Comprehensive DSAT Math Review Notes

Algebra

Linear Equations in One Variable

• General form: $ax + b = c$

  • Step 1 (Isolation – subtraction): Subtract $b$ from both sides → $ax = c - b$

  • Step 2 (Isolation – division): Divide both sides by $a$ → $x = \frac{c-b}{a}$

• Tactic: Always isolate the unknown variable.

• Check: Substitute the obtained $x$ back into the original equation to verify.

Linear Functions

• Slope ($m$): $\frac{rise}{run}$

  • Definition from two points $$ $\frac{y_2-y_1}{x_2-x_1}$

• Key equation forms:

  • Slope–intercept: $y = mx + b$ (intercept $b$ is the $y$–intercept).

  • Point–slope: $y-y1=m\left(x-x_1\right)$

• Graphing procedure:

  • Plot the $y$–intercept $(0,b)$.

  • Use slope $\displaystyle m = \frac{\text{rise}}{\text{run}}$ (e.g.
    $m = 2$ → “up $2$, right $1$”) to locate a second point, then draw the line.

Linear Equations in Two Variables

• Standard form: $Ax + By = C$

• Intercepts:

  • $x$–intercept: set $y = 0$, then solve $Ax = C$ → $x = \tfrac{C}{A}$.

  • $y$–intercept: set $x = 0$, then solve $By = C$ → $y = \tfrac{C}{B}$.

• Slope from standard form: $m = -\tfrac{A}{B}$.

Systems of Linear Equations

• Solution methods:

  • Substitution – solve one equation for $x$ or $y$, substitute into the other.

  • Elimination – multiply equations (if needed) so coefficients align; add/subtract to eliminate one variable.

  • Graphing – plot both lines (recommend Desmos) and read intersection.

• Outcome types:

  • One solution → lines intersect once.

  • No solution → parallel lines (same slope, different intercepts).

  • Infinite solutions → coincident lines (identical equations).

Linear Inequalities

• Solving: same operations as equations, but flip the inequality sign when multiplying/dividing by a negative.

• Graphing conventions:

  • Solid boundary line for $\le$ or $\ge$.

  • Dashed boundary line for < or >.

  • Shade the region that satisfies the inequality (use test point $(0,0)$ if not on the line).

Function Transformations (Shifts & Reflections)

• Horizontal shift: $f(x) \to f(x-h)$

• Vertical shift: $f(x) \to f(x)+k$ → shift up by $k$ if k>0.

• Vertical reflection (flip over $x$-axis): $f(x) \to -f(x)$.

Advanced Math

Quadratic Functions

• General form: $y = ax^2 + bx + c$.

• Roots information:

  • Sum of roots = $-\tfrac{b}{a}$.

  • Product of roots = $\tfrac{c}{a}$.

• Vertex: $x\text{-coordinate }= -\tfrac{b}{2a}$, then $y = f\bigl(-\tfrac{b}{2a}\bigr)$.

• Completing the square:
  $y = a\bigl(x^2 + \tfrac{b}{a}x\bigr) + c$
  $\;\; = a\Bigl[\bigl(x + \tfrac{b}{2a}\bigr)^2 - \bigl(\tfrac{b}{2a}\bigr)^2\Bigr] + c$
  $\;\; = a\bigl(x + \tfrac{b}{2a}\bigr)^2 + \bigl(c - \tfrac{b^2}{4a}\bigr)$
  → Vertex $(h,k)$ where $h = -\tfrac{b}{2a}$ and $k = c - \tfrac{b^2}{4a}$.

  a)^2$</p></li><li><p><strong>Direction:</strong> opens up if$a>0$, down if$a<0$.</p></li><li><p><strong>Factoring tactic:</strong> find two numbers multiplying to$ac$and summing to$b$.</p></li><li><p><strong>Quadratic formula:</strong> <br>$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.</p></li><li><p><strong>Discriminant:</strong>$D = b^2 - 4ac$</p><ul><li><p>$D>0$→ two distinct real solutions.</p></li><li><p>$D=0$→ one (repeated) real solution.</p></li><li><p>$D<0$→ no real solutions.</p></li></ul></li></ul><h3 id="b1cdc86b-2e85-41c6-a22d-0a93a7e777ff" data-toc-id="b1cdc86b-2e85-41c6-a22d-0a93a7e777ff" collapsed="false" seolevelmigrated="true">Exponential Functions</h3><ul><li><p>Form:$y = a \cdot b^x$.</p><ul><li><p>Growth if$b>1$.</p></li><li><p>Decay if$0<b<1.

Nonlinear Equation Techniques

• Quadratics – solve by factoring, completing the square, or quadratic formula.

• Systems containing a nonlinear equation – substitute one equation into the other or graph both curves to find intersection points.

• Equivalent Expressions - Simplify by combining like terms, factoring, expanding.

• Rational expressions – factor numerator & denominator, then cancel common factors.

Problem-Solving and Data Analysis

Ratios, Rates, Proportions

• Ratio formats: a:b$or$\tfrac{a}{b}$.</p></li><li><p>Proportion:$\tfrac{a}{b} = \tfrac{c}{d}$→ cross-multiply:$ad = bc$.</p></li><li><p>Unit conversion example:$1\text{ km} = 0.62\text{ mi}$(multiply by appropriate factor).</p></li></ul><h3 id="162a18c1-b9ca-4b8c-9228-12bdb5c371dc" data-toc-id="162a18c1-b9ca-4b8c-9228-12bdb5c371dc" collapsed="false" seolevelmigrated="true">Percentages</h3><ul><li><p>Part formula:$\text{Part} = \text{Whole} \times \tfrac{\text{Percentage}}{100}$.</p></li><li><p>Whole formula:$\text{Whole} = \tfrac{\text{Part}}{\text{Percentage}/100}$.</p></li><li><p>Percentage formula:$\text{Percentage} = \tfrac{\text{Part}}{\text{Whole}} \times 100$.</p></li><li><p>Percent change:$\tfrac{\text{New} - \text{Old}}{\text{Old}} \times 100\%$.</p></li></ul><h3 id="c7c1858c-0d59-43bb-af39-940261405407" data-toc-id="c7c1858c-0d59-43bb-af39-940261405407" collapsed="false" seolevelmigrated="true">One-Variable Data</h3><ul><li><p>Mean:$\displaystyle \bar{x} = \tfrac{1}{n}\sum{i=1}^{n}xi$.</p></li><li><p>Median: middle value after ordering.</p></li><li><p>Mode: most frequent value.</p></li><li><p>Range:$\text{max} - \text{min}$.</p></li><li><p>Standard deviation: quantitative spread measure (greater SD = data more spread out).</p></li></ul><h3 id="f96e4e76-4238-4f18-b015-6941cb86ee59" data-toc-id="f96e4e76-4238-4f18-b015-6941cb86ee59" collapsed="false" seolevelmigrated="true">Two-Variable Data (Scatterplots)</h3><ul><li><p>Line of best fit → approximate with linear regression (Desmos).</p></li><li><p>Correlation coefficient ($r$):</p><ul><li><p>$r = 1$→ perfect positive.</p></li><li><p>$r = -1$→ perfect negative.</p></li><li><p>$r = 0$→ no linear relationship.</p></li></ul></li></ul><h3 id="24bf5380-d19d-4b0a-af46-fc8d55f5bbb4" data-toc-id="24bf5380-d19d-4b0a-af46-fc8d55f5bbb4" collapsed="false" seolevelmigrated="true">Probability</h3><ul><li><p>Basic probability:$P(A) = \tfrac{\text{favorable outcomes}}{\text{total outcomes}}$.</p></li><li><p>Conditional probability:$P(A\mid B) = \tfrac{P(A \text{ and } B)}{P(B)}.

Inference & Statistics

• Margin of error for 95\%$confidence:$\text{MOE} \approx \tfrac{1}{\sqrt{n}}.

• Observational study: measure without intervention (shows correlation).

• Experimental study: apply treatment (supports causation).

Geometry and Trigonometry

Area & Perimeter / Surface Area & Volume

• Rectangle – area: A = lw$; perimeter:$P = 2l + 2w$.</p></li><li><p>Triangle – area:$A = \tfrac{1}{2}bh$.</p></li><li><p>Circle – area:$A = \pi r^2$; circumference:$C = 2\pi r$.</p></li><li><p>Rectangular prism – volume:$V = lwh$.</p></li><li><p>Pyramid – volume:$V=\tfrac{1}{3}Bh$; slant-height via Pythagorean theorem$s = \sqrt{h^2 + r^2}$.</p></li><li><p>Right circular cylinder – volume:$V = \pi r^2 h$.</p></li><li><p>Hollow cylinder – volume:$V = \pi (R^2 - r^2)h$.</p></li><li><p>Right circular cone – volume:$V = \tfrac{1}{3}\pi r^2 h$.</p></li><li><p>Sphere – volume:$V = \tfrac{4}{3}\pi r^3$.</p></li></ul><h3 id="0169a6cf-e4f1-4486-9850-ff1952585776" data-toc-id="0169a6cf-e4f1-4486-9850-ff1952585776" collapsed="true" seolevelmigrated="true">Fundamental Angle Facts</h3><ul><li><p>Angles on a line sum to$180^{\circ}$.</p></li><li><p>Vertical angles are congruent.</p></li><li><p>Interior angles of any triangle sum to$180^{\circ}$.</p></li></ul><h3 id="643f0ef7-ecf3-4291-9cd7-1ab1cb585aa1" data-toc-id="643f0ef7-ecf3-4291-9cd7-1ab1cb585aa1" collapsed="true" seolevelmigrated="true">Pythagorean Theorem</h3><ul><li><p>Right triangle relationship:$a^2 + b^2 = c^2$.</p></li></ul><h3 id="6c93b510-fa64-494d-92e3-fead61a2567d" data-toc-id="6c93b510-fa64-494d-92e3-fead61a2567d" collapsed="true" seolevelmigrated="true">Special Right Triangles</h3><ul><li><p><strong>45-45-90:</strong> sides in ratio$1:1:\sqrt{2}$.</p></li><li><p><strong>30-60-90:</strong> sides in ratio$1:\sqrt{3}:2$.</p></li><li><p>Common integer triples:$3 ext{-}4 ext{-}5$,$6 ext{-}8 ext{-}10$,$5 ext{-}12 ext{-}13$,$7 ext{-}24 ext{-}25$,$8 ext{-}15 ext{-}17$.</p></li></ul><h3 id="7aa37614-8b83-487c-a31b-75499fe25942" data-toc-id="7aa37614-8b83-487c-a31b-75499fe25942" collapsed="true" seolevelmigrated="true">Right-Triangle Trigonometry (SOH-CAH-TOA)</h3><ul><li><p>$\sin\theta = \tfrac{\text{Opposite}}{\text{Hypotenuse}}$.</p></li><li><p>$\cos\theta = \tfrac{\text{Adjacent}}{\text{Hypotenuse}}$.</p></li><li><p>$\tan\theta = \tfrac{\text{Opposite}}{\text{Adjacent}}$.</p></li></ul><h3 id="eeb9e87f-1dd7-455f-8c01-5a68eedf167a" data-toc-id="eeb9e87f-1dd7-455f-8c01-5a68eedf167a" collapsed="true" seolevelmigrated="true">Circles in the Coordinate Plane</h3><ul><li><p>Equation with center$(h,k)$and radius$r$:$(x-h)^2 + (y-k)^2 = r^2.

Circle Angle & Sector Facts

• Inscribed angle measure = half its intercepted arc measure.

• Arc length for central angle \theta^{\circ}$:$L = \tfrac{\theta}{360} \cdot 2\pi r$.</p></li><li><p>Sector area:$A = \tfrac{\theta}{360} \cdot \pi r^2$$.