Concept: $m$ = change in $y$ when $x$ increases by 1. $b$ = starting value when $x=0$.

Mistake: Students mix “initial value” with “y-intercept”. DSAT uses wording like “At time 0…” or “Originally…”, all meaning $b$.

Mental Trick: Replace $m$ with “per”. Example: “\$5 per hour + 10 base fee” → $y = 5x + 10$.

Graph Interpretation: If $m$ is positive → rising line. If $m$ is negative → decreasing line.

DSAT Master Tip: Whenever the problem uses “each”, “every”, or “per”, that number is **always** the slope.

Concept: Measures steepness: “vertical change per horizontal change”.

Mistake: Mixing the order of points. Use the same order for numerator & denominator.

Mental Trick: Read slope as “rise over run”.

Graph Insight: Slope = “how quickly the line moves up or down”.

DSAT Tip: If $x$ increases steadily (like time), slope tells you how fast something is changing.

Concept: A formula for lines when a point and slope are known.

Common Error: Students forget that $(x_1,y_1)$ are constants, not variables.

Mental Trick: Insert the point like a “plug”. If the line goes through (4,7), replace $x_1=4$, $y_1=7$.

DSAT Tip: Most DSAT linear models are given as “at $x = A$, value is B”. This is point slope structure.

Concept: This is the most general form. You can see the “shape” (depending on $a$), the direction (up or down), and the y-intercept ($c$).

Graph Meaning:
• If $a > 0$ → parabola opens upward (minimum). • If $a < 0$ → parabola opens downward (maximum). • $c$ = y-intercept = the value at $x = 0$.

Common Mistake: Students think $b$ shifts the graph horizontally. It does NOT. It affects the slope at the intercept, not the position directly.

DSAT Master Tip: Standard form is used for quick evaluation and plugging values. If the question wants roots → convert to factored form or use the quadratic formula.

Concept: The easiest way to “see” a quadratic’s key point: the vertex. $(h,k)$ is always the turning point.

Graph Insight:
• If $a>0$ → vertex = minimum • If $a<0$ → vertex = maximum • Larger $|a|$ → narrower parabola

Mental Trick:
Think: “Shift, then stretch.” $(x-h)$ shifts; $a$ stretches.

DSAT Tip: Maximum profit, minimum distance, minimum cost — these ALWAYS use vertex form or require completing the square.

Concept: Factored form exposes the roots immediately.

Axis of symmetry: $x = \frac{r_1 + r_2}{2}$

Graph Meaning: Each root is where the graph touches/crosses the x-axis.

Common Mistake: Students forget to divide the sum by 2 for the axis.

DSAT Trick: If roots are messy, DSAT often expects conceptual reasoning, not full factorization.

Concept: This formula finds intercepts by solving $ax^2+bx+c=0$.

Discriminant:
$b^2 – 4ac$ tells how many roots exist: • > 0 → two real roots • = 0 → one real root • < 0 → no real roots

Mental Trick: Compute discriminant FIRST. Saves time.

Common Mistake: Forgetting denominator $2a$ applies to the whole numerator.

DSAT Master Tip: Many DSAT items only ask about the sign of the discriminant, not actual roots.

Concept: The parabola is symmetric. This formula gives the center of symmetry.

Mental Trick: Think of it as “where the slope becomes zero”.

Graph Insight: Vertex is always on this line.

DSAT Tip: If DSAT gives you a table where values repeat (e.g., y-values mirror), the axis is exactly between them.

Concept: When $a>0$, it’s the minimum; when $a<0$, it's the maximum.

Interpretation: The real-world turning point (best value).

DSAT Trick: DSAT loves vertex problems disguised as “best deal”, “maximum height”, “minimum cost”.

Meaning:
• $\Delta>0$ → 2 intersections with x-axis • $\Delta=0$ → tangent to x-axis • $\Delta<0$ → no real intersections

Mental Trick:
Use discriminant before attempting full algebra.

DSAT Tip:
Many DSAT questions only require comparing discriminants, not computing exact roots.

Concept:
Exponential functions change by constant percent, not constant amount.

Graph Insight:
• If $b > 1$: curve rises, accelerating.
• If $0 < b < 1$: curve decays, flattening out.

Mental Trick:
Think: “Multiply each step.” Linear = add, Exponential = multiply.

DSAT Master Tip:
If the problem says “increases by X% each year” → the base is $1 + \frac{X}{100}$. If it says “decreases by X%” → the base is $1 – \frac{X}{100}$.

Common Mistake:
Mixing up $b$ with rate → $b$ is the multiplier, NOT the percent.

Meaning:
$r$ is the percent rate (e.g., 12% → $0.12$). $(1+r)$ is the multiplier (e.g., 12% growth → $1.12$).

Mental Trick:
Percent → Multiplier is the KEY DSAT skill.

Common Mistake:
Students incorrectly add percent repeatedly instead of compounding.

DSAT Tip:
If DSAT asks “after 3 years”, ALWAYS use exponent 3 — NOT multiply by 3.

Meaning:
$1 – r$ = multiplier for decay (e.g., 20% loss → $0.80$ multiplier).

Graph Insight:
Always decreases but never reaches 0.

Common Mistake:
Doing “subtract 20 three times” → DSAT expects compounding (multiply by $0.8^3$).

DSAT Master Tip:
Two consecutive 20% losses is NOT 40% loss — it’s $0.8 \times 0.8 = 0.64$ (36% total).

Concept:
Doubling happens when repeated multiplication catches up to ×2.

Solving:
$t = \dfrac{\log 2}{\log b}$

Mental Trick:
Faster growth → smaller doubling time.

DSAT Tip:
DSAT rarely asks for exact answers — approximate using calculator.

Concept:
Half-life measures how long it takes for repeated decay to reduce value by half.

Formula:
$t = \dfrac{\log(1/2)}{\log b}$

Graph Insight:
Same shape for all decays — only speed changes.

DSAT Tip:
If $b = 0.5$, half-life = 1 step. If $b = 0.8$, half-life is longer.

Step 1: Divide both sides → $b^x = \frac{c}{a}$

Step 2: Take logs → $x = \dfrac{\log(c/a)}{\log(b)}$

Mental Trick:
Logarithm = “undo exponential”.

Common Mistake:
Taking log too early (before isolating exponential term). DSAT bunu çok dener.

DSAT Master Tip:
Use LN or LOG — DSAT does not care which, because ratio cancels.

Increase = original + part of original → multiply by $(1+r)$.

DSAT Tip: If growth happens annually, use exponent = number of years.

Decay = original minus part of original.

Common Mistake: Students subtract percent repeatedly instead of compounding.

DSAT Master Rule:
Repeated change ALWAYS → exponent.

Mental Trick:
“Out of what?” is the key DSAT question — many items change the whole.

Common Mistake:
Students use the wrong “whole” when two sets of data are compared.

DSAT Tip:
If “percent increase” is asked, the whole is the original value. If “percent of total” is asked, the whole is the total.

Meaning:
The base of comparison is always the old value.

Common Mistake:
Using new value as denominator → DSAT bunu bilerek tuzak olarak kullanır.

DSAT Master Tip:
If A → B is +30%, the reverse (B → A) is NOT -30%. It is $\frac{1}{1.3} – 1 = -23.08\%$.

Mental Trick:
• +20% → ×1.20 • -30% → ×0.70

Key Insight:
Repeated percent changes always multiply, never add.

DSAT Tip:
Two consecutive discounts 20% and 30% is: $0.8 \times 0.7 = 0.56$ → total 44% decrease.

Mental Trick:
A ratio is a “relative multiplier.”

DSAT Tip:
If ratio changes but total stays same → find scaling factor.

Cross Multiplying: $ad = bc$ solves all proportion questions.

Common Mistake:
Writing ratios in wrong order (DSAT bunu deniyor).

DSAT Tip:
Always align units: (boys/girls) = (boys/girls).

Graph Insight:
Scale factor multiplies dimensions but area multiplies by factor².

DSAT Master Tip:
If DSAT mentions “similar figures,” scale factor is the entire problem.

Mental Trick:
ALWAYS convert to unit rate (“per 1”).

Connection:
In data tables, slope = unit rate.

Common Mistake:
Not converting minutes → hours or cm → m.

DSAT Tip:
If a car travels 150 miles in 3 hours → DSAT expects: $150/3 = 50$ mph — NOT a proportion setup.

Meaning:
Lower unit cost = more efficient.

DSAT Trap:
Comparing totals directly instead of per-unit costs.

Real Strategy:
Always divide total by quantity even if numbers look ugly.

Key Rule:
Work rates add: $R_{\text{total}} = R_1 + R_2$

Mental Trick:
Convert “hours per job” → “jobs per hour.”

DSAT Tip:
If two workers together finish in less time → rate must increase, NOT average.

Mental Trick: Mean = “fair share.”

Common Mistake:
Thinking mean must be one of the values — wrong.

DSAT Tip:
If one value changes, mean shifts by $\frac{\text{change}}{n}$.

Meaning:
Larger weight → more influence.

DSAT Trap:
Averaging averages without weights — always incorrect.

Mental Trick:
Treat weights as “how many times the value counts.”

Strategy:
1. Compute total needed for mean 2. Subtract known amounts 3. Remaining = missing value

DSAT Tip:
If mean increases after adding a value → new value must be above old mean.

Mental Model: Counts how many 1×1 squares fit into the region.

DSAT Tip: If a rectangle is scaled by factor $k$: • area scales by $k^2$ • perimeter scales by $k$

Concept: Height is always perpendicular to base.

Common Mistake: Students mistakenly use a slanted side as height.

DSAT Tip: DSAT often hides the right angle for the height.

Mental Trick:
Area grows with the square of the radius. Doubling $r$ → area ×4.

DSAT Tip:
In an annulus (ring shape), subtract two areas: $A = \pi (R^2 – r^2)$.

Insight: Equivalent formula: $C = \pi d$.

DSAT Tip: Many DSAT items expect converting rotations → distance.

Fraction of full circle area using angle proportion.

DSAT Tip: If angle is in radians → use $A = \frac12 r^2 \theta$.

Insight:
Same proportion idea: fraction of circumference.

DSAT Tip:
Radians formula: $L = r\theta$. Much faster.

Concept:
Directly from Pythagorean theorem.

Graph Insight:
Horizontal distance = Δx Vertical distance = Δy

DSAT Tip:
If the question asks for “shortest path”, distance formula always applies.

Mental Trick:
Average the x’s, average the y’s.

Connection:
Directly tied to mean formula.

Insight:
Slope = unit rate = steepness.

DSAT Tip:
If denominator = 0 → slope undefined (vertical).

Mental Trick:
Perpendicular slope = “negative reciprocal.”

DSAT Tip:
Parallel lines → no intersection → same slope.

Key Idea:
Works only for right triangles.

DSAT Trap:
DSAT often hides the right angle in composite shapes.

Mental Trick:
Diagonal of a square = side × $\sqrt{2}$.

DSAT Tip:
If two sides are equal → automatically 45-45-90.

Key Insight:
Hypotenuse is always twice the short leg.

DSAT Tip:
If you see $\sqrt{3}$, assume 30-60-90 immediately.

Mental Trick:
SOH-CAH-TOA → memorize as a chant.

DSAT Tip:
Angle of elevation = angle from ground up. Angle of depression = angle from horizontal down.

Based on unit circle and Pythagorean theorem.

DSAT Tip:
Useful when DSAT gives one trig ratio and asks another.