SAT Math — quadratic functions: reading a parabola without a picture

A parabola can be written three ways, and each one hands you a different fact for nothing. Written as a product of two brackets, it shows where the curve crosses the horizontal axis. Written as a squared bracket plus a constant, it shows the turning point. Written out flat, it shows the height at which the curve meets the vertical axis. Knowing which form answers which question is most of the skill, and it is why a question that looks like it needs a graph usually needs no calculation at all.

The recurring trap in every form is the sign. A bracket reading minus four vanishes at positive four, and the wrong options throughout this set are built from taking the numbers as printed — one sign reversed, both reversed, neither. The turning-point form has the same trap with a twist, because the number inside the bracket reverses and the constant added on the end does not, so a learner who applies one rule to both gets one coordinate right and the other wrong.

Two questions are about symmetry, and they are the ones worth dwelling on. If a function gives the same output at two different inputs, those inputs sit at equal distances either side of the turning point, so the turning point is at their midpoint. That is enough to answer the question without knowing the function, without knowing what the common output was, and without any algebra whatsoever — which is a surprise the first time a learner meets it.

Two more convert between forms, one by completing the square, where the wrong options come from squaring the middle coefficient instead of half of it, or from building the right square and then forgetting to correct the constant for what squaring added.

The last pair puts a quadratic on a thrown ball, and they are deliberately a matched pair: one asks how high it gets, the other asks when. Both facts come from the same turning point, and each question carries the other's answer among its wrong options, with the unit attached so the mismatch is visible to anyone who reads carefully. Getting these two confused is among the most common ways to lose a mark on an otherwise easy question, and it is a reading error rather than a mathematical one.

Every explanation names the specific mistake behind each wrong option, and every turning point and crossing is checked by substitution. All quantities are plausible, every place is invented, and nothing is drawn from any official publication.

• Read the crossings from a factored quadratic, reversing the sign of the number in each bracket
• Read the turning point from a vertex form, reversing the bracket's number but not the added constant
• Say what the constant term and the sign of the leading coefficient tell you about a graph
• Find a turning point from two inputs that share an output, without knowing the function
• Complete the square correctly, halving the middle coefficient and correcting the constant
• Distinguish when a thrown object peaks from how high it gets, and notice which the question asked for

Written for this catalogue, with no source document. The content is the nonlinear-function material named in the College Board's own public description of the Advanced Math domain — quadratic functions, their equivalent forms and their graphs; every question, option and explanation here is original, and nothing is reproduced from any official publication or released test. Note that the digital test presents some of this material with a graph on screen, which a text-only bank cannot reproduce: every feature here is described in words instead. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the SAT, and it is not an exam centre.