Linear, Quadratic, and Exponential Models — ACT Math

Introduction

Real-world modeling is where ACT Math meets science — you need to pick the right function family before you can solve the problem.

Modeling questions appear 4-8 times per ACT, often in the medium-hard range. They test conceptual understanding, not just computation, so they reward pattern recognition.

By the end of this lesson you will be able to:

Identify whether a scenario is best modeled by a linear, quadratic, or exponential function
Write the equation of a model given key features (slope, vertex, growth rate)
Interpret the parameters of each model in a real-world context

You'll determine whether a bacteria population growing by 50% each hour is linear or exponential, and write the correct equation.

The Concept

The Core Rule

Linear: f(x) = mx + b — constant rate of change (slope m). Quadratic: f(x) = ax² + bx + c — parabola, constant second differences. Exponential: f(x) = a·bˣ — constant percent rate of change; b > 1 is growth, 0 < b < 1 is decay. Vertex of parabola: x = −b/(2a), y = f(−b/(2a)).

How the ACT tests this

Giving a table of values and asking which function type fits
Asking for the vertex, zeros, or y-intercept of a quadratic
Asking for the time to double or half-life in an exponential model

Linear Models

Slope-intercept: y = mx + b where m = rise/run and b = y-intercept. Point-slope: y − y₁ = m(x − x₁). Two-point slope: m = (y₂ − y₁)/(x₂ − x₁). Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals (m₁ × m₂ = −1).

Constant first differences in a table signal a linear model
y-intercept: set x = 0; x-intercept (zero): set y = 0
Increasing slope → steeper graph; negative slope → falls left to right

Quadratic Models

Standard form: ax² + bx + c. Vertex form: a(x − h)² + k where (h, k) is the vertex. Factored form: a(x − r₁)(x − r₂) where r₁, r₂ are zeros. Axis of symmetry: x = −b/(2a). Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a).

a > 0: parabola opens upward, vertex is minimum
a < 0: parabola opens downward, vertex is maximum
Discriminant: b²−4ac > 0 (2 real roots), = 0 (1 real root), < 0 (no real roots)

Exponential Models

f(x) = a·bˣ: a = initial value, b = base (growth factor). Percent growth rate r: b = 1 + r. Decay: b = 1 − r. Compound interest: A = P(1 + r/n)^(nt). Doubling time: when bˣ = 2.

Constant ratios between consecutive terms in a table signal an exponential model
Exponential growth eventually outpaces any linear or quadratic model
Half-life: time for quantity to halve; solved by setting a·bᵗ = a/2 → bᵗ = 1/2

Your strategy

Check a table for constant differences (linear), constant second differences (quadratic), or constant ratios (exponential)
For vertex problems, use x = −b/(2a), then substitute back to find y
For exponential models, write out a·bˣ and identify a from the initial condition and b from the rate
Backsolve: plug x-values from the problem into each answer choice to see which matches

Worked Examples

Easy Example 1 Miscalculating Slope By Using Y₁/x₁ Instead Of (y₂−y₁)/(x₂−x₁)

A line passes through (0, 4) and (3, 10). What is the equation of the line?

A. y = 2x + 4 (Correct answer)
B. y = 2x − 4
C. y = 3x + 4
D. y = 3x + 1
E. y = 4x + 2

Step 1

Find slope: m = (10 − 4)/(3 − 0) = 6/3 = 2

Step 2

y-intercept from (0, 4): b = 4

Step 3

Equation: y = 2x + 4

Correct answer: A

Why A is correct

Correct: slope 2, y-intercept 4

Why other options are wrong

B: Correct slope but wrong y-intercept sign

C: Wrong slope: used 10/3 ≈ 3.3

D: Miscalculated both slope and intercept

E: Swapped rise and run in slope calculation

⚠ Trap: Miscalculating slope by using y₁/x₁ instead of (y₂−y₁)/(x₂−x₁)

Medium Example 2 Using −b/a (forgetting The 2 In The Denominator Of The Vertex Formula)

The vertex of the parabola y = 2x² − 8x + 5 is at which point?

A. (2, −3) (Correct answer)
B. (2, 3)
C. (4, 5)
D. (−2, 25)
E. (1, −1)

Step 1

x-coordinate of vertex: x = −b/(2a) = −(−8)/(2·2) = 8/4 = 2

Step 2

y-coordinate: y = 2(2)² − 8(2) + 5 = 8 − 16 + 5 = −3

Step 3

Vertex: (2, −3)

Correct answer: A

Why A is correct

Correct: x = 2, y = −3

Why other options are wrong

B: Correct x but wrong y: arithmetic error in substitution

C: Used x = −b/a instead of x = −b/(2a) → x = 8/2 = 4

D: Sign error: used x = −b/2a as −2

E: Plugged in x = 1 without finding vertex x-coordinate first

⚠ Trap: Using −b/a (forgetting the 2 in the denominator of the vertex formula)

Hard Example 3 Using 0.08 As The Base Instead Of 1.08 (confusing Rate With Growth Factor)

A town's population was 5,000 in 2010 and grows by 8% per year. Which equation gives the population P after t years?

A. P = 5000 + 8t
B. P = 5000(0.08)^t
C. P = 5000(1.08)^t (Correct answer)
D. P = 5000(1.8)^t
E. P = 5000 + 0.08t

Step 1

Exponential growth model: P = a · bᵗ where a = initial value and b = 1 + growth rate

Step 2

a = 5000, growth rate = 8% = 0.08, so b = 1 + 0.08 = 1.08

Step 3

P = 5000(1.08)^t

Correct answer: C

Why C is correct

Correct: exponential growth at 8% per year

Why other options are wrong

A: Linear model: grows by a flat 8 people per year, ignores compounding

B: Uses 0.08 as base: the population would shrink to nearly zero each year

D: Uses 1.8: implies 80% growth per year, not 8%

E: Linear model with 0.08 people per year — extremely slow growth

⚠ Trap: Using 0.08 as the base instead of 1.08 (confusing rate with growth factor)

Strategy Tips

Memorize: constant first differences → linear; constant ratios → exponential; constant second differences → quadratic
For exponential growth/decay: b = 1 + rate (growth) or b = 1 − rate (decay), NOT just the rate
Vertex formula x = −b/(2a) has a 2 in the denominator — write it large to avoid forgetting it
Completing the square converts standard to vertex form if needed for harder problems
Backsolve by testing a specific input value against each answer choice

Common pitfalls

Using the rate r as the base in exponential models instead of (1 + r)

Mixing up the vertex x formula: using −b/a instead of −b/(2a)

Assuming all 'growing' situations are exponential — check if the increase is constant (linear) or proportional (exponential)

Model identification from a table: 30 seconds. Writing the equation: 30 seconds. Vertex calculation: 60 seconds. Just use x = −b/(2a) and plug back in — it's faster than completing the square.

Summary

Exponential models use b = 1 ± r, not r itself, as the base
The vertex of y = ax² + bx + c is at x = −b/(2a), y = f(−b/(2a))
To identify model type from a table: check first differences, second differences, and ratios

A table shows x: 0, 1, 2, 3 and y: 3, 6, 12, 24. Identify the model type, find the equation, and predict y when x = 5.

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