Solving Nonlinear Functions Questions on the SAT

The method: Name the family, then act

Wrong family, wrong plan. A constant add each step is linear; a constant multiply is exponential; an x² term is quadratic. Name the family first, then use that family’s tools.

Use Name → Interpret → Evaluate/Solve. For f(x) = a(x − h)² + k, vertex is (h, k). For f(x) = ab^x, a is the start and b is the growth/decay factor (b > 1 growth, 0 < b < 1 decay). Solve quadratics by factoring, square root, or quadratic formula; for exponentials, often evaluate or compare factors rather than “solve for t” with logs (rarely needed on the SAT).

Go faster without getting sloppy

For vertex form, read (h, k) cold. For standard ax² + bx + c, vertex x-coordinate is −b/(2a). Factor when integers are friendly. On MC exponential questions, compute one more year by multiplying the previous value by b — don’t restart from a every time if the table already gives a nearby year.

Spot the traps in the choices

Growth of 5% is factor 1.05, not 0.05 and not 5. Decay of 20% is factor 0.8. For quadratics, forgetting that (x − 3)² opens the same way as x² but shifted. Mixing linear “add 5 each year” with exponential “×1.05 each year” is the story trap.

When the model hides in a story

Read whether the quantity is multiplied or added each period. Projectile and area problems are usually quadratic; compound interest and population are exponential; constant-speed distance is linear.

Practice until it’s automatic

Practice questions

Here are five practice problems that help you hone your skills. Use the same method we learned earlier above to solve these problems. Remember, practice makes perfect.