Quadratic and exponential word problems

How the SAT tests this, with worked examples — plus the 4 skills it covers. Free to read, no account needed.

The first decision in these questions is which kind of model the words describe, and one test settles it. If a quantity changes by the same amount each period it is linear; if it changes by the same factor or percentage, it is exponential.
et_linear_vs_exponential.png
Phrases such as doubles, halves, grows by 8% a year and decays all signal exponential behaviour, written as y=a⋅bxy = a \cdot b^x where aa is the starting amount and bb the multiplier. Growth of 8%8\% means b=1.08b = 1.08; decay of 8%8\% means b=0.92b = 0.92.
Quadratic models appear whenever something rises and then falls — a thrown object, a profit that peaks — or when two varying lengths multiply to give an area.
For a quadratic, the vertex answers the most common questions: the maximum height, the time it occurs, the price giving the greatest revenue. The roots answer when the quantity reaches zero, such as when a ball lands.
As always, read what is being asked. A question giving h=−5t2+20th = -5t^2 + 20t may want the maximum height, the time at that height, or the time it hits the ground — three different parts of the same model.

Worked examples

A population of 500500 grows by 4%4\% a year. Write a model for the population after tt years.
Growth by a percentage is exponential, with multiplier 1.041.04
P=500(1.04)t{P = 500(1.04)^t}
A sample of 8080 grams halves every 66 years. Write a model.
Halving is a constant factor, and the exponent counts how many 66-year periods have passed
m=80(12)t/6{m = 80\left(\frac{1}{2}\right)^{t/6}}
A ball follows h=−5t2+20th = -5t^2 + 20t. When does it land?
Landing means h=0h = 0, so factor
−5t(t−4)=0  ⇒  t=0 or t=4{-5t(t - 4) = 0 \;\Rightarrow\; t = 0 \text{ or } t = 4}
t=0t = 0 is the throw, so it lands at t=4t = 4 seconds. Note the maximum height occurs at t=2t = 2, halfway between the roots.

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