Nonlinear functions are one of the most heavily tested areas on the Digital SAT Math section (44 questions, 70 minutes). These questions fall under the Advanced Math category and focus on quadratic, exponential, polynomial, and rational functions. You need to move fluently between equations, tables, graphs, and real-world contexts—and interpret what each part of a function means in a given scenario.

more resources to help you study

practice questions cheatsheets

Quadratic Functions: Forms and Features

Quadratic functions produce parabolas when graphed. The Digital SAT tests whether you can recognize key features from different algebraic forms and connect those features to context.

Standard form: $f(x) = ax^2 + bx + c$

The y-intercept is $c$ (set $x = 0$ and you get $f(0) = c$ ).
The sign of $a$ tells you direction: $a > 0$ opens upward, $a < 0$ opens downward.
The vertex x-coordinate is $x = -\frac{b}{2a}$ .

Vertex form: $f(x) = a(x - h)^2 + k$

The vertex is $(h, k)$ , which gives you the minimum (if $a > 0$ ) or maximum (if $a < 0$ ).
Use this form when a question asks about the minimum or maximum value.

Factored form: $f(x) = a(x - r)(x - s)$

The x-intercepts are $x = r$ and $x = s$ .
Use this form when a question asks about zeros or x-intercepts.

The Digital SAT frequently asks you to identify the most suitable form to display a particular feature. If a question asks "what is the minimum value," vertex form shows it directly. If it asks "at what values of $x$ does the function equal zero," factored form is what you want.

Worked Example 1: Choosing the Right Form

A ball is launched upward. Its height in feet after $t$ seconds is modeled by $h(t) = -16t^2 + 64t + 80$ . What is the maximum height?

Find the t-coordinate of the vertex:

$t = -\frac{b}{2a} = -\frac{64}{2(-16)} = 2$

Plug $t = 2$ back in:

$h(2) = -16(4) + 64(2) + 80 = -64 + 128 + 80 = 144$

The maximum height is 144 feet. In vertex form, this is $h(t) = -16(t - 2)^2 + 144$ , which displays the maximum directly.

Worked Example 2: Interpreting Parts in Context

Using the same function $h(t) = -16t^2 + 64t + 80$ :

The constant 80 is $h(0)$ —the height when $t = 0$ . The ball was launched from 80 feet.
The coefficient −16 relates to gravitational acceleration.
The x-intercepts tell you when the ball is at ground level ( $h = 0$ ).

If the question asks "what does the 80 represent," the answer is the height of the ball at the moment it was launched.

Exponential Functions: Growth and Decay

Exponential functions model quantities that change by a constant percentage over equal intervals. The general form is $f(x) = a \cdot b^x$ , where $a$ is the initial value and $b$ is the growth factor.

When $b > 1$ : exponential growth (quantity increases).
When $0 < b < 1$ : exponential decay (quantity decreases).
The y-intercept is always $a$ , because $f(0) = a \cdot b^0 = a$ .

A common real-world model uses $f(t) = a(1 + r)^t$ , where $r$ is the rate of change per time period. For growth, $r > 0$ ; for decay, $r < 0$ (or written as $1 - r$ ).

Worked Example 3: Building an Exponential Model

A researcher counts 200 bacteria. The population doubles every 3 hours. Which function models the population after $t$ hours?

The initial value is 200. Doubling every 3 hours means the exponent counts 3-hour periods:

$P(t) = 200 \cdot 2^{t/3}$

After 9 hours:

$P(9) = 200 \cdot 2^{9/3} = 200 \cdot 2^3 = 1600$

Worked Example 4: Interpreting Exponential Constants

A car's value is modeled by $V(t) = 25000(0.85)^t$ , where $t$ is years after purchase.

25,000 is the initial value (y-intercept, when $t = 0$ ).
0.85 is the decay factor. Each year the car retains 85% of its previous value, losing 15% per year.

If asked "what does 0.85 represent," say: each year, the car's value is 0.85 times its value the previous year.

Interpreting Graphs and Points in Context

Points on the graph: Every point $(a, b)$ on $y = f(x)$ means the input $a$ produces output $b$ . Translate this into the scenario's units.

Intercepts:

The y-intercept (where $x = 0$ ) typically represents the starting value or initial condition.
The x-intercepts (where $f(x) = 0$ ) represent when the output quantity equals zero—for example, when a projectile hits the ground or a balance reaches zero.

Other graph features:

For a quadratic, the vertex is the maximum or minimum. In context, this could be the highest point of a trajectory or the lowest cost.
For an exponential, the curve's steepness shows how fast the quantity changes.
The Digital SAT may ask about intervals: "Over what interval is the function increasing?"

Worked Example 5: Interpreting a Graph Point

A company's profit in thousands of dollars is modeled by $P(x) = -2x^2 + 20x - 32$ , where $x$ is the product price in dollars. The point $(6, 16)$ is on the graph. What does this point mean?

When the price is $6, the company's profit is $16,000. Translate the coordinates into the context's units—that's all this question requires.

Connecting Tables, Graphs, and Equations

The Digital SAT frequently gives you one representation and asks you to identify another.

Table to equation—look at the pattern in outputs:

Constant first differences → linear (not this topic)
Constant second differences → quadratic
Constant ratios between consecutive outputs → exponential

Worked Example 6: Identifying Function Type from a Table

$x$	$f(x)$
0	5
1	15
2	45
3	135

Check ratios: $\frac{15}{5} = 3$ , $\frac{45}{15} = 3$ , $\frac{135}{45} = 3$ . Constant ratio of 3 → exponential. Initial value is 5:

$f(x) = 5 \cdot 3^x$

Equation to graph: Identify key features (intercepts, vertex, asymptotes, direction), then match to the graph.

Graph to equation: Read off key features and build the equation. For a parabola with vertex $(3, -7)$ opening upward: $f(x) = a(x - 3)^2 - 7$ with $a > 0$ .

Function Notation and Input/Output

Evaluating: Given $f(x) = 3x^2 - 2x + 1$ , find $f(4)$ :

$f(4) = 3(16) - 2(4) + 1 = 48 - 8 + 1 = 41$

Finding the input: Given $f(x) = x^2 + 2x$ , find $x$ when $f(x) = 15$ :

$x^2 + 2x = 15$ $x^2 + 2x - 15 = 0$

$(x + 5)(x - 3) = 0$

$x = -5 \text{ or } x = 3$

Worked Example 7: Rational Function Input

Given $g(x) = \frac{12}{x - 1}$ , find $x$ when $g(x) = 4$ :

$\frac{12}{x - 1} = 4$

$12 = 4(x - 1)$

$16 = 4x$ $x = 4$

The point $(4, 4)$ lies on the graph of $g$ .

Polynomial and Rational Functions, and Transformations

Polynomial functions of degree 3 or higher have smooth, continuous curves. A degree-3 polynomial can have up to 2 turning points and up to 3 x-intercepts. The Digital SAT may show a table of values and ask which equation could produce those values, or show a graph and ask you to identify the function.

Rational functions are ratios of polynomials, such as $f(x) = \frac{1}{x}$ or $f(x) = \frac{x + 2}{x - 3}$ . Key features include vertical asymptotes (where the denominator equals zero) and horizontal asymptotes.

Transformations apply to any function type. For $y = f(x)$ :

$f(x) + k$ shifts the graph up by $k$
$f(x - h)$ shifts the graph right by $h$
$-f(x)$ reflects over the x-axis
$af(x)$ stretches vertically by a factor of $a$

Worked Example 8: Transformation in Context

$f(x) = x^2$ is transformed to $g(x) = (x + 4)^2 - 1$ . What are the vertex and intercepts of $g$ ?

The vertex shifts from $(0, 0)$ to $(-4, -1)$ . For x-intercepts, set $g(x) = 0$ :

$(x + 4)^2 = 1$ $x + 4 = \pm 1$ $x = -3 \text{ or } x = -5$

The y-intercept is $g(0) = 16 - 1 = 15$ .

What to Watch For on Test Day

Match the form to the question. Minimum or maximum → vertex form. Zeros → factored form. Y-intercept → standard form's constant term. Don't convert unless you need to.
Exponential vs. quadratic in tables. Check ratios for exponential (constant ratio) and second differences for quadratic (constant second differences). This distinction appears often.
Context interpretation is about meaning, not just math. When asked "what does 0.97 represent," don't just say "it's the base." Say "each year, the value is 97% of the previous year's value."
$f(0)$ is the y-intercept; solutions to $f(x) = 0$ are the x-intercepts. The Digital SAT tests this connection directly.
Check that your answer makes sense in context. Negative time, negative populations, or unreasonable prices are signals to recheck. The Digital SAT includes trap answers that are mathematically valid but contextually meaningless.