Nonlinear Equations & Functions — 700-800

Nonlinear Equations & Functions: The 700-800 Patterns

Part 1 of 3 — The Archetypes Hard-Tier Items Are Built From

Four archetypes generate almost every hard nonlinear item.

Archetype 1: Tangency via the Discriminant

"The line and the parabola intersect at exactly one point" — set equal, collect on one side, discriminant $= 0$. The parameter you solve for is almost never the final answer: the item asks for the $y$-coordinate of the tangent point, or a rejected case filters the parameter ("$m$ is negative").

Worked example. $y = x^{2} - 6x + 13$ meets $y = mx + 4$ once, $m < 0$. Then $x^{2} - (6+m)x + 9 = 0$ needs $(6+m)^{2} = 36$, so $m = 0$ or $m = -12$; take $m = -12$. The tangent point: $x^{2} + 6x + 9 = 0$ gives $x = -3$, and $y = 9 + 18 + 13 = 40$.

Archetype 2: Radical Equations and Extraneous Roots

Squaring both sides manufactures fake solutions. After solving, check each root in the ORIGINAL equation — any root that makes the right side negative is dead on arrival, because $\sqrt{\phantom{x}}$ never outputs a negative. The item then asks for a value built from the surviving root, and the extraneous root's version is a planted choice.

Archetype 3: Line–Parabola Intersections and Symmetry

Two tools:

• Vieta on the intersection equation: the sum of the $x$-coordinates of the intersection points is $-\frac{b}{a}$ of the combined quadratic. But if the question asks about $y$-coordinates, push each $x$ through the LINE (cheaper than the parabola).
• Symmetry for horizontal lines: if $y = k$ cuts a parabola at two points $d$ apart, those points sit $\frac{d}{2}$ on each side of the axis $x = -\frac{b}{2a}$. Find the axis, step out, evaluate once.

Archetype 4: Absolute-Value Models

$P = 400 - 25|q - 18|$ style. Isolate the absolute value, and remember: dividing by a negative flips the inequality. $|x - c| \le r$ means the interval $[c - r,\, c + r]$ — length $2r$, and it contains $2r + 1$ integers when $c$ and $r$ are integers.

Nonlinear Equations & Functions: Traps & Speed

Part 2 of 3 — Distractor Autopsy and Faster Routes

The Four Standard Distractors

1. The extraneous root. Any answer built from the root that fails the original radical equation. Fast filter: a root making the non-radical side negative is extraneous — no substitution needed.
2. The wrong coordinate. Vieta gives the sum of $x$-coordinates in one step, so the test asks for the sum of $y$-coordinates. The $x$-sum is always a choice. Push each $x$ through the LINE (never the parabola) to get the $y$-values.
3. The unflipped inequality. Isolating $|x - c|$ usually requires dividing by a negative. Forgetting the flip turns "inside the interval" into "outside" and produces a mirror-image answer set.
4. The half-width / endpoint / count confusion. From $|x - 6| \le 4$: the half-width is $4$, the interval is $[2, 10]$, its length is $8$, and it contains $9$ integers. All four numbers appear in the choices — know which one was asked.

Speed Techniques

• Extraneous pre-check: before solving $\sqrt{ax + b} = x - c$, note that any valid root must satisfy $x \ge c$. This tells you in advance which quadratic root will survive.
• Symmetry beats substitution: points on a parabola with equal $y$-values straddle the axis symmetrically. Distance conditions become one axis computation plus one half-step.
• Desmos exploits: graph both sides of a radical equation — the intersections are the TRUE solutions only, extraneous roots never appear. For tangency, add a slider and watch the two intersection points merge. For absolute-value models, graph the expression and the threshold line; the answer is read off the intersection $x$-values.
• Integer counting: the integers in $[a, b]$ number $b - a + 1$. The "$+1$" is the whole trap — the test plants $b - a$ every time.

Nonlinear Equations & Functions: Timed Drill

Part 3 of 3 — Four Questions at Full Difficulty

Budget about 75 seconds per question. Opening moves: tangency $\rightarrow$ discriminant; radical $\rightarrow$ note which sign the non-radical side must have; symmetric points $\rightarrow$ find the axis first; absolute value $\rightarrow$ isolate, flip if you divide by a negative. Before clicking, confirm your number is the QUANTITY asked — coordinate, parameter, length, or count.