📘 College of Engineering Entrance Test Math Review
🟦 1. Function Concepts
1-1 Domain and Range
• Domain: Set of all real x values for which the function is defined.
• For rational functions: Exclude values that make the denominator zero.
• For square roots: Ensure the expression inside is ≥ 0.
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• Example: f (x) = x−2
⇒ Domain : x  2 =
• Range: Set of all possible output values.
• Use algebra or graphing to determine.
• Example: f (x) = x2 ⇒ Range : [0, ∞)
1-2 Polynomials
• General form: f (x) = an xn + an−1 xn−1 + … + a0
• Characteristics:
• Domain: all real numbers
• Degree = highest power of x
• End behavior depends on leading term
• Number of turning points ≤ degree − 1
• Example: f (x) = x3 − 3x + 2
1-3 Absolute Value
• Definition:
∣x∣ = {
x
if x ≥ 0
−x if x < 0
• To solve ∣A∣ = B , split into cases:
• A = B or A = −B
• Example: ∣x − 3∣ = 5 ⇒ x − 3 = 5 or x − 3 = −5 ⇒ x = 8 or − 2
1-4 Graph of a Function
• Know shapes of common functions:
• Linear: straight line
• Quadratic: parabola (opens up/down)
• Absolute value: V-shape
• Square root: starts from origin, curves right
• Transformations:
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• f (x) + c : shift up/down
• f (x + c) : shift left/right
• −f (x) : reflect over x-axis
• f (−x) : reflect over y-axis
• af (x) : stretch vertically
🟦 2. Exponents and Logarithms
2-1 Exponential Functions
• General form: f (x) = ax , a > 0, a  1
=
• Properties:
• ax ⋅ ay = ax+y
x
• aay = ax−y
• (ax )n = axn
• Graph is always above x-axis, increasing if a > 1
2-2 Logarithmic Functions
• Inverse of exponentials: y = loga (x) ⇔ ay = x
• Properties:
• log(ab) = log a + log b
• log ( ab ) = log a − log b
• log(an ) = n log a
• Domain: x > 0 , Range: all real numbers
🟦 3. Trigonometric Functions
3-1 Acute Angles
• Right triangle definitions:
opposite
• sin θ = hypotenuse
adjacent
• cos θ = hypotenuse
opposite
• tan θ = adjacent
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• Pythagorean identity: sin θ + cos2 θ = 1
3-2 General Angles
• Measured from x-axis, counterclockwise is positive
• Use unit circle definitions
y
• sin θ = y , cos θ = x , tan θ = x
3-3 Graphs
• sin x , cos x : wave pattern
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• Amplitude = max height
• Period = 2π
• tan x : repeats every π , has vertical asymptotes
3-4 Sum and Difference Formulas
• Derive or memorize:
• sin(a ± b) = sin a cos b ± cos a sin b
• cos(a ± b) = cos a cos b ∓ sin a sin b
3-5 Double and Half Angle
• sin(2x) = 2 sin x cos x
2
2
• cos(2x) = cos2 x − sin x = 2 cos2 x − 1 = 1 − 2 sin x
• Half-angle:
2
x
• sin x2 = 1−cos
2
x
• cos2 x2 = 1+cos
2
3-6 Product-to-Sum and Sum-to-Product
• Examples:
• sin a sin b = 12 [cos(a − b) − cos(a + b)]
• cos a cos b = 12 [cos(a − b) + cos(a + b)]
🟦 4. Vectors (Focus on Space Vectors)
4-1 Plane Vectors
• Represented as: v = ⟨x, y⟩
• Operations:
• Addition: component-wise
• Scalar multiplication: multiply each component
• Dot product: a ⋅ b = ax bx + ay by
4-2 Space Vectors
• Represented as: v = ⟨x, y, z⟩
• Magnitude: ∣v ∣ =
x2 + y 2 + z 2
• Direction cosines: angles with coordinate axes
• Unit vector: ∣vv ∣
4-3 Operations
• Dot product: measures alignment
• a ⋅ b = ∣a∣∣b∣ cos θ
• If dot product = 0, vectors are perpendicular
• Cross product: gives vector ⟂ to both
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• a × b = ⟨ay bz − az by , az bx − ax bz , ax by − ay bx ⟩
• Angle between vectors:
• Use dot product formula above
4-4 Equations in Space
• Line (parametric form):
• x = x0 + at, y = y0 + bt, z = z0 + ct
• Plane:
• General: ax + by + cz = d
• Normal vector: n = ⟨a, b, c⟩
• Sphere:
• (x − a)2 + (y − b)2 + (z − c)2 = r 2
4-5 Matrices
•
2×2 matrix: [
a
c
b
]
d
• Addition: add corresponding elements
• Multiplication:
ae + bg af + bh
•
AB = [
ce + dg cf + dh
• Determinant: ad − bc
]
• Inverse: exists if det ≠ 0
4-6 Linear Transformations
• Represented by matrices
• Common types:
• Rotation
• Reflection
• Scaling
• Applied using matrix-vector multiplication
4-7 Polar Coordinates
• Coordinates: (r, θ)
• x = r cos θ
• y = r sin θ
• Conversion:
• Rectangular to polar: r =
x2 + y 2 , θ = tan−1 ( xy )
🟦 5. Conic Curves
5-1 Parabola
• Standard form:
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• Horizontal: y 2 = 4ax
• Vertical: x2 = 4ay
• Vertex: (0,0) if in standard form
• Focus: at distance a from vertex
• Directrix: line y = −a or x = −a
5-2 Ellipse
• Standard form:
2
2
• x2 + y2 = 1 , where a > b
a
b
• Center: (0,0) if in standard form
• Foci: distance c = a2 − b2 from center along major axis
• Major axis: length = 2a; Minor axis: length = 2b
5-3 Hyperbola
• Standard form:
2
2
• Horizontal: x2 − y2 = 1
2
a
2
b
• Vertical: y2 − x2 = 1
b
a
• Asymptotes:
• Horizontal: y = ± ab x
• Vertical: y = ± ab x
• Foci: c =
a2 + b2
📌 Tips for the Test
• Master all formulas and identities (especially trig and conics)
• Practice simplifying expressions and solving equations under time
• Use diagrams to visualize vectors, angles, and conic sections
• Double-check domain restrictions, especially with logs and roots
Let me know if you'd like practice problems, summary sheets, or worked examples!
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