Nonlinear functions

Official Digital SAT skill — Advanced Math domain.

What this question tests

Nonlinear function questions test whether you can evaluate a function at a given input and interpret key features like minimum or maximum values, intercepts, and growth behavior from an equation or a graph. The test commonly uses quadratics and exponentials, and it expects you to choose the form of the function that makes the requested feature easiest to read. For quadratics, you may need to identify the vertex and decide whether the function has a minimum or maximum based on the leading coefficient. For exponentials, you may need to use points to determine parameters and then infer the function’s behavior or compute a value. This skill is tested because real-world modeling often uses nonlinear relationships, and interpreting them correctly requires more than just plugging in numbers—you must understand what the structure of the function is telling you.

What to know

A quadratic can be written in vertex form as $f(x)=a(x-h)^2+k$, where the vertex is $(h,k)$, and the function has a minimum value of $k$ if $a>0$ or a maximum value of $k$ if $a<0$.
A quadratic can also be written in standard form as $f(x)=ax^2+bx+c$, and you can find the vertex’s x-coordinate with $x=-\frac{b}{2a}$, then plug that value back in to get the corresponding function value.
In vertex form, $(x-h)^2\ge 0$ for all real $x$, so the sign of $a$ determines whether the parabola opens up or down and whether $k$ is a minimum or maximum value.
Factored form $f(x)=a(x-r_1)(x-r_2)$ makes zeros (x-intercepts) easy to read as $x=r_1$ and $x=r_2$, which helps when the question asks where the function equals zero or where the graph crosses the x-axis.
An exponential function often appears as $f(x)=a^x+b$ (or as $f(x)=A\cdot r^x$), where $b$ represents a vertical shift and the value at $x=0$ is $f(0)=1+b$ for $a^x$ form, which you can use to solve for parameters from a given point.
Evaluating a function means substituting the input into the formula exactly, using parentheses and order of operations, and interpreting a function means connecting the formula’s parameters to graph features like shifts, intercepts, and growth direction.

How to approach it

First, identify the function type (quadratic or exponential) and the specific feature the question asks for, because different forms reveal different features quickly.
Next, choose the most useful form for the task: use vertex form $a(x-h)^2+k$ to read a minimum/maximum value, factored form to read zeros, and standard form when you can compute the vertex using $-\frac{b}{2a}$.
If the function is quadratic and given in standard form, compute the vertex x-coordinate with $x=-\frac{b}{2a}$ and then evaluate the function at that x-value to get the minimum or maximum value, because the question may ask for the function value rather than the x-value.
If the function is quadratic and you see a completed square, read the vertex directly as $(h,k)$ and remember that the minimum or maximum value is the y-value $k$, not the x-coordinate $h$.
If the function is exponential and you are given points, use $f(0)$ to pin down a vertical shift when possible, then use another point to solve for the base or rate, because small algebra steps can reveal the parameters cleanly.
When evaluating at a specific x, substitute carefully with parentheses to avoid sign errors, especially when squaring negatives or applying exponents.
Finally, check your result against the graph behavior implied by the form (opening up/down for quadratics, increasing/decreasing for exponentials) to catch answers that are numerically possible but conceptually inconsistent.

Common traps

Confusing the minimum or maximum value with the x-coordinate where it occurs happens because the vertex is $(h,k)$ and students sometimes grab $h$ instead of $k$; avoid it by explicitly labeling $h$ as x-location and $k$ as function value.
Squaring or doubling the intended answer occurs when students perform an extra operation after identifying the vertex value; avoid it by stopping once you have the requested quantity and re-reading what the question asks for.
Misreading exponential parameters happens when students treat the base as the rate or ignore the vertical shift; avoid it by substituting a given point into the exact form and solving systematically for each parameter.
Choosing a form that makes the problem harder leads to algebra mistakes; avoid it by rewriting into vertex form for extrema or factored form for zeros before doing heavy computation.
Plugging in without parentheses causes sign and exponent errors, especially with negative inputs; avoid it by writing the substitution explicitly as $(x-h)$ or $(\text{input})$ in parentheses before computing.

Tips & shortcuts

If a quadratic is already in vertex form, the minimum or maximum value is usually one quick read: it is the $k$ value, and the direction depends on the sign of $a$.
If you only need zeros, try to factor first, because factored form turns the question into simple root reading.
For exponential forms with a vertical shift, use $f(0)$ early because it often reveals the shift immediately and reduces the remaining algebra.
Always re-check what the question asks (value versus location) because many wrong choices come from swapping the y-value and x-value of the vertex.

Worked example

For the function $h(x)=2\cdot 3^{x}+5$, what is the value of $h(2)-h(1)$?

A. $12$ ✓ (correct answer)
B. $6$
C. $24$
D. $18$

Why: Compute each value from the function. $h(2)=2\cdot 3^2+5=2\cdot 9+5=23$. $h(1)=2\cdot 3+5=6+5=11$. Therefore, $h(2)-h(1)=23-11=12$, so the correct choice is $B$.

Use the Practice Questions for this skill to drill it, then attempt a Timed Practice Test.