SAT Math – Advanced Math
Quadratic and Exponential Word Problems
Applying parabolic and growth/decay models to real-world scenarios
Word problems translate real-world situations into mathematical equations that require quadratic or exponential models. On the SAT, you'll analyze projectile motion, area optimization, population growth, compound interest, and radioactive decay—identifying when relationships involve squared terms or repeated multiplication, then solving to answer questions about maximum values, break-even points, or future quantities.
Success requires recognizing problem types, extracting relevant information, setting up appropriate equations, solving using vertex formulas or exponential properties, and interpreting results in context. These aren't just textbook exercises—they model physics (projectile trajectories), business (profit optimization), biology (population dynamics), finance (compound growth), and any scenario involving acceleration, squared relationships, or exponential change over time.
Understanding Problem Types
Quadratic Word Problems
Quadratic problems involve squared relationships, often height over time or area optimization.
Area problems: Dimensions that create rectangles, gardens, fences
Profit/revenue: \(P(x) = -ax^2 + bx - c\) (opens down, has max)
Key questions: Maximum height, time to hit ground, optimal dimensions
Exponential Growth Problems
Growth problems involve quantities that increase by a constant percentage each period.
Population: Bacteria, people, animals growing over time
Compound interest: Money growing with interest
Key insight: Multiplying by same factor repeatedly
Exponential Decay Problems
Decay problems involve quantities that decrease by a constant percentage.
Radioactive decay: Half-life problems
Depreciation: Car or equipment value decreasing
Cooling: Temperature approaching room temperature
Setting Up Equations from Context
Translate word problems into mathematical equations.
Define variables: Clearly state what x, t, h represent
Identify relationships: Quadratic (squared) or exponential (repeated %)
Solve and interpret: Answer must make sense in context
Essential Formulas and Models
Projectile Motion (Gravity)
\(h(t) = -16t^2 + v_0t + h_0\)
\(h(t)\) = height at time t (feet)
\(v_0\) = initial velocity (feet/second)
\(h_0\) = initial height (feet)
-16 is gravity constant (use -4.9 for meters)
Maximum/Minimum of Quadratic
For \(f(x) = ax^2 + bx + c\):
Vertex x-coordinate: \(x = -\frac{b}{2a}\)
Vertex y-coordinate: \(y = f\left(-\frac{b}{2a}\right)\)
Maximum if \(a < 0\), minimum if \(a > 0\)
Exponential Growth and Decay
Growth: \(A(t) = A_0(1 + r)^t\)
Decay: \(A(t) = A_0(1 - r)^t\)
\(A_0\) = initial amount
\(r\) = growth/decay rate (as decimal)
\(t\) = time (in consistent units)
Compound Interest
\(A = P(1 + \frac{r}{n})^{nt}\)
P = principal (initial amount)
r = annual interest rate (decimal)
n = times compounded per year
t = time in years
Common Pitfalls & Expert Tips
❌ Using wrong units
If time is in hours but rate is per year, convert! Exponential formulas require consistent time units.
❌ Forgetting negative sign in projectile formula
\(h(t) = -16t^2 + v_0t + h_0\) has negative coefficient for \(t^2\) because gravity pulls down!
❌ Confusing growth rate with growth factor
5% growth means multiply by 1.05, not 0.05. Factor = 1 + rate.
❌ Not interpreting answer in context
If question asks "when does it hit the ground," answer is the time value, not the height! Read carefully.
✓ Expert Tip: Identify the model type first
Look for keywords: "thrown/dropped" → quadratic projectile, "grows by 5% each" → exponential, "decreases by half" → decay.
✓ Expert Tip: Use vertex formula for max/min questions
For "maximum height" or "optimal value," find vertex using \(x = -\frac{b}{2a}\), then plug back in.
✓ Expert Tip: Check if answer makes sense
Negative time? Population of -500? If answer doesn't fit real-world context, recheck your work!
Fully Worked SAT-Style Examples
A ball is thrown upward from a height of 5 feet with an initial velocity of 48 feet per second. The height h(t) of the ball after t seconds is given by \(h(t) = -16t^2 + 48t + 5\). What is the maximum height reached by the ball?
Solution:
Step 1: Identify coefficients
\(a = -16\), \(b = 48\), \(c = 5\)
Step 2: Find time at maximum (vertex)
\(t = -\frac{b}{2a} = -\frac{48}{2(-16)} = \frac{48}{32} = 1.5\) seconds
Step 3: Calculate maximum height
\(h(1.5) = -16(1.5)^2 + 48(1.5) + 5\)
\(= -16(2.25) + 72 + 5\)
\(= -36 + 72 + 5 = 41\) feet
Answer: 41 feet
An object is dropped from a 144-foot tall building. Its height after t seconds is given by \(h(t) = -16t^2 + 144\). How many seconds until the object hits the ground?
Solution:
Step 1: Ground level means h = 0
\(0 = -16t^2 + 144\)
Step 2: Solve for t
\(16t^2 = 144\)
\(t^2 = 9\)
\(t = 3\) (take positive root only)
Context note:
Time cannot be negative, so discard \(t = -3\)
Answer: 3 seconds
A farmer has 200 feet of fencing to enclose a rectangular garden. The area A of the garden as a function of width w is \(A(w) = w(100 - w)\). What width maximizes the area?
Solution:
Step 1: Expand to standard form
\(A(w) = 100w - w^2 = -w^2 + 100w\)
\(a = -1\), \(b = 100\)
Step 2: Find vertex (maximum since a < 0)
\(w = -\frac{b}{2a} = -\frac{100}{2(-1)} = 50\) feet
Verification:
Width = 50 ft, Length = 100 - 50 = 50 ft (square!)
Perimeter: 2(50 + 50) = 200 ft ✓
Answer: 50 feet
A bacteria population starts with 500 bacteria and doubles every 3 hours. How many bacteria will there be after 12 hours?
Solution:
Step 1: Set up exponential model
Initial: \(A_0 = 500\)
Doubles every 3 hours: growth factor = 2
Number of 3-hour periods in 12 hours: \(\frac{12}{3} = 4\)
Step 2: Calculate
\(A(12) = 500 \times 2^4 = 500 \times 16 = 8{,}000\)
Answer: 8,000 bacteria
A car purchased for $30,000 depreciates at 15% per year. What will its value be after 5 years?
Solution:
Step 1: Identify decay model
Initial value: \(A_0 = 30{,}000\)
Decay rate: \(r = 0.15\)
Decay factor: \(1 - 0.15 = 0.85\)
Step 2: Apply formula
\(A(5) = 30{,}000(0.85)^5\)
\(= 30{,}000 \times 0.4437\)
\(\approx 13{,}311\)
Answer: Approximately $13,311
Sara invests $2,000 in an account that earns 6% annual interest compounded annually. How much will she have after 10 years?
Solution:
Use growth formula:
\(A_0 = 2{,}000\)
Growth rate: \(r = 0.06\)
Time: \(t = 10\) years
Calculate:
\(A(10) = 2{,}000(1.06)^{10}\)
\(= 2{,}000 \times 1.7908\)
\(\approx 3{,}581.60\)
Answer: Approximately $3,581.60
A company's revenue R (in thousands) from selling x thousand units is modeled by \(R(x) = -2x^2 + 20x - 18\). How many units should be sold to maximize revenue?
Solution:
Identify coefficients:
\(a = -2\), \(b = 20\), \(c = -18\)
Since \(a < 0\), parabola opens down → maximum exists
Find maximum:
\(x = -\frac{b}{2a} = -\frac{20}{2(-2)} = \frac{20}{4} = 5\)
Interpret:
x represents thousands of units
5 thousand = 5,000 units
Answer: 5,000 units (or 5 thousand units)
A population of 1,000 animals grows at 8% per year. The population P after t years is given by \(P(t) = 1{,}000(1.08)^t\). After how many years will the population reach 2,000?
Solution:
Set up equation:
\(2{,}000 = 1{,}000(1.08)^t\)
Solve for t:
\((1.08)^t = 2\)
Take log or test values:
Try \(t = 9\): \(1.08^9 \approx 2.0\) ✓
Alternative approach:
Use logarithms: \(t = \frac{\log 2}{\log 1.08} \approx 9.0\) years
Answer: Approximately 9 years
Problem Type Recognition
Use Quadratic When...
• Thrown/dropped objects (gravity)
• Area/perimeter optimization
• Profit/revenue maximization
• Variable squared appears
Use Exponential When...
• "Grows/decreases by X% per..."
• Population over time
• Compound interest
• Doubles/halves repeatedly
Word Problems: Connecting Mathematics to Reality
Quadratic and exponential word problems bridge abstract mathematics and tangible phenomena, requiring translation skills that extend far beyond classroom exercises. The SAT tests these applications because they represent mathematical literacy essential for informed citizenship and professional success—understanding that projectile motion follows \(h(t) = -16t^2 + v_0t + h_0\) explains why basketball arcs peak at predictable moments, why engineers calculate optimal launch angles, and why safety regulations specify maximum heights for amusement park rides. Quadratic optimization models business decisions: maximizing profit by finding the vertex of revenue functions, determining optimal production levels where marginal costs equal marginal revenue, designing containers that minimize material while maximizing volume. Exponential models capture compounding processes fundamental to finance, biology, and physics—compound interest transforms modest savings into substantial retirement funds, bacterial populations explode under favorable conditions, and radioactive decay provides carbon dating for archaeological discoveries. Master the recognition skills that distinguish model types: keywords like "thrown upward" or "maximum height" signal quadratics with vertex calculations, while "grows by 5% each year" or "doubles every hour" indicates exponential with repeated multiplication. Develop the habit of defining variables explicitly, writing equations that match units consistently, solving systematically using appropriate formulas, and most critically—interpreting answers in context. When calculations yield negative time or populations exceeding planetary capacity, mathematical solutions must bow to physical constraints. These problem-solving skills transcend test preparation, equipping you to analyze mortgage amortization schedules, evaluate investment opportunities, understand epidemiological projections, and question claims about resource sustainability or climate trends. Every time you model a real situation mathematically, solve the resulting equation, and interpret results meaningfully, you're exercising the quantitative reasoning that distinguishes informed decision-makers from passive consumers of numerical claims.
