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Digital SAT Math Last Minutes Revision Sheet

Digital SAT Math Last Minutes Revision Sheet - New Syllabus

DSAT Math Blue book reference

DSAT Math Formula Sheet: Algebra

Linear Equations in One Variable

  • Standard form: \( ax + b = 0 \)
  • Solution: \( x = -\dfrac{b}{a} \)

Linear Equations in Two Variables

  • General form: \( ax + by = c \)
  • Slope-intercept form: \( y = mx + b \)
  • Slope: \( m = \dfrac{\Delta y}{\Delta x} = \dfrac{y_2 – y_1}{x_2 – x_1} \)
  • Point-slope form: \( y – y_1 = m(x – x_1) \)

Linear Functions

  • Function form: \( f(x) = mx + b \)
  • Rate of change (slope): \( m \)
  • Intercept: value of \( f(x) \) when \( x = 0 \): \( f(0) = b \)

Systems of Two Linear Equations in Two Variables

  • General system:
    \( a_1x + b_1y = c_1 \)
    \( a_2x + b_2y = c_2 \)
  • Solutions:
    • One solution (lines intersect): consistent and independent
    • No solution (parallel lines): inconsistent
    • Infinite solutions (same line): dependent
  • Substitution or elimination methods used to solve

Linear Inequalities in One or Two Variables

  • One variable: \( ax + b < c \), \( \leq, \geq, > \)
  • Two variables: \( y < mx + b \), etc.
  • Flip inequality sign when multiplying/dividing by a negative number
  • Graphical region: shade above/below the line depending on inequality

DSAT Math Formula Sheet: Advanced Math

Equivalent Expressions

  • Distributive Property: \( a(b + c) = ab + ac \)
  • Combining like terms: \( ax + bx = (a + b)x \)
  • Factoring: \( ax^2 + bx + c = (mx + n)(px + q) \) (where applicable)
  • Difference of squares: \( a^2 – b^2 = (a – b)(a + b) \)
  • Perfect square trinomial: \( a^2 \pm 2ab + b^2 = (a \pm b)^2 \)

Nonlinear Equations in One Variable

  • Quadratic form: \( ax^2 + bx + c = 0 \)
  • Quadratic formula: \( x = \dfrac{-b \pm \sqrt{b^2 – 4ac}}{2a} \)
    • $\text{Discriminant} =b^2-4 a c$ ,
    • VIETA’S FORMULAS $x_1+x_2=-\frac{b}{a} \quad \text { and } \quad x_1 x_2=\frac{c}{a}$
  • Minimum and Maximum: Quadratic Functions
      • 1. If $a>0$, then $f$ has a minimum that occurs at $x=-\frac{b}{2 a}$. This minimum value is $f\left(-\frac{b}{2 a}\right)$.
      • 2. If $a<0$, then $f$ has a maximum that occurs at $x=-\frac{b}{2 a}$. This maximum value is $f\left(-\frac{b}{2 a}\right)$.
      • In each case, the value of $x$ gives the location of the minimum or maximum value. The value of $y$, or $f\left(-\frac{b}{2 a}\right)$, gives that minimum or maximum value.
  • Square root property: \( x^2 = k \Rightarrow x = \pm \sqrt{k} \)
  • Cubic or higher degree polynomials solved by factoring or substitution 

Systems of Equations Involving Nonlinear Functions

  • Example system:
    \( y = ax^2 + bx + c \)
    \( y = mx + b \)
  • Substitute one equation into the other to solve for variable(s)
  • Solutions correspond to points of intersection of graphs

Nonlinear Functions

  • Quadratic function: \( f(x) = ax^2 + bx + c \)
  • Vertex form: \( f(x) = a(x – h)^2 + k \)
  • Absolute value function: \( f(x) = |x| \)
  • Piece wise function : $y=f(x)= \begin{cases}-x, & x<0 \\ x^2, & 0 \leq x \leq 1 \\ 1, & x>1\end{cases}$
  • Exponential functions (basic): \( f(x) = a \cdot b^x \), \( b > 0, b \neq 1 \)
  • Rules for Exponents:
    • If $a>0$ and $b>0$, the following hold for all real numbers $x$ and $y$.
      1.  $a^x \cdot a^y=a^{x+y}$
      2. $\frac{a^x}{a^y}=a^{x-y}$
      3.  $\left(a^x\right)^y=\left(a^y\right)^x=a^{x y}$
      4. $a^x \cdot b^x=(a b)^x$
      5. $\left(\frac{a}{b}\right)^x=\frac{a^x}{b^x}$

DSAT Math Formula Sheet: Problem-Solving & Data Analysis

Ratios, Rates, Proportional Relationships, and Units

  • Ratio: \( \dfrac{a}{b} \) means “a to b”
  • Rate: ratio comparing two quantities with different units (e.g., miles per hour)
  • Proportion equation: \( \dfrac{a}{b} = \dfrac{c}{d} \) → cross-multiply: \( ad = bc \)
  • Unit conversions: multiply by conversion factor \( \dfrac{\text{desired unit}}{\text{given unit}} \)

Percentages

  • Percent to decimal: \( \% = \dfrac{\text{percent}}{100} \)
  • Percent increase/decrease: \(\text{Percent change} = \dfrac{\text{New} – \text{Original}}{\text{Original}} \times 100\% \)
  • Percent of a number: \( \text{Part} = \text{Percent} \times \text{Whole} \)

1-Variable Data: Distributions and Measures of Center and Spread

  • Mean (average): \( \bar{x} = \dfrac{\sum x_i}{n} \)
  • Median: middle value when data is ordered
  • Mode: most frequent value
  • Range: \( \text{Max} – \text{Min} \)
  • Interquartile Range (IQR): \( Q_3 – Q_1 \)
  • Standard deviation (informal): measure of spread around mean

2-Variable Data: Models and Scatterplots

  • Scatterplot shows relationship between two variables
  • Correlation describes strength and direction of linear relationship (positive, negative, none)
  • Line of best fit (trend line) models approximate linear relation
  • Equation of line: \( y = mx + b \)

Probability and Conditional Probability

  • Probability of event \( A \): \( P(A) = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \)
  • Complement rule: \( P(\text{not } A) = 1 – P(A) \)
  • For independent events \( A \) and \( B \): \( P(A \text{ and } B) = P(A) \times P(B) \)
  • Conditional probability: \( P(A|B) = \dfrac{P(A \text{ and } B)}{P(B)} \)

Inference from Sample Statistics and Margin of Error

  • Margin of error (approximate): \( \text{ME} = \dfrac{1}{\sqrt{n}} \) (where \( n \) is sample size)
  • Increasing sample size decreases margin of error
  • Sample statistics estimate population parameters

Evaluating Statistical Claims: Observational Studies and Experiments

  • Observational study: researcher observes without intervention
  • Experiment: researcher imposes treatment or intervention
  • Correlation does not imply causation
  • Beware of bias, confounding variables, and sampling errors

DSAT Math Formula Sheet: Geometry & Trigonometry

Area and Volume

  • Area of rectangle: \( A = l \times w \)
  • Area of square: \( A = s^2 \)
  • Area of triangle: \( A = \dfrac{1}{2} \times \text{base} \times \text{height} \)
  • Area of parallelogram: \( A = b \times h \)
  • Area of trapezoid: \( A = \dfrac{1}{2} (b_1 + b_2) \times h \)
  • Area of circle: \( A = \pi r^2 \)
  • Outer Surface Area of a Cylinder:
    • Hollow cylinder $A=2 \pi r h$ (no ends)
    • Open cylinder $A=2 \pi r h+\pi r^2$ (one end)
    • Solid cylinder $A=2 \pi r h+2 \pi r^2$ (two ends)
  • Volume of rectangular prism: \( V = l \times w \times h \)
  • Volume of cube: \( V = s^3 \)
  • Volume of cylinder: \( V = \pi r^2 h \)
  • Volume of cone: \( V = \dfrac{1}{3} \pi r^2 h \)
  • Volume of sphere: \( V = \dfrac{4}{3} \pi r^3 \)

Lines, Angles, and Triangles

  • Line Formulas:
    • Slope formula: \( m = \dfrac{y_2 – y_1}{x_2 – x_1} \)
    • Point-slope form: \( y – y_1 = m(x – x_1) \)
    • Slope-intercept form: \( y = mx + b \)
    • Standard form: \( Ax + By = C \)
    • Parallel lines: same slope
    • Perpendicular lines: slopes are negative reciprocals (\( m_1 \cdot m_2 = -1 \))
  • Distance and Midpoint:
    • Distance between two points: \(d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} \)
    • Midpoint of a segment: \(M = \left( \dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2} \right) \)
  • Sum of interior angles of triangle: \( 180^\circ \)
  • Sum of interior angles of polygon: \( (n – 2) \times 180^\circ \)
  • Triangle inequality: \( \text{sum of any two sides} > \text{third side} \)
  • Pythagorean theorem (right triangle): \( a^2 + b^2 = c^2 \)
  • Special right triangles:
    • 45°-45°-90° triangle sides: \( x, x, x\sqrt{2} \)
    • 30°-60°-90° triangle sides: \( x, x\sqrt{3}, 2x \)
  • Angle relationships:
    • Complementary angles: sum to \( 90^\circ \)
    • Supplementary angles: sum to \( 180^\circ \)
    • Vertical angles: equal
    • Alternate interior angles (parallel lines): equal

Trigonometry (Right Triangles)

  • Sine: \( \sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}} \)
  • Cosine: \( \cos \theta = \dfrac{\text{adjacent}}{\text{hypotenuse}} \)
  • Tangent: \( \tan \theta = \dfrac{\text{opposite}}{\text{adjacent}} \)
  • Basic identities: \( \tan \theta = \dfrac{\sin \theta}{\cos \theta} \)

Circles

  • Circumference: \( C = 2 \pi r \)
  • Area: \( A = \pi r^2 \)
  • Arc length: \( \text{Arc length} = \dfrac{\theta}{360^\circ} \times 2 \pi r \)
  • Area of sector: \( \text{Sector area} = \dfrac{\theta}{360^\circ} \times \pi r^2 \)
  • Equation of circle with center \( (h,k) \) and radius \( r \): \( (x – h)^2 + (y – k)^2 = r^2 \)

DSAT Math Test Content Breakdown

Algebra

  • Linear Equations & Inequalities
  • Systems of Linear Equations/Inequalities

~35%  |  13–15 questions

Advanced Math

  • Polynomial, Quadratic, Nonlinear & Exponential Functions
  • Radical, Rational, & Absolute Value Equations

~35%  |  13–15 questions

Problem Solving & Data 

  • Rates, Ratios, Proportions
  • Percentages & Probability
  • Data Representations & Statistical Claims

~15%  |  5–7 questions

Geometry & Trigonometry

  • Area, Perimeter, and Volume
  • Lines, Angles, Triangles, and Circles
  • Trigonometry

~15%  |  5–7 questions

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