SAT Advanced Math: Exponentials and Nonlinear Functions

Exponent rules, the growth/decay master template y = a·b^x, and how to translate "increases by 5% per year" into a math model. Plus the extraneous-solution trap on radical equations.

8 分钟 TEKS ADV SAT Math

Exponential growth, exponent rules, and nonlinear functions show up 3–5 times per Digital SAT — often disguised as population, interest, or radioactive-decay word problems. Learn the rules and the templates, not the stories.

Exponent rules you must know cold

x^a · x^b = x^a+b x^a / x^b = x^a−b (x^a)^b = x^ab x⁰ = 1 (for any nonzero x) x^−a = 1 / x^a x^1/n = ⁿ√x SAT will combine 2 or 3 of these in a single question. Practice until automatic.

Exponential growth and decay model

The base r decides direction: r > 1 explodes upward, 0 < r < 1 decays toward zero.

The master template

y = a · b^x where:

a = starting amount (when x = 0)
b = growth factor per unit time
b > 1 means growth; 0 < b < 1 means decay

Percent growth → exponential base

Translate percent language to the b in y = a · b^x:

"increases by 5% per year" → b = 1.05 "decreases by 12% per year" → b = 0.88 "doubles every 7 years" → b = 2, time unit = 7-year "halves every 10 days" → b = 0.5, time unit = 10-day Anything compounding is exponential, not linear.

Worked example

A bacteria colony starts at 200 cells and triples every 4 hours. y = 200 · 3^(t/4) After 12 hours: y = 200 · 3³ = 200 · 27 = 5400 The exponent is (t/4), not t — divide by the time-unit length.

Other nonlinear functions on the SAT

Square root: y = √x — grows but slows down
Cubic: y = x³ — through the origin, S-shape
Rational: y = 1/x — has asymptotes (lines the curve approaches but never touches)
Absolute value: y = |x| — V-shape opening upward

Radical equations

To solve a radical equation, isolate the radical and square both sides. Always check answers — squaring can introduce extraneous (fake) solutions.

√(x + 7) = x − 5 x + 7 = (x − 5)² = x² − 10x + 25 0 = x² − 11x + 18 0 = (x − 2)(x − 9) → x = 2 or 9 Check: x = 2 gives √9 = −3? No (LHS positive, RHS negative). Reject. x = 9 x = 2 is extraneous. Skipping the check costs students an entire question.

Desmos for nonlinear

Graph the function and read off values. For exponential word problems, type the model and use the table feature to read values at specific x's — faster than algebra.

Polynomials and end behavior

For higher-degree polynomials, the SAT mostly tests end behavior and roots.

Even-degree, positive leading coefficient → both ends go up
Even-degree, negative leading coefficient → both ends go down
Odd-degree, positive leading coefficient → down-left, up-right
Odd-degree, negative leading coefficient → up-left, down-right

A polynomial has roots at x = 1, x = 3, x = −2. Equation form: y = a(x − 1)(x − 3)(x + 2) If the graph crosses the y-axis at y = 12, find a: 12 = a(−1)(−3)(2) = 6a → a = 2 Factored form makes roots and y-intercept calculations one line each.

Common mistakes

Treating "5% increase per year" as linear (+5 per year) instead of exponential (×1.05)
Forgetting to divide the exponent by the time-unit length
Not checking extraneous solutions after squaring
Confusing x^−a = 1/x^a with x^−a = −x^a
Forgetting that x⁰ = 1 (a common trick in exponent expressions)

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