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3.3 Modeling with Linear Functions
ACCUPLACER Advanced Algebra and Functions Exam
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1 Introduction & Exam Overview
1.1 Current Exam Facts 1.2 Scoring, Skills Insight Bands, and Placement 1.3 How to Study for AAF
2 Linear Equations
2.1 Writing Linear Equations in One and Two Variables 2.2 Solving Linear Equations 2.3 Linear Inequalities 2.4 Systems of Two Linear Equations
3 Linear Applications and Graphs
3.1 Slope, Intercepts, and Graphing Lines 3.2 Parallel and Perpendicular Lines 3.3 Modeling with Linear Functions 3.4 Graphing Linear Inequalities and Systems
4 Factoring
4.1 GCF, Grouping, and Special Products 4.2 Factoring Quadratic Trinomials 4.3 Factoring Cubics and Higher Polynomials
5 Quadratics
5.1 Quadratic Forms, Graphs, Vertex, and Intercepts 5.2 Solving Quadratics by Factoring 5.3 Quadratic Formula and Completing the Square 5.4 Quadratic Inequalities and Quadratic-Linear Systems
6 Functions
6.1 Function Notation, Evaluation, Domain, and Range 6.2 Graphs of Functions, Vertical Line Test, Maxima and Minima 6.3 Function Operations, Composition, and Inverses 6.4 Transformations and Interpreting Functions in Context
7 Radical and Rational Equations
7.1 Simplifying Radical Expressions 7.2 Solving Radical Equations and Extraneous Roots 7.3 Rational Expressions and Equations 7.4 Graphs, Domain, and Range of Radical and Rational Functions
8 Polynomial Equations
8.1 Polynomial Operations and Structure 8.2 Solving Polynomial Equations 8.3 Graphing Polynomials, Roots, and End Behavior
9 Exponential and Logarithmic Equations
9.1 Exponential Functions, Growth, and Decay 9.2 Graphs of Exponential Functions 9.3 Logarithms and Logarithmic Properties 9.4 Solving Exponential and Logarithmic Equations
10 Geometry Concepts for Algebra 1
10.1 Area, Perimeter, and Prism Volume 10.2 Distance Formula and Pythagorean Theorem 10.3 Dilations, Rotations, Translations, and Reflections
11 Geometry Concepts for Algebra 2
11.1 Volume of Cylinders, Cones, and Spheres 11.2 Intersecting Lines and Angle Relationships 11.3 Triangle Similarity and Congruence 11.4 Circles in the Coordinate Plane
12 Trigonometry
12.1 Right-Triangle Trigonometry and Special Triangles 12.2 Unit Circle, Radians, and Arc Length 12.3 Trigonometric Graphs and Identities 12.4 Solving Trig Equations; Law of Sines and Cosines
3.3 Modeling with Linear Functions
Key Takeaways Explain This Topic AI
In y = mx + b, m is a rate with units (dollars per text, dollars per day, miles per hour) and b is the starting value when the input is 0.
Cell-phone Plan A y = 0.10x + 25 and unlimited Plan B y = 40 break even at 150 texts; Plan A is cheaper below 150 texts and Plan B is cheaper above.
Campus lot cost 2.25d + 40 versus garage 6d breaks even at d = 32/3 ≈ 10.67 days, so the lot is cheaper from 11 days onward.
A tutor charging 15 + 28h costs $99 for 3 hours; a $200 eight-hour package is a single point, not a line you can evaluate at 4 hours.
College Board Skills Insight band 200–236 on AAF expects you to evaluate a linear function from a real-world context; creating the equation from the story is the same skill run backward.
Last updated: August 2026
3.3 Modeling with Linear Functions
Quick Answer: A linear model is y = m x + b y = mx + b y = m x + b where m m m is a rate with units (dollars per hour, dollars per credit) and b b b is the starting value when the input is 0 . Translate the story into m m m and b b b , then evaluate, solve, or compare two models. The cheaper (or larger) option after n n n units is the one with the smaller (or larger) output at that n n 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 = m x + b y = mx + b y = m x + b from Section 3.1; the new skill is naming what x x x , y y y , m m m , and b b 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 m m , including the sign (increase vs decrease) and the units.
A starting amount , fee, or value at zero input → intercept b b b .
Then decide what the question wants: a value of y y y (evaluate), a value of x x x (solve), or a comparison of two models.
Story type Typical m m m Typical b b b Typical question
Cost / plan price per unit signup or monthly fee Which plan costs less after n n n units?
Rate of pay dollars per hour flat booking fee How many hours to earn Y Y Y ?
Distance speed (mi/h) starting mile marker When do two travelers meet?
Mixture / inventory units added or used per day initial amount When does stock hit 0?
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 b = 120 b = 120 dollars and m = 1.50 m = 1.50 m = 1.50 dollars/day — not 121.50 121.50 121.50 lumped together. If a shuttle is already at mile 12, that 12 is b b 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 x x be the number of texts in a month and y y y the monthly bill in dollars. y A = 0.10 x + 25 , y B = 40. y_A = 0.10x + 25, \qquad y_B = 40. y A = 0.10 x + 25 , y B = 40.
Plan B is a horizontal line: cost does not depend on x x x . The plans cost the same when 0.10 x + 25 = 40 0.10x + 25 = 40 0.10 x + 25 = 40 , so 0.10 x = 15 0.10x = 15 0.10 x = 15 and x = 150 x = 150 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 m_A = 0.10 m A = 0.10 dollars per text , b A = 25 b_A = 25 b A = 25 dollars (the bill at 0 texts). m B = 0 m_B = 0 m B = 0 dollars per text, b B = 40 b_B = 40 b B = 40 dollars. After 80 texts, Plan A costs 0.10 ( 80 ) + 25 = 33 0.10(80) + 25 = 33 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 m m as rise/run: ( 30 − 25 ) / ( 50 − 0 ) = 5 / 50 = 0.10 (30 - 25)/(50 - 0) = 5/50 = 0.10 ( 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 d d be the number of parking days in the semester. Lot: C L = 2.25 d + 40 , Garage: C G = 6 d . \text{Lot: } C_L = 2.25d + 40, \qquad \text{Garage: } C_G = 6d. Lot: C L = 2.25 d + 40 , Garage: C G = 6 d .
Set them equal to find the break-even day count: 2.25 d + 40 = 6 d ⇒ 40 = 3.75 d ⇒ d = 40 / 3.75 = 32 / 3 ≈ 10.67. 2.25d + 40 = 6d \Rightarrow 40 = 3.75d \Rightarrow d = 40/3.75 = 32/3 \approx 10.67. 2.25 d + 40 = 6 d ⇒ 40 = 3.75 d ⇒ d = 40/3.75 = 32/3 ≈ 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 2.25(10) + 40 = 62.50 2.25 ( 10 ) + 40 = 62.50 , garage $60. After 20 days the lot costs 2.25 ( 20 ) + 40 = 85 2.25(20) + 40 = 85 2.25 ( 20 ) + 40 = 85 and the garage costs $120, so the lot wins by $35.
Interpretations: m L = 2.25 m_L = 2.25 m L = 2.25 dollars/day is the incremental cost of one more parking day in the lot; b L = 40 b_L = 40 b L = 40 dollars is the cost of zero days (you already bought the sticker). The garage has b G = 0 b_G = 0 b G = 0 and a steeper slope. On a CAT item you will usually be given a whole-number d d 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 = 28 h + 15 C_T = 28h + 15 C T = 28 h + 15 . After 3 hours, C T = 84 + 15 = 99 C_T = 84 + 15 = 99 C T = 84 + 15 = 99 dollars. After 5 hours, C T = 140 + 15 = 155 C_T = 140 + 15 = 155 C T = 140 + 15 = 155 dollars. The 8-hour package is $200 regardless of whether you attend every hour; it is not C = 25 h C = 25h C = 25 h unless you commit to all 8 hours. Comparing C T C_T C T at h = 8 h = 8 h = 8 gives 28 ( 8 ) + 15 = 239 28(8) + 15 = 239 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 28(4) + 15 = 127 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 h = 4 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 = 45 t + 12 s = 45t + 12 s = 45 t + 12 , with t t t in hours and s s s in miles. A second shuttle starts at mile marker 80 and travels toward campus at 40 mph on the same road: s = 80 − 40 t s = 80 - 40t s = 80 − 40 t . They meet when 45 t + 12 = 80 − 40 t 45t + 12 = 80 - 40t 45 t + 12 = 80 − 40 t , so 85 t = 68 85t = 68 85 t = 68 and t = 68 / 85 = 0.8 t = 68/85 = 0.8 t = 68/85 = 0.8 hours (48 minutes), at mile 45 ( 0.8 ) + 12 = 48 45(0.8) + 12 = 48 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 = − 2.5 t + 18 g = -2.5t + 18 g = − 2.5 t + 18 . It runs out when g = 0 g = 0 g = 0 , t = 18 / 2.5 = 7.2 t = 18/2.5 = 7.2 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 t = 4 t = 4 hours, g = − 2.5 ( 4 ) + 18 = 8 g = -2.5(4) + 18 = 8 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 y_1 = m_1 x + b_1 y 1 = m 1 x + b 1 and y 2 = m 2 x + b 2 y_2 = m_2 x + b_2 y 2 = m 2 x + b 2 compete:
Set m 1 x + b 1 = m 2 x + b 2 m_1 x + b_1 = m_2 x + b_2 m 1 x + b 1 = m 2 x + b 2 and solve for the break-even x x x .
Test one x x x on each side of the break-even to see which model is smaller (cheaper) or larger.
Watch for horizontal competitors ( m = 0 m = 0 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 m_1 > m_2 m 1 > m 2 and b 1 < b 2 b_1 < b_2 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 h h .
Keep units glued to every number you write. A slope of 2.25 2.25 2.25 without dollars per day is how students grab the wrong intercept on the next item. Once the story is y = m x + b y = mx + b y = m x + b , the rest is Section 3.1 arithmetic.