Nonlinear Functions

Nonlinear functions show up throughout the Digital SAT's Advanced Math section. This topic focuses on two function families in particular: quadratic functions and exponential functions, though you'll also need to work with polynomial functions, rational functions, and other nonlinear types. The core skill is connecting different representations (equations, tables, graphs, and real-world contexts) and interpreting what the parts of a function mean. Expect roughly 6–8 questions on nonlinear functions across both math modules, making this one of the most heavily tested areas on the exam.

Your job on these questions isn't just to solve equations. You need to read a scenario, pick the right model, interpret constants and variables in context, and move fluently between a table of values, a graph, and an algebraic expression. The SAT rewards students who can look at a function and immediately see what each piece tells you.

Quadratic Functions: Forms and Features

Quadratic functions produce parabolas when graphed. The 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 $a < 0$).
• This form is best when a question asks about the minimum or maximum value.

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

• The intercepts are $x = r$ and $x = s$.
• This form is best when a question asks about zeros or x-intercepts.

The SAT frequently asks you to determine the most suitable form of the expression to display a particular feature. If a question asks "what is the minimum value of the function," 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 the ball reaches?

You need the vertex. Find the t-coordinate:

$t = -\frac{b}{2a} = -\frac{64}{2(-16)} = -\frac{64}{-32} = 2$

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 function 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 initial height when $t = 0$. The ball was launched from 80 feet.
• The coefficient $-16$ relates to gravitational acceleration.
• The intercepts of the graph tell you when the ball is at ground level ($h = 0$).

If the question says "what does the 80 represent in this context," 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 $b > 1$, you have exponential growth. The quantity increases.
• When $0 < b < 1$, you have exponential decay. The quantity decreases.
• The y-intercept is always $a$, because $f(0) = a \cdot b^0 = a$.

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

Worked Example 3: Building an Exponential Model

A researcher counts 200 bacteria in a sample. 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 growth factor of 2 applies every 3-hour period, so the number of doubling periods in $t$ hours is $\frac{t}{3}$:

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

If the question asks for the population after 9 hours:

$P(9) = 200 \cdot 2^{9/3} = 200 \cdot 2^3 = 200 \cdot 8 = 1600$

Worked Example 4: Interpreting Exponential Constants

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

• 25,000 is the initial value of the car (the y-intercept, when $t = 0$).
• 0.85 is the decay factor. Each year, the car retains 85% of its previous value, meaning it loses 15% per year.
• The function never reaches zero, but the value keeps decreasing.

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

Interpreting Graphs and Points in Context

Several learning objectives focus on reading graphs of quadratic and exponential functions and explaining what specific points or features mean.

Points on the graph: Every point $(a, b)$ on the graph of $y = f(x)$ means that when the input is $a$, the output is $b$. In context, you translate this into the scenario's units.

Intercepts:

• The y-intercept is where $x = 0$. For $y = f(x)$, this is $f(0)$. It typically represents the starting value or initial condition.
• The x-intercepts are where $f(x) = 0$. These are the solutions to $f(x) = 0$. In context, they represent when the output quantity equals zero (e.g., when a projectile hits the ground, or when a balance reaches zero).

Other graph features:

• For a quadratic, the vertex represents the maximum or minimum value. 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 is changing. An increasing section of the graph means the output is growing; a decreasing section means it's shrinking.
• The SAT may ask about an interval: "Over what interval is the function increasing?" or "What does the decreasing portion of the graph represent?"

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 price in dollars of their product. 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. That's it. Translate the coordinates into the context's units.

Connecting Tables, Graphs, and Equations

The SAT frequently gives you one representation and asks you to identify another. Here's how to move between them.

Table to equation: Look at the pattern in the outputs.

• If the differences between consecutive outputs are constant, it's linear (not this topic).
• If the second differences are constant, it's quadratic.
• If the ratios between consecutive outputs are constant, it's exponential.

Worked Example 6: Identifying the Function Type from a Table

Check ratios: $\frac{15}{5} = 3$, $\frac{45}{15} = 3$, $\frac{135}{45} = 3$. The ratio is constant at 3, so this is exponential. The initial value (when $x = 0$) is 5, and the base is 3:

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

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

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

Function Notation and Input/Output

Function notation like $f(x)$ is how the SAT communicates input/output relationships. You need to be comfortable in both directions.

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 $f(x) = 15$:

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

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

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

For exponential, polynomial, and rational functions, finding the input for a given output follows the same idea: set the function equal to the output and solve.

Worked Example 7: Rational Function Input

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

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

$12 = 4(x - 1)$

$12 = 4x - 4$

$16 = 4x$ $x = 4$

This connects to the graph: the point $(4, 4)$ lies on the graph of $g$.

Polynomial and Rational Functions, and Transformations

Beyond quadratics and exponentials, the SAT tests your ability to work with polynomial functions of higher degree and simple rational functions, especially when matching tables, graphs, and equations.

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 SAT may show you 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, like $f(x) = \frac{1}{x}$ or $f(x) = \frac{x + 2}{x - 3}$. Their 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 $a$

When transformations are involved, the SAT may give you the graph of $f(x)$ and ask about $f(x - 2) + 3$, or give you a table for $f(x)$ and ask you to produce the table for a transformed version.

Worked Example 8: Transformation in Context

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

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

$(x + 4)^2 - 1 = 0$

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

The y-intercept is $g(0) = (4)^2 - 1 = 15$.

What to Watch For on Test Day

1. Match the form to the question. If they ask about the minimum, think vertex form. If they ask about zeros, think factored form. If they ask about the y-intercept, standard form shows it as the constant term. Don't waste time converting unless you need to.

2. Exponential vs. quadratic identification. In tables, check ratios for exponential (constant ratio) and second differences for quadratic (constant second differences). This distinction comes up often.

3. Context interpretation questions are about units and meaning. When a question asks "what does the number 0.97 represent," your answer should reference the real-world scenario, not just say "it's the base." Say something like "each year, the value is 97% of the previous year's value."

4. Don't forget $f(0)$ is the y-intercept. The SAT explicitly tests whether you know that $f(0)$ gives the y-intercept and that solutions to $f(x) = 0$ give the x-intercepts. These are quick points if you're clear on the connection.

5. Check that your answer makes sense in context. Negative time, negative populations, or prices above a reasonable range should make you pause. The SAT includes trap answers that are mathematically valid but contextually meaningless.