Non-linear Functions & Graphs

1. Introduction

Once you leave straight lines behind, the SAT asks you to think about functions whose graphs curve: parabolas, exponential curves, higher-degree polynomials, and the occasional absolute-value or square-root shape. These Passport to Advanced Math questions test something deeper than computation — they test whether you can fluently translate between a function's algebraic form and its graph. Can you look at $f(x) = (x-2)(x+3)$ and instantly know where the curve crosses the $x$-axis? Can you see a parabola dipping below the axis and know what that means about its solutions?

This translation skill is enormously valuable because the SAT often gives you information in one representation (say, a graph) and asks a question that is easiest to answer in another (say, the factored equation). The students who score highest move freely in both directions, and they know the handful of "structural facts" — zeros are $x$-intercepts, the $y$-intercept is the value at $x = 0$, transformations shift and reflect predictably — that make these questions quick.

This article covers reading and interpreting non-linear graphs, the meaning of zeros and intercepts, finding intersections of two curves, the full set of function transformations (shifts, reflections, stretches), and the behavior of exponential graphs including asymptotes. Throughout, we emphasize the SAT-specific reasoning and the traps — especially the backwards horizontal shift — that catch unprepared students.

2. Core Concepts

2.1 Functions and their graphs

A function assigns to each input $x$ exactly one output $f(x)$. Its graph is the set of points $(x, f(x))$. Because each input has one output, a graph passes the vertical line test: no vertical line crosses it more than once. Non-linear functions simply have graphs that bend — but every interpretive tool (intercepts, intersections, transformations) still applies.

2.2 Zeros and $x$-intercepts

The zeros of a function are the inputs where $f(x) = 0$ — and these are exactly the $x$-intercepts of the graph, the points where it crosses or touches the $x$-axis. If $f(x) = (x - 2)(x + 3)$, the zeros are $x = 2$ and $x = -3$. The number of real zeros equals the number of times the graph meets the $x$-axis. A factor that repeats, like $(x-1)^2$, gives a point where the curve touches the axis and turns back rather than crossing.

2.3 The $y$-intercept

The $y$-intercept is the single point where the graph crosses the $y$-axis, found by evaluating $f(0)$. Don't confuse it with the zeros: the $y$-intercept is one output at input $0$, while zeros are inputs giving output $0$. For a polynomial in standard form, the $y$-intercept is just the constant term.

2.4 Intersections of two graphs

When two functions $y = f(x)$ and $y = g(x)$ are graphed together, their intersection points are the $(x, y)$ pairs satisfying both — the simultaneous solutions of the system. Algebraically you find them by setting $f(x) = g(x)$. The number of intersection points equals the number of real solutions of that equation, which is itself a common SAT question.

2.5 Transformations

Starting from a parent function $f(x)$, you can shift, stretch, and reflect it predictably:

• Vertical shift: $f(x) + k$ moves the graph up by $k$ (down if $k < 0$).
• Horizontal shift: $f(x - h)$ moves the graph right by $h$ — note this is opposite the sign inside, the most common point of confusion.
• Vertical stretch/compression: $a \cdot f(x)$ stretches away from the $x$-axis if $|a| > 1$, compresses if $0 < |a| < 1$.
• Reflection: $-f(x)$ flips over the $x$-axis; $f(-x)$ flips over the $y$-axis. A negative leading coefficient on a parabola flips it to open downward.

2.6 Exponential graphs and asymptotes

An exponential function $y = a \cdot b^x$ (with $b > 0$, $b \neq 1$) grows rapidly when $b > 1$ (growth) and shrinks toward zero when $0 < b < 1$ (decay). Its defining graphical feature is a horizontal asymptote: the curve approaches the line $y = 0$ ever more closely but never touches it (until you add a vertical shift, which moves the asymptote to $y = k$). The initial value $a$ is the $y$-intercept, since $b^0 = 1$ makes $y = a$ at $x = 0$.

2.7 Multiplicity and local graph behavior

If $(x - r)^n$ is a factor with odd $n$, the graph crosses the $x$-axis at $r$. If $n$ is even, the graph touches and turns. Example: $f(x) = (x - 2)^2(x + 1)$ crosses at $x = -1$ (odd power $1$) and touches at $x = 2$ (even power $2$).

2.8 End behavior of polynomials

For large $|x|$, the highest-degree term dominates. A quadratic with $a > 0$ rises on both ends (opens up); with $a < 0$ it falls on both ends. Cubics with positive leading coefficient fall left, rise right. End behavior rarely needs calculation on the SAT — but it helps you eliminate wrong graph choices quickly.

2.9 Absolute value graphs

$y = |x|$ is V-shaped with vertex at the origin. $y = |x - 3| + 2$ shifts right $3$ and up $2$, vertex at $(3, 2)$. Solving $|2x - 1| = 5$ splits into $2x - 1 = 5$ or $2x - 1 = -5$.

2.10 Square root graphs

$y = \sqrt{x}$ starts at $(0, 0)$, defined only for $x \ge 0$. $y = \sqrt{x - 4}$ shifts right $4$; domain $x \ge 4$. The SAT tests domain and range reading from graphs more than symbolic manipulation of radicals.

2.11 Average rate of change on a curve

Between $x = a$ and $x = b$, the average rate of change is $\dfrac{f(b) - f(a)}{b - a}$ — the slope of the secant line. On a nonlinear graph this differs from the instantaneous rate, but the SAT only asks for this secant slope, often from a table or two labeled points.

2.12 Function notation on graphs

$f(3)$ is the $y$-coordinate when $x = 3$ — move vertically to the curve. "For what $x$ is $f(x) = 4$?" means find $x$ where the graph hits height $4$ — move horizontally from $y = 4$. Students reverse these constantly.

2.13 Piecewise and restricted-domain graphs

A graph may show only part of a function — a parabola drawn for $x \ge 0$ only, or an exponential starting at $t = 0$. Read the visible domain from the endpoints. Questions about "all $x$" vs. "shown portion" differ.

2.14 Comparing function values from a graph

"For which $x$ is $f(x) > g(x)$?" means where the graph of $f$ lies above $g$. Shade mentally between intersection points; the SAT may ask for an interval description or a count of integer $x$ values in that region.