SAT Math · Advanced Math

Nonlinear Functions

Under 5% correct on real student attempts — the largest weak-skill sample in the system. One habit fixes most of it: set the right variable to zero.

The rule

Nonlinear functions bend — they aren't lines. To find key points, plug in zero for the right variable. x-intercepts come from setting y=0; y-intercepts and starting values come from setting x=0. In a quadratic, the vertex x-coordinate is exactly midway between the two x-intercepts. In an exponential, the multiplier out front is always the starting amount, and the base controls growth or decay.

Worked example

The function $f$ is defined by $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. The graph of $y = f(x)$ in the $xy$ -plane passes through the points $(10, 0)$ and $(-5, 0)$ . If $a$ is an integer greater than 1, which of the following could be the value of $a + b$ ?

A.
$-8$
B.
$-4$
C.
$5$
D.
$6$

Walk through it

Since the graph passes through $(10,0)$ and $(-5,0)$ , $f(x) = a(x-10)(x+5) = ax^2 - 5ax - 50a$ , so $b=-5a$ and $a+b = a-5a = -4a$ . The smallest integer $a>1$ is $a=2$ , giving $a+b=-8$ , choice A.

The traps

Swapping x and y roles.

A point like (1, 88) means the output is 88 when the input is 1. Wrong answers flip those numbers or pretend the change is constant.

Sign errors with roots.

When a quadratic is factored, wrong choices use the constant term as an intercept or flip the sign of a root.

Overlooking symmetry.

Given a vertex and one x-intercept, traps use a point on the wrong side instead of the midpoint.

The procedure

Identify what's asked — y-intercept, x-intercept, starting value, or meaning of a piece — and set the right variable to zero.
If factored, set each bracket to zero and solve. If standard quadratic, factor or use the quadratic formula.
For missing x-intercepts, average the known intercept with the vertex's x-coordinate to find the other.
In stories, check that units and roles match the equation before picking an interpretation.

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Drill this skill

Rule, traps, and procedure drawn from the verified skill playbook for “Nonlinear Functions” (drafted against real questions, verified by a second independent pass — see lib/content/playbooks/nonlinear-functions.md). Worked example is a real, verified SAT practice question from the live question bank, not generated for this lesson.