# Nonlinear Functions on the SAT: Examples & Practice | 1600.now Source: https://1600.now/sat-skill/nonlinear-functions --- Math · Advanced Math ## Nonlinear Functions The form of a nonlinear function tells you what you can read directly. Factored form exposes zeros, vertex form exposes a quadratic maximum or minimum, and exponential form exposes an initial value and multiplier. Written by [Luke Finigan](https://1600.now/about) 2 min read Updated Oct 2, 2026 ### Choose the quadratic form that reveals the requested feature The function $f(x)=2(x-3)^2-8$ has vertex $(3,-8)$. Because the coefficient 2 is positive, $-8$ is its minimum output. To find zeros, set it equal to zero: $(x-3)^2=4$, giving $x=1$ and $x=5$. In factored form, $f(x)=2(x-1)(x-5)$ makes those zeros visible. The vertical intercept is found separately: $f(0)=10$. A root is an input; a minimum value is an output. ### Use the axis of symmetry as a check The zeros 1 and 5 are equally far from 3, so the axis is $x=3$. For a quadratic in standard form $ax^2+bx+c$, the axis is $x=-\frac{b}{2a}$. Substitute that input to find the vertex's output. Completing the square can rewrite a standard-form expression when a question asks for the minimum or maximum. Keep the leading coefficient attached to the entire squared term. ### Read exponential multipliers over the stated interval In $P(t)=200(1.08)^t$, where $t$ counts years, 200 is the initial amount and 1.08 means an 8% increase each year. For $200(0.92)^t$, the amount decreases by 8% per year. If the exponent is $t/3$, the multiplier applies every three years. The annual multiplier is $(1.08)^{1/3}$, not 1.08. Read the input unit and exponent together. ### Separate transformations from individual values Changing $f(x)$ to $f(x)+4$ raises every output by 4. Changing it to $f(x-4)$ shifts its graph right by 4. For the vertex expression $(x-h)^2$, the input reaches the vertex when $x=h$. Test a transformed expression at a convenient input if the direction is unclear. That check is more reliable than trying to remember a sign rule in isolation. ### Check your understanding Try it yourself: What is the minimum value of $f(x)=3(x+2)^2+7$? · A $-2$ B $3$ C $7$ D $2$ ### Sample nonlinear functions questions Nonlinear Functions · Easy $f(x)=(x-1)^{2}+7$ What is the minimum value of the given function? A 1 B 6 C 7 D 8 - [Open the question and explanation](https://1600.now/bank/math/2f951bd1) Nonlinear Functions · Easy $p(x)=2x^{2}+20x+15$ What is the minimum value of the given function? A -35 B -15 C -5 D 15 - [Open the question and explanation](https://1600.now/bank/math/9f43d2b6) ### Practice this skill 1. Identify whether the prompt asks for an input, output, rate, or parameter. 2. Rewrite only when the current form hides the requested feature. 3. Check graph readings with a substitution or symmetry calculation. [Practice nonlinear functions](https://1600.now/bank/math/skill/Nonlinear%20functions) [Print the nonlinear functions worksheet and worked answers](https://1600.now/sat-nonlinear-functions-worksheet). Then use a [mixed practice module](https://1600.now/modules) to check whether you can choose this method among other question types. ### Source and question labels The bank label for this guide is "Nonlinear functions" in Advanced Math. The worked examples above are written for this guide. - [College Board: Types of Math tested](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types) ### Related skills - **[Nonlinear Equations and Systems](https://1600.now/sat-skill/nonlinear-equations-and-systems)** Choose a method that fits the equation: factor a quadratic, isolate a radical before squaring, split an absolute value into cases, or substitute to find graph intersections. - **[Equivalent Expressions](https://1600.now/sat-skill/equivalent-expressions)** Equivalent expressions give the same value for every input where both are defined. - **[Linear Functions](https://1600.now/sat-skill/linear-functions)** A linear function changes by the same amount for each equal increase in its input.