3.3 Modeling with Linear Functions

3.3 Modeling with Linear Functions

Quick Answer: A linear model is $y = mx + b$ where $m$ is a rate with units (dollars per hour, dollars per credit) and $b$ is the starting value when the input is 0. Translate the story into $m$ and $b$, then evaluate, solve, or compare two models. The cheaper (or larger) option after $n$ units is the one with the smaller (or larger) output at that $n$.

Skills Insight scores in the 200–236 band expect you to evaluate a linear function that represents a real-world context — and creating the equation that fits the story is the same fluency in reverse. College Board Table 11 lists applying linear equations to real-life contexts and elementary linear functions under Linear applications and graphs (10–15% of AAF, 2–3 CAT items). The algebra is the same $y = mx + b$ from Section 3.1; the new skill is naming what $x$, $y$, $m$, and $b$ mean, with units. FREE practice is at /practice/accuplacer-advanced-algebra.

From a Sentence to y = mx + b

Read the story for two numbers:

• A constant rate → slope $m$, including the sign (increase vs decrease) and the units.
• A starting amount, fee, or value at zero input → intercept $b$.

Then decide what the question wants: a value of $y$ (evaluate), a value of $x$ (solve), or a comparison of two models.

Do not invent a slope from the intercept or the other way around. If a campus permit costs $120 plus $1.50 per day, $b = 120$ dollars and $m = 1.50$ dollars/day — not $121.50$ lumped together. If a shuttle is already at mile 12, that 12 is $b$, not a speed.

Cell-Phone Plans

Plan A charges $25 per month plus $0.10 per text. Plan B charges $40 per month with unlimited texts (so its per-text rate is $0). Let $x$ be the number of texts in a month and $y$ the monthly bill in dollars.

$$y_A = 0.10x + 25, \qquad y_B = 40.$$

Plan B is a horizontal line: cost does not depend on $x$. The plans cost the same when $0.10x + 25 = 40$, so $0.10x = 15$ and $x = 150$ texts. For fewer than 150 texts, Plan A is cheaper; for more than 150, Plan B is cheaper. At exactly 150 texts both bills are $40.

Units check: $m_A = 0.10$ dollars per text, $b_A = 25$ dollars (the bill at 0 texts). $m_B = 0$ dollars per text, $b_B = 40$ dollars. After 80 texts, Plan A costs $0.10(80) + 25 = 33$ dollars, which is $7 less than Plan B. After 200 texts, Plan A costs $45, which is $5 more than Plan B.

If the stem instead gives a table of Plan A bills — 0 texts → $25, 50 texts → $30, 100 texts → $35 — recover $m$ as rise/run: $(30 - 25)/(50 - 0) = 5/50 = 0.10$, matching the story. Skills Insight 200–236 is the evaluation half of this translation: story or table in, output out.

Campus Parking

A commuter lot charges a $40 semester sticker plus $2.25 per day you park. A garage across the street charges $6.00 per day with no sticker. Let $d$ be the number of parking days in the semester.

$$\text{Lot: } C_L = 2.25d + 40, \qquad \text{Garage: } C_G = 6d.$$

Set them equal to find the break-even day count:

$$2.25d + 40 = 6d \Rightarrow 40 = 3.75d \Rightarrow d = 40/3.75 = 32/3 \approx 10.67.$$

After 11 days the lot is cheaper (the sticker has been paid off by the $3.75 daily savings). After 10 days the garage is still slightly cheaper: lot $2.25(10) + 40 = 62.50$, garage $60. After 20 days the lot costs $2.25(20) + 40 = 85$ and the garage costs $120, so the lot wins by $35.

Interpretations: $m_L = 2.25$ dollars/day is the incremental cost of one more parking day in the lot; $b_L = 40$ dollars is the cost of zero days (you already bought the sticker). The garage has $b_G = 0$ and a steeper slope. On a CAT item you will usually be given a whole-number $d$ and asked which option costs less, or asked for the smallest whole number of days that makes the lot cheaper.

Tutoring Rates

A campus tutor charges a $15 booking fee plus $28 per hour. A learning-center package is $200 for 8 hours of group sessions (a constant average of $25 per hour, but only if you buy the whole package).

The individual tutor is $C_T = 28h + 15$. After 3 hours, $C_T = 84 + 15 = 99$ dollars. After 5 hours, $C_T = 140 + 15 = 155$ dollars. The 8-hour package is $200 regardless of whether you attend every hour; it is not $C = 25h$ unless you commit to all 8 hours. Comparing $C_T$ at $h = 8$ gives $28(8) + 15 = 239$, so the package saves $39 if you will actually use 8 hours. If you only need 4 hours, the tutor costs $28(4) + 15 = 127$, which is less than $200.

Linear models are only valid on the domain the story allows — here, the package is a single point, not a line you can evaluate at $h = 4$. AAF word items punish the student who prorates a bundle that the story does not allow you to split.

Distance and Mixture, Still Linear

A shuttle leaves campus at mile marker 12 and travels 45 mph. Its position is $s = 45t + 12$, with $t$ in hours and $s$ in miles. A second shuttle starts at mile marker 80 and travels toward campus at 40 mph on the same road: $s = 80 - 40t$. They meet when $45t + 12 = 80 - 40t$, so $85t = 68$ and $t = 68/85 = 0.8$ hours (48 minutes), at mile $45(0.8) + 12 = 48$. The slopes are the velocities, signed by direction; the intercepts are the starting positions.

A campus café starts the morning with 18 gallons of coffee and sells 2.5 gallons per hour. Remaining coffee is $g = -2.5t + 18$. It runs out when $g = 0$, $t = 18/2.5 = 7.2$ hours. The negative slope is not a trick — it is the rate of decrease, and its units are gallons per hour. At $t = 4$ hours, $g = -2.5(4) + 18 = 8$ gallons remain.

Comparing Two Linear Models

Which Option Is Cheaper After n Units

When two models $y_1 = m_1 x + b_1$ and $y_2 = m_2 x + b_2$ compete:

1. Set $m_1 x + b_1 = m_2 x + b_2$ and solve for the break-even $x$.
2. Test one $x$ on each side of the break-even to see which model is smaller (cheaper) or larger.
3. Watch for horizontal competitors ($m = 0$, a flat fee) and for domain restrictions (whole days, a minimum purchase, a package that cannot be prorated).

If $m_1 > m_2$ and $b_1 < b_2$, model 1 starts cheaper and eventually becomes more expensive. That is the cell-phone and parking pattern. If the slopes are equal, the lines are parallel: the one with the smaller intercept is always cheaper, and there is no break-even. Two tutors who both charge $28/hour, one with a $15 fee and one with a $40 fee, never swap rank — the $15-fee tutor is cheaper at every positive $h$.

Keep units glued to every number you write. A slope of $2.25$ without dollars per day is how students grab the wrong intercept on the next item. Once the story is $y = mx + b$, the rest is Section 3.1 arithmetic.