation and extrapolation.
Mathematical modeling turns real-world situations into equations you can solve. Linear models appear in economics, science, and engineering — and building them from data is a core skill for any quantitative field.
Essential Question
How do you translate a real-world situation into a linear equation, and what do the slope and y-intercept tell you about the situation?
Lesson Overview
A linear model is a linear function f(x) = mx + b used to represent a real-world relationship. To build one: (1) identify the input variable x and output variable y; (2) extract two data points from the problem; (3) compute slope m = Δy/Δx; (4) write the equation using point-slope form; (5) interpret and use the model. The slope always represents the rate of change — how much y changes per one unit increase in x — and must include units (e.g., dollars per year, miles per hour). The y-intercept represents the initial value — the value of y when x = 0 — and also carries units. Interpolation means predicting within the range of the data (more reliable). Extrapolation means predicting outside the data range (less reliable — the linear trend may not continue). Always check whether your answer makes sense in context: negative quantities, impossibly large values, or values outside a reasonable domain are red flags. The domain of a real-world linear model is often restricted to values that make physical sense (e.g., time ≥ 0, quantity ≥ 0).
Interpreting m and b
Slope m = Δy/Δx
"For each additional [x-unit], y [increases/decreases] by [|m|] [y-units]."
y-intercept b
"When x = 0, y = b [y-units]." This is the starting value or fixed cost.
x-intercept
"The value of x when y = 0." Often the break-even point or when a quantity runs out.
Model-Building Checklist
- Define variables with units
- Identify two known (x, y) pairs
- Compute slope m (with units)
- Write equation in slope-intercept form
- State domain restrictions
- Interpret slope and y-intercept in context
- Use model to answer the question
- Check: does the answer make sense?
The Linear Modeling Process
1.Identify Variables
What changes? What is input (x)? Output (y)?
2.Find Two Data Points
From table, graph, or verbal description
3.Calculate Slope
m = (y₂ − y₁) / (x₂ − x₁)
4.Write the Equation
Use point-slope → slope-intercept form
5.Interpret & Predict
Slope = rate of change; b = initial value
Linear Model: Temperature vs. Ice Cream Sales
Sales (hundreds $) vs. Temperature (°F)
Interpolation vs. Extrapolation
Interpolation
Predicting within the data range. More reliable — the model was built from this region.
Extrapolation
Predicting outside the data range. Less reliable — the linear trend may not continue.
Worked Examples
A car rental company charges a flat fee of $45 plus $0.30 per mile driven. Write a linear model for the total cost C(m) after m miles. Find the cost for 200 miles. How many miles can you drive for $90?
Variables: m = miles driven (input), C = total cost in dollars (output).
Flat fee = $45 → y-intercept b = 45. Rate = $0.30/mile → slope m = 0.30.
Model: C(m) = 0.30m + 45.
C(200) = 0.30(200) + 45 = 60 + 45 = $105.
Set C = 90: 90 = 0.30m + 45 → 45 = 0.30m → m = 150 miles.
A town's population was 12,400 in 2010 and 15,100 in 2020. Assuming linear growth, write a model P(t) where t = years since 2010. Predict the population in 2035. When will the population reach 20,000?
Two points: (0, 12400) and (10, 15100). (t = 0 corresponds to 2010.)
Slope: m = (15100 − 12400)/(10 − 0) = 2700/10 = 270 people/year.
y-intercept: b = 12400 (population in 2010).
Model: P(t) = 270t + 12400.
2035 → t = 25: P(25) = 270(25) + 12400 = 6750 + 12400 = 19,150.
Set P = 20000: 20000 = 270t + 12400 → 7600 = 270t → t ≈ 28.1 → year 2038.
The table shows hours studied (x) and exam scores (y): (1, 62), (3, 74), (5, 86), (7, 98). Verify linearity, write the model, and predict the score for 4 hours of study.
Check slope between consecutive points: (74−62)/(3−1)=12/2=6, (86−74)/(5−3)=12/2=6, (98−86)/(7−5)=12/2=6. Constant slope → linear ✓.
Slope m = 6 points per hour.
Use point (1, 62): y − 62 = 6(x − 1) → y = 6x + 56.
Predict x = 4: y = 6(4) + 56 = 24 + 56 = 80.
A candle is 20 cm tall when lit. After 3 hours it is 14 cm tall. Write a linear model H(t) for height after t hours. Find when the candle burns out. State the domain and range.
Two points: (0, 20) and (3, 14).
Slope: m = (14 − 20)/(3 − 0) = −6/3 = −2 cm/hour.
y-intercept: b = 20. Model: H(t) = −2t + 20.
Burns out when H = 0: 0 = −2t + 20 → t = 10 hours.
Domain: [0, 10] hours (candle exists from lighting to burnout).
Range: [0, 20] cm (height from 0 to full height).
A company's profit P (in thousands of dollars) is modeled by P(x) = 8x − 120, where x is the number of units sold (in hundreds). Interpret the slope and y-intercept. Find the break-even point. What profit is earned selling 25 hundred units?
Slope = 8: for each additional hundred units sold, profit increases by $8,000.
y-intercept = −120: when 0 units are sold, the company loses $120,000 (fixed costs).
Break-even: P = 0 → 0 = 8x − 120 → x = 15 hundred units = 1,500 units.
P(25) = 8(25) − 120 = 200 − 120 = 80 → profit of $80,000.
Guided Practice
A gym charges a $30 sign-up fee plus $25 per month. Write a linear model C(m) for total cost after m months. Find the cost after 1 year.
Hint: Flat fee = y-intercept. Monthly rate = slope. C(m) = 25m + 30. For 1 year, m = 12.
A city's average temperature was 58°F in March and 82°F in July (months 3 and 7). Write a linear model T(m) for temperature in month m. Predict the temperature in September (month 9).
Hint: Two points: (3, 58) and (7, 82). Find slope m = Δy/Δx. Use point-slope form. Then evaluate at m = 9.
A phone battery is at 100% at 8:00 AM and 40% at 2:00 PM. Write a model B(t) where t = hours after 8 AM. When will the battery die (reach 0%)?
Hint: Two points: (0, 100) and (6, 40). Slope = (40−100)/6. Set B(t) = 0 and solve for t.
A linear model predicts that a house worth $180,000 in 2015 will be worth $240,000 in 2025. Write the model V(t) where t = years since 2015. Interpret the slope.
Hint: Two points: (0, 180000) and (10, 240000). Slope = Δvalue/Δyears. Interpret: 'The house value increases by $__ per year.'
The cost to produce x items is C(x) = 15x + 800. Revenue is R(x) = 35x. Find the break-even point and interpret it.
Hint: Break-even: C(x) = R(x). Set 15x + 800 = 35x and solve for x. Interpret: at this quantity, cost equals revenue.
Key Vocabulary
Linear Model
A linear function f(x) = mx + b used to represent a real-world relationship. Assumes constant rate of change between variables.
Rate of Change (Slope)
How much the output changes per unit increase in input. Always has units: e.g., $/mile, people/year, °F/hour.
Initial Value (y-intercept)
The value of the output when the input is zero. Represents a starting amount, fixed cost, or base value.
Break-Even Point
The x-value where cost equals revenue (profit = 0), or where a quantity reaches zero. Found by setting two expressions equal or setting y = 0.
Interpolation
Using a model to predict values within the range of the original data. Generally more reliable.
Extrapolation
Using a model to predict values outside the range of the original data. Less reliable — the linear trend may not continue.
Interactive Practice — 5 Questions
A taxi charges $2.50 per mile plus a $4 base fee. Which model represents total cost C for m miles?
A population model is P(t) = 500t + 8000, where t = years since 2000. What does the slope 500 represent?
A candle model is H(t) = −3t + 18. When does the candle burn out?
Cost C(x) = 12x + 300 and Revenue R(x) = 20x. What is the break-even quantity?
A model built from data for years 2010–2020 is used to predict a value for 2050. This is an example of:
Independent Practice
Independent Practice
A plumber charges a $65 service fee plus $90 per hour. Write C(h) for total cost after h hours. Find the cost for 3.5 hours.
A town had 5,200 residents in 2005 and 6,800 in 2015. Write a linear model P(t) where t = years since 2005. Predict the population in 2030.
A car depreciates linearly from $28,000 when new to $16,000 after 6 years. Write V(t) for value after t years. When is the car worth $10,000?
Cost: C(x) = 18x + 540. Revenue: R(x) = 30x. Find the break-even point and interpret it in context.
A spring stretches 4 cm with a 10 N force and 10 cm with a 25 N force. Write a linear model L(F) for length vs. force. Find the natural length (F = 0) and the length at 40 N.
Common Mistakes
Forgetting to attach units to slope and intercept interpretations: saying 'slope = 5' instead of 'slope = 5 dollars per hour'.
Always include units in your interpretation. Slope = Δ(output units)/Δ(input units). The y-intercept has the same units as the output variable.
Using the model outside its reasonable domain without noting the limitation — e.g., predicting negative time or negative quantity.
Always state the domain of your model. A candle height model H(t) = −2t + 20 only applies for t ∈ [0, 10]. After t = 10 the candle is gone.
Confusing the break-even point with the y-intercept. Setting x = 0 to find break-even.
Break-even is where C(x) = R(x) or where profit P(x) = 0. Set the two expressions equal and solve for x — it is NOT x = 0.
Treating extrapolation as equally reliable as interpolation.
Extrapolation (predicting outside the data range) is less reliable. The linear trend may not continue. Always flag extrapolated predictions as estimates.
Math Tips
When building a model from a verbal description, look for two key numbers: a fixed amount (y-intercept) and a per-unit rate (slope). "Plus $X" → y-intercept. "$Y per unit" → slope.
Interpretation template: "For each additional [x-unit], [output] [increases/decreases] by [|m|] [y-units]." Practice this sentence until it is automatic.
To find when a quantity runs out (reaches zero), set the model equal to zero and solve for x. This gives the x-intercept, which is the "zero crossing" of the model.
Always define your variables clearly at the start: "Let t = years since 2010" or "Let x = number of items produced." This prevents unit errors and makes interpretation easier.
Sanity check your model: plug in the original data points and verify they satisfy the equation. If they do not, recheck your slope and intercept calculations.