Digital SAT Math Formulas Cheat Sheet

1. Introduction to SAT Math Formulas

Many students believe that because the Digital SAT provides a built-in math reference sheet, they do not need to memorize any formulas. This is a dangerous misconception. The official College Board reference sheet only provides basic geometry formulas—specifically, area and volume equations for common shapes, along with special right triangle side-length ratios. It completely omits the algebra, coordinate geometry, quadratic, exponential, statistical, and trigonometric formulas that make up the vast majority of the test.

Furthermore, relying on the reference sheet during the exam is a major time sink. Every second counts on the Digital SAT. Having to open the reference sheet overlay, find the correct formula, and apply it is far slower than having that formula instantly accessible in your memory.

To secure a perfect 800 on the Math section, you must build a comprehensive, highly active mathematical library in your head. This guide serves as both a detailed conceptual review of every formula tested and a printable-style cheat sheet. We will cover why each formula is useful, when to apply it, common traps, and how to verify your results using coordinate methods and the Desmos graphing calculator.

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2. Algebra and Linear Equations

Linear equations are the foundation of SAT Math, representing more than 30% of the questions. To navigate these, you must master the coordinate equations of lines and their properties.

The Slope Formula

The slope \(m\) of a line measures its steepness and direction. Given two coordinate points \((x_1, y_1)\) and \((x_2, y_2)\), the slope is defined as: \[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x} = \frac{\text{Rise}}{\text{Run}}\]

• Positive Slope: The line rises from left to right.
• Negative Slope: The line falls from left to right.
• Zero Slope: The line is horizontal (represented by the equation \(y = c\)).
• Undefined Slope: The line is vertical (represented by the equation \(x = c\)).

Midpoint Formula

The midpoint \(M\) of a line segment connecting points \((x_1, y_1)\) and \((x_2, y_2)\) is the exact numerical average of their coordinates: \[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]

Distance Formula

The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is derived from the Pythagorean theorem: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

                       (x2, y2)
                         /|
                        / |
                     d /  | y2 - y1
                      /   |
                     /____|
             (x1, y1)  x2 - x1

Forms of Linear Equations

You must be comfortable translating between three primary forms of linear equations:

1. Slope-Intercept Form: \[y = mx + b\]
   • Useful when you know the slope \(m\) and the \(y\)-intercept \((0, b)\).
2. Point-Slope Form: \[y - y_1 = m(x - x_1)\]
   • Useful when you know the slope \(m\) and a single coordinate point \((x_1, y_1)\). This form is highly recommended for writing equations of lines quickly.
3. Standard Form: \[Ax + By = C\]
   • In this form, you can find the slope and intercepts quickly using these shortcuts:
     • Slope: \(m = -\frac{A}{B}\)
     • \(y\)-intercept: \((0, \frac{C}{B})\)
     • \(x\)-intercept: \((\frac{C}{A}, 0)\)

Parallel and Perpendicular Lines

• Parallel Lines: Never intersect. They have the exact same slope. \[m_1 = m_2\]
• Perpendicular Lines: Intersect at a right angle (\(90^\circ\)). Their slopes are negative reciprocals of each other (their product is \(-1\)). \[m_2 = -\frac{1}{m_1} \quad \implies \quad m_1 \cdot m_2 = -1\]

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3. Systems of Linear Equations

Systems of equations represent pairs of lines. The SAT will frequently ask you to determine the number of solutions a system has without actually solving it.

A system of two linear equations: \[\begin{cases} A_1x + B_1y = C_1 \ A_2x + B_2y = C_2 \end{cases}\] can have zero, one, or infinitely many solutions.

1. Zero Solutions (Parallel Lines)

The lines are parallel and never intersect. They have the same slope but different \(y\)-intercepts.

• Condition: \[\frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2}\] If equations are in slope-intercept form: \(m_1 = m_2\) and \(b_1 \neq b_2\).

2. Infinitely Many Solutions (Identical Lines)

The two equations represent the exact same line. Every point on the line is a solution.

• Condition: \[\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2}\] If equations are in slope-intercept form: \(m_1 = m_2\) and \(b_1 = b_2\).

3. One Unique Solution (Intersecting Lines)

The lines intersect at exactly one coordinate point \((x, y)\). They have different slopes.

• Condition: \[\frac{A_1}{A_2} \neq \frac{B_1}{B_2}\] If equations are in slope-intercept form: \(m_1 \neq m_2\).

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4. Ratios, Percentages, and Financial Math

The SAT tests real-world modeling through ratios, unit rates, percentages, and interest equations.

Percentage Formulas

• Percentage Value: \[\text{Value} = \text{Percent (as a decimal)} \times \text{Base}\]
• Percentage Increase/Decrease: \[\text{New Value} = \text{Original Value} \times (1 \pm r)\]
  • (where \(r\) is the percentage rate written as a decimal. For example, a \(25%\) increase uses a scale factor of \(1.25\); a \(15%\) discount uses a scale factor of \(0.85\)).
• Percent Change Formula: \[\text{Percent Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100%\]

Exponential Growth and Decay Models

Exponential equations model populations, radioactive substances, or financial investments that grow or decay by a constant percentage rate over time. \[A(t) = P(1 \pm r)^t\]

• \(P\) = initial value (y-intercept)
• \(r\) = growth/decay rate (as a decimal)
• \(t\) = time periods

If the growth or decay happens over a specific interval \(d\): \[A(t) = P(1 \pm r)^{\frac{t}{d}}\] For example, if a population doubles every \(5\) years: \[A(t) = P(2)^{\frac{t}{5}}\]

Simple and Compound Interest

• Simple Interest: Interest is earned only on the initial principal amount. \[A = P(1 + rt)\]
  • \(P\) = principal (initial investment)
  • \(r\) = annual interest rate (as a decimal)
  • \(t\) = time in years
• Compound Interest: Interest is earned on the principal plus previously accumulated interest. \[A = P\left(1 + \frac{r}{n}\right)^{nt}\]
  • \(n\) = number of times interest is compounded per year (e.g., semi-annually: \(n=2\), quarterly: \(n=4\), monthly: \(n=12\)).

Ratios, Proportions, and Dimensional Analysis

The SAT heavily features conversion questions requiring you to translate rates from one unit system to another (e.g., miles per hour to feet per second). To do this without errors, you must set up conversion factor equations: \[\text{Target Rate} = \text{Initial Rate} \times \frac{\text{Conversion Factor 1 (Numerator)}}{\text{Conversion Factor 1 (Denominator)}} \times \dots\]

For example, to convert \(60\) miles per hour to feet per second, you multiply by the conversion ratios so that intermediate units cancel: \[\frac{60 \text{ miles}}{1 \text{ hour}} \times \frac{5280 \text{ feet}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}} = 88 \text{ feet per second}\]

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5. Exponents and Radicals

Exponents and radicals follow strict operational rules. You must memorize these identities to simplify algebraic expressions.

Rules of Exponents

For any base \(x\) and \(y\), and exponents \(a\) and \(b\):

Rule Name	Formula
Product Rule	\(x^a \cdot x^b = x^{a+b}\)
Quotient Rule	\(\frac{x^a}{x^b} = x^{a-b}\)
Power of a Power Rule	\((x^a)^b = x^{ab}\)
Power of a Product	\((xy)^a = x^a \cdot y^a\)
Power of a Quotient	\(\left(\frac{x}{y}\right)^a = \frac{x^a}{y^a}\)
Negative Exponents	\(x^{-a} = \frac{1}{x^a}\)
Zero Exponent	\(x^0 = 1 \quad (x \neq 0)\)

Rational Exponent Conversion

The index of a radical corresponds to the denominator of a fractional exponent, and the power corresponds to the numerator: \[x^{\frac{m}{n}} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m\]

Example: \(x^{\frac{3}{2}} = \sqrt{x^3} = x\sqrt{x}\).

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6. Quadratic and Polynomial Formulas

Quadratic functions describe parabolas in the coordinate plane. They are represented in three standard forms, each highlighting a different key feature of the graph.

The Three Forms of Quadratic Functions

1. Standard Form: \[f(x) = ax^2 + bx + c\]
   • The \(y\)-intercept is at \((0, c)\).
   • The \(x\)-coordinate of the vertex is given by the formula: \[h = -\frac{b}{2a}\]
2. Vertex Form: \[f(x) = a(x - h)^2 + k\]
   • The coordinates of the vertex are explicitly visible as \((h, k)\).
   • The vertical axis of symmetry is the line \(x = h\).
3. Factored Form: \[f(x) = a(x - r_1)(x - r_2)\]
   • The \(x\)-intercepts (roots or zeros) are at \((r_1, 0)\) and \((r_2, 0)\).
   • The axis of symmetry lies exactly halfway between the roots: \[h = \frac{r_1 + r_2}{2}\]

The Quadratic Formula and the Discriminant

For any standard quadratic equation \(ax^2 + bx + c = 0\), the roots can be calculated using: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

The value inside the square root is the discriminant (\(\Delta\)): \[\Delta = b^2 - 4ac\]

The value of the discriminant determines the number and type of solutions:

Discriminant Value	Graph Interpretation	Number of Real Solutions
\(\Delta > 0\)	Parabola crosses the \(x\)-axis twice	2 unique real solutions
\(\Delta = 0\)	Parabola’s vertex lies exactly on the \(x\)-axis	1 unique real solution
\(\Delta < 0\)	Parabola never touches the \(x\)-axis	0 real solutions (2 complex solutions)

Sum and Product of Roots

You can calculate the sum and product of the roots of a quadratic equation \(ax^2 + bx + c = 0\) instantly using its coefficients:

• Sum of Roots: \[r_1 + r_2 = -\frac{b}{a}\]
• Product of Roots: \[r_1 \cdot r_2 = \frac{c}{a}\]

These formulas are extremely useful when a question asks for the sum of solutions to a quadratic equation, allowing you to skip using the quadratic formula or factoring.

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7. Circles and Coordinate Circle Geometry

Circle geometry covers both two-dimensional circumference/area measurements and coordinate equations.

Basic Properties

For a circle with radius \(r\) and diameter \(d = 2r\):

• Circumference: \(C = 2\pi r = \pi d\)
• Area: \(A = \pi r^2\)

Arcs and Sectors

Arc length and sector area are fractions of the total circumference and area.

Degree Measures (Central angle \(\theta^\circ\))

• Arc Length (\(s\)): \[s = \frac{\theta}{360} \times 2\pi r\]
• Sector Area (\(A_{\text{sector}}\)): \[A_{\text{sector}} = \frac{\theta}{360} \times \pi r^2\]

Radian Measures (Central angle \(\theta\) rad)

• Arc Length (\(s\)): \[s = r\theta\]
• Sector Area (\(A_{\text{sector}}\)): \[A_{\text{sector}} = \frac{1}{2}r^2\theta\]

Coordinate Equation of a Circle

In the \(xy\)-plane, the standard form equation of a circle is: \[(x - h)^2 + (y - k)^2 = r^2\]

• Center: \((h, k)\)
• Radius: \(r\) (Note: the right side of the equation is \(r^2\), not \(r\).)

If the circle’s equation is expanded, you must complete the square to find its center and radius.

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8. Geometry of Polygons and Triangles

Triangles and polygons are tested for their angle properties, congruence, and similarity.

Polygons Interior Angle Sum

For any polygon with \(n\) sides, the sum of the interior angles \(S\) is: \[S = (n - 2) \times 180^\circ\int\]

Triangle Inequality Theorem

For a triangle with sides \(a\), \(b\), and \(c\): \[|a - b| < c < a + b\] The third side must be strictly greater than the difference and strictly less than the sum of the other two sides.

Similar Figures scaling

If two figures are similar with a side length scale factor of \(k = \frac{\text{Side}_2}{\text{Side}_1}\):

• Perimeter Ratio: \(\frac{\text{Perimeter}_2}{\text{Perimeter}_1} = k\)
• Area Ratio: \(\frac{\text{Area}_2}{\text{Area}_1} = k^2\)
• Volume Ratio (for 3D): \(\frac{\text{Volume}_2}{\text{Volume}_1} = k^3\)

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9. 3D Solids: Volume and Surface Area

While volume formulas are provided on the reference sheet, surface area equations must be understood conceptually, especially Cylinder Surface Area.

Volume Equations

• Rectangular Prism: \(V = lwh\)
• Cylinder: \(V = \pi r^2 h\)
• Sphere: \(V = \frac{4}{3}\pi r^3\)
• Cone: \(V = \frac{1}{3}\pi r^2 h\)
• Pyramid: \(V = \frac{1}{3}lwh\)

Surface Area Equations

• Rectangular Prism: \(A = 2(lw + lh + wh)\)
• Cylinder: \(A = 2\pi r^2 + 2\pi rh\)
• Sphere: \(A = 4\pi r^2\)

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10. Right-Triangle Trigonometry and Radians

Trigonometry on the SAT is centered on right triangles and complementary angle relationships.

SOH-CAH-TOA

For an acute angle \(\theta\) in a right triangle:

• \(\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}\)
• \(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
• \(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin(\theta)}{\cos(\theta)}\)

Complementary Angle Identities

The sine of an acute angle is equal to the cosine of its complement, and vice versa: \[\sin(\theta) = \cos(90^\circ - \theta)\) \[\cos(\theta) = \sin(90^\circ - \theta)\)

In radians: \[\sin(x) = \cos\left(\frac{\pi}{2} - x\right)\] \[\cos(x) = \sin\left(\frac{\pi}{2} - x\right)\]

Pythagorean Identity

For any angle \(\theta\): \[\sin^2(\theta) + \cos^2(\theta) = 1\]

Degree-Radian Conversion

• Degrees to Radians: Multiply by \(\frac{\pi}{180}\)
• Radians to Degrees: Multiply by \(\frac{180}{\pi}\)

The Unit Circle & Coordinate Trigonometry

In coordinate geometry, the unit circle is centered at the origin \((0,0)\) with a radius of \(1\). Any point on the unit circle has coordinates: \[(x, y) = (\cos(\theta), \sin(\theta))\]

                         y-axis
                           | (0, 1)
                           |   
                           |     (cosθ, sinθ)
                           |     / 
                           |    /  
                           |   / r=1
                           |  /    
                           | / θ   
             --------------+-------------- x-axis
             (-1, 0)       |       (1, 0)
                           |
                           |
                           | (0, -1)

The sign of trigonometric ratios depends on the quadrant containing the angle:

• Quadrant I (All positive): \(\sin, \cos, \tan > 0\)
• Quadrant II (Sine positive): \(\sin > 0\), \(\cos, \tan < 0\)
• Quadrant III (Tangent positive): \(\tan > 0\), \(\sin, \cos < 0\)
• Quadrant IV (Cosine positive): \(\cos > 0\), \(\sin, \tan < 0\)
• Mnemonic: All Students Take Calculus (ASTC).

Here is a summary table of unit circle trigonometric values for key angles:

Degrees	Radians	\(\sin(\theta)\)	\(\cos(\theta)\)	\(\tan(\theta)\)
\(0^\circ\)	\(0\)	\(0\)	\(1\)	\(0\)
\(30^\circ\)	\(\frac{\pi}{6}\)	\(\frac{1}{2}\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{\sqrt{3}}\)
\(45^\circ\)	\(\frac{\pi}{4}\)	\(\frac{\sqrt{2}}{2}\)	\(\frac{\sqrt{2}}{2}\)	\(1\)
\(60^\circ\)	\(\frac{\pi}{3}\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{2}\)	\(\sqrt{3}\)
\(90^\circ\)	\(\frac{\pi}{2}\)	\(1\)	\(0\)	Undefined
\(180^\circ\)	\(\pi\)	\(0\)	\(-1\)	\(0\)
\(270^\circ\)	\(\frac{3\pi}{2}\)	\(-1\)	\(0\)	Undefined

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11. Reference Sheet Comparison: Provided vs. Memorized

Let us contrast the formulas provided in the Bluebook app’s reference sheet with the critical formulas you must memorize.

Provided on the SAT Reference Sheet

• Area of a circle: \(A = \pi r^2\)
• Circumference of a circle: \(C = 2\pi r\)
• Area of a rectangle: \(A = lw\)
• Area of a triangle: \(A = \frac{1}{2}bh\)
• Area of a parallelogram: \(A = bh\)
• Volume of a rectangular solid: \(V = lwh\)
• Volume of a cylinder: \(V = \pi r^2 h\)
• Volume of a sphere: \(V = \frac{4}{3}\pi r^3\)
• Volume of a cone: \(V = \frac{1}{3}\pi r^2 h\)
• Volume of a pyramid: \(V = \frac{1}{3}lwh\)
• Special right triangle ratios:
  • \(45^\circ\text{-}45^\circ\text{-}90^\circ\) (\(1:1:\sqrt{2}\))
  • \(30^\circ\text{-}60^\circ\text{-}90^\circ\) (\(1:\sqrt{3}:2\))
• Total degrees in a circle: \(360^\circ\)
• Total radians in a circle: \(2\pi\)
• Sum of interior angles of a triangle: \(180^\circ\)

NOT Provided (MUST Memorize)

• Slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
• Midpoint formula: \(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)
• Distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
• Point-slope form: \(y - y_1 = m(x - x_1)\)
• Perpendicular lines slope condition: \(m_1 \cdot m_2 = -1\)
• Quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• Discriminant: \(\Delta = b^2 - 4ac\)
• Sum and Product of roots: \(-\frac{b}{a}\) and \(\frac{c}{a}\)
• Vertex coordinate formula: \(h = -\frac{b}{2a}\)
• Exponential growth/decay models: \(A = P(1 \pm r)^t\)
• Simple and Compound interest formulas: \(A = P(1+rt)\) and \(A = P(1+\frac{r}{n})^{nt}\)
• Standard equation of a circle: \((x-h)^2 + (y-k)^2 = r^2\)
• Radian-Degree conversion formulas
• Trigonometric ratios (SOH-CAH-TOA) and identities (\(\sin^2(\theta)+\cos^2(\theta)=1\), \(\sin(\theta)=\cos(90^\circ-\theta)\))
• Arc length and sector area formulas in radians (\(s=r\theta\), \(A=\frac{1}{2}r^2\theta\))
• Polygon angle sum formula: \(S = (n-2) \times 180^\circ\)

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12. Printable-Style Cheat Sheet Tables

Use these quick reference tables for daily review.

1. Algebra & Coordinate Geometry

Name	Formula	Key Condition / Use Case
Slope	\(m = \frac{y_2 - y_1}{x_2 - x_1}\)	Finds the rate of change between two points
Midpoint	\(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)	Averages coordinate points
Distance	\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)	Finds lengths on coordinate grids
Slope-Intercept	\(y = mx + b\)	Highlights slope \(m\) and y-intercept \(b\)
Point-Slope	\(y - y_1 = m(x - x_1)\)	Best for writing linear equations
Parallel Slopes	\(m_1 = m_2\)	Slopes are identical
Perpendicular Slopes	\(m_1 \cdot m_2 = -1\)	Slopes are negative reciprocals

2. Quadratics & Exponents

Name	Formula	Key Condition / Use Case
Quadratic Formula	\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)	Solves \(ax^2 + bx + c = 0\)
Discriminant	\(\Delta = b^2 - 4ac\)	Determines number of real roots
Sum of Roots	\(-\frac{b}{a}\)	Adds solutions of a quadratic
Product of Roots	\(\frac{c}{a}\)	Multiplies solutions of a quadratic
Vertex \(x\)-coordinate	\(h = -\frac{b}{2a}\)	Finds axis of symmetry and vertex
Fractional Exponents	\(x^{\frac{m}{n}} = \sqrt[n]{x^m}\)	Translates radicals to exponents
Compound Interest	\(A = P\left(1 + \frac{r}{n}\right)^{nt}\)	Computes multi-period financial growth

3. Circles & Trigonometry

Name	Formula	Key Condition / Use Case
Circle Equation	\((x-h)^2 + (y-k)^2 = r^2\)	Standard coordinate circle centered at \((h,k)\)
Arc Length (Rad)	\(s = r\theta\)	Arc length with \(\theta\) in radians
Sector Area (Rad)	\(A = \frac{1}{2}r^2\theta\)	Sector area with \(\theta\) in radians
Trig complementary	\(\sin(\theta) = \cos(90^\circ - \theta)\)	Relates acute angles in right triangles
Pythagorean Identity	\(\sin^2(\theta) + \cos^2(\theta) = 1\)	Simplifies complex trig equations
Degrees to Radians	\(\text{Rad} = \text{Deg} \times \frac{\pi}{180}\)	Converts degree angles to radian units
Radians to Degrees	\(\text{Deg} = \text{Rad} \times \frac{180}{\pi}\)	Converts radian angles to degree units

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13. Elite Desmos Graphing Calculator Strategies for Formulas

The built-in Desmos calculator changes how you can interact with equations on the SAT. You can use it to verify formulas and solve complex algebraic expressions instantly.

1. Equivalent Expression Graphing

If a question asks you to simplify an algebraic expression (e.g., radical conversions or polynomial factoring), you can graph the original expression alongside the options.

• The Strategy: Type the original expression in line 1: e.g., \(y = \frac{x^2 - 9}{x - 3}\). Then, type the four options in lines 2–5. The correct option’s graph will lie exactly on top of the original graph. This provides a foolproof check for factoring and exponent rules.

2. Vertex and Root Extraction

Instead of completing the square or using the quadratic formula, you can simply type a quadratic equation directly into Desmos.

• The Strategy: For \(y = 3x^2 - 12x + 7\), type the function to display the parabola. Click on the lowest/highest point to display the vertex coordinates \((2, -5)\) instantly. Click on the x-intercepts to read the decimal roots.

3. Coordinate Circle Verification

Desmos handles implicit equations seamlessly. You do not need to solve circles for \(y\) to graph them.

• The Strategy: Type \(x^2 + y^2 - 6x + 8y = 11\) directly. Desmos will draw the circle. To find the center visually, type the estimated center: e.g., \((3, -4)\), which will appear exactly in the middle of the circle. You can find the radius by counting grid spaces from the center to the boundary.

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14. Common Pitfalls and Traps to Avoid

Avoid these mistakes to ensure a high score:

Trap 1: Forgetting to Square the Radius

In standard circle equations, the constant on the right side is \(r^2\), not the radius itself.

• The Fix: If the equation is \((x-3)^2 + (y+1)^2 = 49\), the radius is \(\sqrt{49} = 7\). If the radius is \(5\), the constant in the equation must be \(25\).

Trap 2: Incorrect Signs in the Midpoint Formula

Students often subtract coordinates instead of adding them when calculating midpoints, confusing the formula with the slope formula.

• The Fix: Midpoint is a coordinates average. You must add the coordinates: \(\frac{x_1 + x_2}{2}\).

Trap 3: Calculator Mode Slip-ups

Using Degree mode for radian equations or Radian mode for degree equations will yield incorrect results.

• The Fix: Check the angle units in the question. If it contains \(\pi\) or lacks a degree symbol, configure your calculator to Radian mode.

Trap 4: Signs in the Sum of Roots

Students often forget the negative sign in the sum of roots formula, using \(\frac{b}{a}\) instead of \(-\frac{b}{a}\).

• The Fix: Write down \(-\frac{b}{a}\) explicitly on your scratch paper before plugging in coefficients.

Trap 5: Confusing Simple and Compound Interest Compounding

When a question specifies that interest is compounded semi-annually, quarterly, or monthly, students often forget to divide the annual interest rate \(r\) by the number of compounding periods \(n\), or forget to multiply the years \(t\) by \(n\) in the exponent. Alternatively, they might use the compound interest formula when the question explicitly states simple interest is applied.

• The Fix: Underline the compounding frequency in the problem description immediately. If it says “compounded quarterly”, write down \(n = 4\), and use the compound interest formula with \(\frac{r}{4}\) as the period interest rate and \(4t\) as the total number of periods: \(A = P\left(1 + \frac{r}{4}\right)^{4t}\). If it states simple interest, always default to \(A = P(1+rt)\).

Trap 6: Perpendicular Slope Sign Slips

When finding the slope of a line perpendicular to a given line, students frequently perform only one of the two necessary transformations: they either flip the fraction (reciprocal) but keep the sign, or flip the sign but leave the fraction unchanged.

• The Fix: A perpendicular slope is the negative reciprocal. You must perform two distinct operations: first, invert the numerator and denominator; second, multiply by \(-1\) to change the sign. If the original slope is \(-\frac{3}{5}\), its perpendicular slope is \(\frac{5}{3}\). If the original slope is \(2\), its perpendicular slope is \(-\frac{1}{2}\).

Trap 7: Sector Area Denominator vs. Radian Formula Confusion

Students often confuse the degree-based sector area formula with the radian-based sector area formula. For example, they might plug an angle in degrees directly into the radian formula: \(A = \frac{1}{2}r^2\theta\), which yields a value that is vastly too large. Conversely, they might divide a radian measure by \(360\), which is mathematically incorrect because \(360\) is a degree measure (the equivalent divisor in radians is \(2\pi\)).

• The Fix: Check the angle units before selecting your formula. If the angle is given with a degree symbol (\(^\circ\)), use the degree-based ratio: \(A = \frac{\theta}{360} \times \pi r^2\). If the angle is in radians (often containing \(\pi\)), use the radian-based formula: \(A = \frac{1}{2}r^2\theta\). If you are ever unsure, convert the angle to degrees first using the conversion factor \(\frac{180}{\pi}\).

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15. Concept Drills & Worked Examples

Let’s review 8 worked examples showcasing how to apply these formulas on the SAT.

Example 1: Distance Formula

Question: What is the distance between the points \((-3, 2)\) and \((5, -4)\) in the \(xy\)-plane?

Step-by-Step Solution:

1. State the distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
2. Substitute the points \((x_1, y_1) = (-3, 2)\) and \((x_2, y_2) = (5, -4)\): \[d = \sqrt{(5 - (-3))^2 + (-4 - 2)^2}\]
3. Simplify the terms inside the parentheses: \[d = \sqrt{(8)^2 + (-6)^2}\]
4. Square the terms: \[d = \sqrt{64 + 36} = \sqrt{100}\]
5. Take the square root: \[d = 10\]
6. The distance between the points is \(10\) units.

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Example 2: Perpendicular Lines

Question: A line \(L_1\) passes through points \((-1, 4)\) and \((2, -2)\). Line \(L_2\) is perpendicular to Line \(L_1\). What is the slope of Line \(L_2\)?

Step-by-Step Solution:

1. Calculate the slope \(m_1\) of Line \(L_1\) using the slope formula: \[m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 4}{2 - (-1)} = \frac{-6}{3} = -2\]
2. Since Line \(L_2\) is perpendicular to Line \(L_1\), its slope \(m_2\) must be the negative reciprocal of \(m_1\): \[m_2 = -\frac{1}{m_1} = -\frac{1}{-2} = \frac{1}{2}\]
3. The slope of Line \(L_2\) is \(\frac{1}{2}\) (or \(0.5\)).

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Example 3: Sum of Roots

Question: What is the sum of the solutions to the quadratic equation \(3x^2 - 18x + 12 = 0\)?

Step-by-Step Solution:

1. Identify the coefficients in the quadratic equation \(3x^2 - 18x + 12 = 0\):
   • \(a = 3\)
   • \(b = -18\)
   • \(c = 12\)
2. State the sum of roots formula: \[\text{Sum} = -\frac{b}{a}\]
3. Substitute the values of \(a\) and \(b\): \[\text{Sum} = -\frac{-18}{3} = \frac{18}{3} = 6\]
4. The sum of the solutions is \(6\) (avoiding the need to factor or use the quadratic formula).

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Example 4: Quadratic Vertex

Question: What are the coordinates of the vertex of the parabola defined by \(f(x) = 2x^2 - 8x + 5\)?

Step-by-Step Solution:

1. Identify the coefficients of the quadratic function in standard form \(f(x) = ax^2 + bx + c\):
   • \(a = 2\), \(b = -8\), \(c = 5\)
2. Calculate the \(x\)-coordinate \(h\) of the vertex using the vertex formula: \[h = -\frac{b}{2a} = -\frac{-8}{2(2)} = \frac{8}{4} = 2\]
3. Substitute \(h = 2\) back into the function to find the \(y\)-coordinate \(k\) of the vertex: \[k = f(2) = 2(2)^2 - 8(2) + 5\] \[k = 2(4) - 16 + 5 = 8 - 16 + 5 = -3\]
4. The coordinates of the vertex are \((2, -3)\).

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Example 5: Compound Interest

Question: A student deposits \($5,000\) into a savings account that earns an annual interest rate of \(4%\) compounded quarterly. If no other deposits or withdrawals are made, which of the following expressions represents the balance of the account after \(6\) years?

Step-by-Step Solution:

1. State the compound interest formula: \[A = P\left(1 + \frac{r}{n}\right)^{nt}\]
2. Identify the parameters from the problem statement:
   • Principal \(P = 5000\)
   • Annual interest rate \(r = 4% = 0.04\)
   • Compounded quarterly, so \(n = 4\) times per year
   • Time \(t = 6\) years
3. Substitute these values into the formula: \[A = 5000\left(1 + \frac{0.04}{4}\right)^{4(6)}\]
4. Simplify the expression: \[A = 5000(1 + 0.01)^{24} = 5000(1.01)^{24}\]
5. The balance is represented by the expression \(5000(1.01)^{24}\).

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Example 6: Circle Completing the Square

Question: A circle is defined by the equation \(x^2 + y^2 + 10x - 4y + 20 = 0\). What is the area of the circle?

Step-by-Step Solution:

1. Group the \(x\) and \(y\) terms, moving the constant to the right side: \[(x^2 + 10x) + (y^2 - 4y) = -20\]
2. Complete the square for both parts:
   • For the \(x\)-terms: half of \(10\) is \(5\), and \(5^2 = 25\).
   • For the \(y\)-terms: half of \(-4\) is \(-2\), and \((-2)^2 = 4\).
3. Add these values to both sides of the equation: \[(x^2 + 10x + 25) + (y^2 - 4y + 4) = -20 + 25 + 4\]
4. Factor the perfect square trinomials: \[(x + 5)^2 + (y - 2)^2 = 9\]
5. In standard form \((x-h)^2 + (y-k)^2 = r^2\), the constant on the right is \(r^2\): \[r^2 = 9\]
6. The area of a circle is calculated using \(A = \pi r^2\). Substitute \(r^2 = 9\): \[A = 9\pi\]
7. The area of the circle is \(9\pi\) square units.

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Example 7: Complementary Angle Identity

Question: If \(\sin(x) = \frac{5}{13}\) and \(x\) is an acute angle measured in radians, what is the value of \(\cos\left(\frac{\pi}{2} - x\right)\)?

Step-by-Step Solution:

1. Recognize the complementary angle identity in radians: \[\cos\left(\frac{\pi}{2} - x\right) = \sin(x)\]
2. Since we are given that \(\sin(x) = \frac{5}{13}\), it follows that: \[\cos\left(\frac{\pi}{2} - x\right) = \frac{5}{13}\]
3. The answer is \(\frac{5}{13}\) (requiring no calculations).

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Example 8: Sector Area in Radians

Question: A sector of a circle with a radius of \(6\) centimeters has a central angle of \(\frac{\pi}{3}\) radians. What is the area of the sector, in square centimeters?

Step-by-Step Solution:

1. State the sector area formula when the central angle \(\theta\) is in radians: \[A = \frac{1}{2}r^2\theta\]
2. Substitute the given values \(r = 6\) and \(\theta = \frac{\pi}{3}\): \[A = \frac{1}{2}(6)^2\left(\frac{\pi}{3}\right)\]
3. Simplify the expression: \[A = \frac{1}{2}(36)\left(\frac{\pi}{3}\right) = 18 \times \frac{\pi}{3} = 6\pi\]
4. The area of the sector is \(6\pi\) square centimeters.

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15. Practice Quiz

Test your formula recall and application with these 5 practice questions.

Questions

1. Question 1: A line in the \(xy\)-plane passes through the points \((1, 3)\) and \((k, 9)\). If the slope of the line is \(3\), what is the value of \(k\)?

   • A) 2
   • B) 3
   • C) 4
   • D) 5

2. Question 2: What is the sum of the roots of the equation \(2x^2 + 8x - 10 = 0\)?

   • A) -5
   • B) -4
   • C) 4
   • D) 5

3. Question 3: Which of the following equations represents a circle in the \(xy\)-plane centered at \((3, -2)\) with a diameter of \(10\)?

   • A) \((x - 3)^2 + (y + 2)^2 = 100\)
   • B) \((x + 3)^2 + (y - 2)^2 = 25\)
   • C) \((x - 3)^2 + (y + 2)^2 = 25\)
   • D) \((x + 3)^2 + (y - 2)^2 = 10\)

4. Question 4: If the discriminant of the quadratic equation \(2x^2 - 6x + k = 0\) is exactly \(0\), what is the value of \(k\)?

   • A) 3
   • B) 4
   • C) 4.5
   • D) 9

5. Question 5: Convert \(\frac{5\pi}{6}\) radians to degrees.

   • A) \(120^\circ\)
   • B) \(135^\circ\)
   • C) \(150^\circ\)
   • D) \(210^\circ\)

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Answer Key and Explanations

Question 1

• Correct Answer: B) 3
• Detailed Explanation:
  • State the slope formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]
  • Substitute the known values: \(m = 3\), \((x_1, y_1) = (1, 3)\), and \((x_2, y_2) = (k, 9)\): \[3 = \frac{9 - 3}{k - 1}\] \[3 = \frac{6}{k - 1}\]
  • Multiply both sides by \(k - 1\): \[3(k - 1) = 6\]
  • Divide by \(3\): \[k - 1 = 2 \quad \implies \quad k = 3\]
  • Therefore, B is the correct option.

Question 2

• Correct Answer: B) -4
• Detailed Explanation:
  • For the quadratic equation \(2x^2 + 8x - 10 = 0\), the coefficients are \(a = 2\) and \(b = 8\).
  • State the sum of roots formula: \[\text{Sum} = -\frac{b}{a}\]
  • Substitute the coefficients: \[\text{Sum} = -\frac{8}{2} = -4\]
  • Alternatively, factor the equation: \[2(x^2 + 4x - 5) = 0 \quad \implies \quad 2(x + 5)(x - 1) = 0\] The roots are \(x = -5\) and \(x = 1\). The sum of these roots is \(-5 + 1 = -4\).
  • Both methods confirm that B is the correct option.

Question 3

• Correct Answer: C) \((x - 3)^2 + (y + 2)^2 = 25
• Detailed Explanation:
  • The standard circle equation is \((x - h)^2 + (y - k)^2 = r^2\).
  • Substitute the center \((3, -2)\): \[(x - 3)^2 + (y - (-2))^2 = r^2 \quad \implies \quad (x - 3)^2 + (y + 2)^2 = r^2\] This eliminates options B and D.
  • The problem states the diameter of the circle is \(10\). The radius \(r\) is half the diameter: \[r = \frac{10}{2} = 5\]
  • Substitute the radius into the equation as \(r^2\): \[r^2 = 5^2 = 25\]
  • The complete equation is \((x - 3)^2 + (y + 2)^2 = 25\). Option C is correct.

Question 4

• Correct Answer: C) 4.5
• Detailed Explanation:
  • State the discriminant formula for a quadratic equation \(ax^2 + bx + c = 0\): \[\Delta = b^2 - 4ac\]
  • For \(2x^2 - 6x + k = 0\), the coefficients are \(a = 2\), \(b = -6\), and \(c = k\).
  • Set the discriminant equal to \(0\): \[(-6)^2 - 4(2)(k) = 0\] \[36 - 8k = 0\]
  • Solve for \(k\): \[36 = 8k\] \[k = \frac{36}{8} = 4.5\]
  • Therefore, C is the correct option.

Question 5

• Correct Answer: C) \(150^\circ\)
• Detailed Explanation:
  • To convert radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\): \[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]
  • Substitute the given value: \[\text{Degrees} = \frac{5\pi}{6} \times \frac{180}{\pi}\]
  • Simplify the expression by canceling \(\pi\) and dividing \(180\) by \(6\): \[\text{Degrees} = 5 \times 30 = 150^\circ\]
  • Therefore, C is the correct option.

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16. Printable-Style SAT Formulas Cheat Sheet Section

Print or bookmark this section for quick reference on the morning of the exam.

SAT Math Formulas Cheat Sheet Reference

Category	Concept	MathJax Formula
1. Linear Equations & Coordinates	Slope	(m = \frac{y_2 - y_1}{x_2 - x_1})
	Midpoint	(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right))
	Distance	(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2})
	Slope-Intercept Form	(y = mx + b)
	Point-Slope Form	(y - y_1 = m(x - x_1))
	Parallel Slopes	(m_1 = m_2)
	Perpendicular Slopes	(m_1 \cdot m_2 = -1)
2. Quadratics & Functions	Quadratic Formula	(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a})
	Discriminant	(\Delta = b^2 - 4ac)
	* Discriminant outcomes	(\Delta > 0): 2 real roots (\Delta = 0): 1 real root (\Delta < 0): 0 real roots
	Sum of Roots	(\text{Sum} = -\frac{b}{a})
	Product of Roots	(\text{Product} = \frac{c}{a})
	Vertex x-coordinate	(h = -\frac{b}{2a})
3. Percentages & Financials	Percent Change	(\text{Change} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100%)
	Exponential Growth/Decay	(A(t) = P(1 \pm r)^t)
	Simple Interest	(A = P(1 + rt))
	Compound Interest	(A = P\left(1 + \frac{r}{n}\right)^{nt})
4. Exponents & Radicals	Product Rule	(x^a \cdot x^b = x^{a+b})
	Quotient Rule	(\frac{x^a}{x^b} = x^{a-b})
	Power Rule	((x^a)^b = x^{ab})
	Radical Conversion	(x^{m/n} = \sqrt[n]{x^m})
5. Circles & Trigonometry	Circle Equation	((x - h)^2 + (y - k)^2 = r^2)
	Arc Length (Radians)	(s = r\theta)
	Sector Area (Radians)	(A = \frac{1}{2}r^2\theta)
	Degrees to Radians	(\text{Rad} = \text{Deg} \times \frac{\pi}{180})
	Radians to Degrees	(\text{Deg} = \text{Rad} \times \frac{180}{\pi})
	Trig Complementary Identity	(\sin(\theta) = \cos(90^\circ - \theta) \quad \text{and} \quad \sin(x) = \cos\left(\frac{\pi}{2} - x\right))
	Pythagorean Identity	(\sin^2(\theta) + \cos^2(\theta) = 1)