Quick Answer
A nonlinear function is any mathematical relationship where the rate of change is not constant, resulting in a curved graph. On the Digital SAT, these typically appear in Math Modules 1 and 2 as quadratic or exponential models, making up approximately 25-30% of the Advanced Math section questions.
A nonlinear function is a function whose graph is not a straight line, meaning the variable is raised to a power other than one. Common examples include quadratic functions like $f(x) = ax^2 + bx + c$ and exponential functions like $g(x) = ab^x$.
Question: A population of bacteria triples every 4 hours. If the initial population is 200, which function P(t) represents the population after t hours? Solution: This is an exponential (nonlinear) function with the form $P(t) = P_0(r)^{t/k}$. Here, $P_0 = 200$, the rate $r = 3$, and the interval $k = 4$. Therefore, $P(t) = 200(3)^{t/4}$.
Confusing constant change with constant ratio: Students often apply linear growth (adding a fixed amount) to problems describing exponential growth (multiplying by a fixed factor).
Incorrectly identifying the vertex: When working with quadratic functions, students may mistake the y-intercept for the maximum or minimum value of the graph.
Misinterpreting negative exponents: In nonlinear models, students frequently forget that a negative exponent or a fractional base between 0 and 1 indicates decay rather than growth.
Students targeting 750+ should know that the SAT often tests the relationship between the algebraic form of a nonlinear function and its graphical transformations. For instance, understanding how changing the constant 'k' in $f(x) = a(x-h)^2 + k$ shifts the vertex vertically can save significant time compared to manually plugging in points or using a graphing calculator.
Quadratic Equation
A quadratic equation is a second-degree polynomial equation typically written in standard form as ax² + bx + c = 0. On the Digital SAT, these equations appear frequently in the Advanced Math section, accounting for approximately 15% of math questions. Students must solve them using factoring, completing the square, or the quadratic formula.
Polynomial
A polynomial is a mathematical expression consisting of variables, coefficients, and non-negative integer exponents. On the Digital SAT, polynomials frequently appear in the Advanced Math section, typically requiring students to add, subtract, multiply, or factor expressions. These questions often represent approximately 10-15% of the math content across both modules.
Exponential Function
An exponential function is a mathematical relationship where a constant ratio determines the change in the dependent variable. On the Digital SAT, these functions frequently appear in the Math section, specifically within the Advanced Math domain, often requiring students to interpret growth or decay constants in real-world modeling word problems.
A nonlinear function on the SAT refers to any mathematical relationship where the variable's power is not one, such as quadratic, exponential, or radical functions. These appear frequently in the Math section of the Digital SAT, specifically within the 'Advanced Math' category. They are used to model real-world scenarios where change is not constant, such as gravity, compound interest, or area calculations.
To identify a nonlinear function, look at the equation or the graph. If the equation contains an exponent other than 1 (like $x^2$ or $2^x$), it is nonlinear. Graphically, if the plot is a curve, such as a parabola or an exponential swoop, rather than a straight line, the function is nonlinear. On the SAT, keywords like 'doubles' or 'percent increase' usually signal nonlinear exponential functions.
The primary difference lies in the rate of change. A linear equation has a constant rate of change (slope), resulting in a straight-line graph represented by $y = mx + b$. In contrast, a nonlinear function has a variable rate of change, meaning its graph curves. While linear equations describe steady addition or subtraction, nonlinear functions often describe acceleration, compounding, or squaring relationships.
While the exact number varies by test version, nonlinear functions typically appear in approximately 20% to 35% of the Math section. These questions are categorized under 'Advanced Math' and 'Problem Solving and Data Analysis.' You will likely encounter several questions involving quadratics and at least two or three involving exponential growth or decay in each testing session.
