Week 8 · EVERYTHING COVERED · 15–20 hrs

Last new topics: coordinate geometry, boldface & TPA

Coordinate geometry

Quantitative

On GMAT Focus, coordinate geometry is less a topic than a translation service. The exam rarely hands you a labelled graph. It gives you two vertices of a square in the $xy$-plane, a linear cost formula, or the statement $\mathrm{\mid }x-4\mathrm{\mid }<1$, and expects you to convert that into a line, a distance, or a region.

The disguises are consistent. Linear functions are lines: in $C=40+3n$, the $3$ is a slope and the $40$ is an intercept with meaning attached. Absolute value is distance from a coordinate, so $\mathrm{\mid }x-4\mathrm{\mid }\le 1$ describes a vertical band, not an equation to solve. Inequalities describe regions, and the question then asks whether a named point sits inside, above, or on the boundary. Data Sufficiency leans hard on loci: one statement may define an entire line or circle of possible points, which is rarely enough to pin down a single point.

Three decisions separate right from tempting. First, decide whether the question wants a signed value or a distance — midpoints and coordinate differences keep their signs, distances never do. Second, before using the perpendicular-slope rule, confirm neither line is vertical; the horizontal/vertical pair is the trap and the rule simply fails there. Third, subtract coordinates in the same order in numerator and denominator; flipping only one reverses the sign of the slope.

For memory: $m=\frac{y_2-y_1}{x_2-x_1}$ and $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$, with perpendicular slopes satisfying $m_1{m}_2=-1$ whenever both slopes are defined.

Functions and invented notation

Quantitative

GMAT Focus does not ask you to analyse functions. It asks whether you can execute a rule you have never seen, exactly, under time pressure. The rule arrives as an invented symbol ($a\mathrm{\square }b$), a named function with a piecewise definition, or a table. The mathematics is trivial; the discipline is not.

Disguises: a function defined only for positive integers, so $f(-1)$ is undefined; a rule that looks symmetric but is not ($a\star b=a-b$); composition written as $f(f(x))$ or $f(g(2))$ with $g$ defined separately; a piecewise rule where the boundary value belongs to one branch only. Read the domain sentence before you substitute anything.

The decisive moves are: substitute the entire input expression, not just the variable; preserve order when the operation is not commutative; and check whether the output feeds back as an input (composition) or whether the question asks for $x$ given $f(x)$ — a different task.

Strong candidates lose points mechanically by distributing: assuming $f(a+b)=f(a)+f(b)$ for an arbitrary invented rule. Nothing licenses that. Compute $f(a+b)$ directly.

For any function, $f(\text{input})$ means replace every instance of the variable in the definition with that entire input. That is the whole method. The condition: the input must lie in the stated domain.

Critical Reasoning: boldface and argument roles

Verbal

Boldface questions do not ask you to evaluate an argument; they ask you to map its architecture. The exam presents an argument with one or two boldfaced statements and asks what role each plays. The content is irrelevant except as a cue: a boldfaced sentence may be the main conclusion, an intermediate conclusion, a premise, a concession, an opposing view, a prediction, or a background fact. The same sentence type can play different roles depending on what supports what.

Disguises: a boldfaced claim attributed to "critics" or "some scientists" is not the author's conclusion unless the author adopts it. A concession introduced by "although" or "granted" is a premise the author accepts but does not rely on as the main support. A prediction or recommendation can be the main conclusion. An intermediate conclusion is supported by other statements and in turn supports the main point; it is neither the main conclusion nor a bare premise.

Two decisions separate the correct answer from the tempting one. First, who asserts the boldfaced statement, and does the author agree? Second, does the statement support another statement (premise) or is it supported by others (conclusion)? If it is supported by some and supports the main point, it is an intermediate conclusion. The tempting wrong answer often mislabels an intermediate conclusion as the main conclusion, or an opposing claim as the author's premise.

Strong candidates lose points by labelling from content—"this sounds like a conclusion"—without tracing support arrows. Read the argument twice: first for the author's main point, then for the role of each boldface.

Rule: a statement is a conclusion if other statements are offered as reasons for it, and a premise if it is offered as a reason for another statement. In "Critics say the policy will fail. But their evidence is flawed, so the policy will succeed," the first boldface is an opposing claim; the second is the main conclusion.

Graphics Interpretation

Data Insights

Graphics Interpretation is not a chart-reading test. Every question is a short sentence with two or three blanks, each behind a dropdown, and exactly one combination makes the sentence true. The graph is a constraint, not a story.

What it tests: whether you can pull a specific value, compare two series, or reason about a plotted relationship — then express it in the units the sentence demands.

The disguises. Dual-axis line charts where the left and right scales differ, so a series that appears to overtake another never does. Axes that do not start at zero, making a 3% change look like a doubling. Units swapped between graph and sentence (per-capita against total, thousands against millions). A line of best fit drawn through the points, inviting a precise claim the data cannot support.

Three decisions. Read axis labels, units and the plotted range of $x$ before reading shape. Choose the dropdown options jointly: the first blank often has two defensible answers until the second eliminates one. Separate interpolation from extrapolation — a fitted line licenses claims only within the plotted domain, so "the model predicts" for an unplotted $x$ is the wrong option, not a calculation to attempt.

The mechanical loss: candidates pick the first dropdown by eyeballing the trend, then hunt for a second option that fits, instead of testing candidate pairs against a specific plotted point.

Worth memorising: for a fitted line, $m={\displaystyle \frac{y_2-y_1}{x_2-x_1}}$, using two clear points on the line itself, not data points.

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