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College Algebra Midterm Key Concepts and Problems

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Terms in this set (24)

  • Evaluate 2(x + 3) − 12 when x = −4
    Substitute x = −4: 2(−4 + 3) − 12 = 2(−1) − 12 = −2 − 12 = −14.
  • Write 3.1642 × 10^6 in standard notation
    Move decimal 6 places right: 3,164,200.
  • Simplify 8x^3 / (2x)^2
    Rewrite denominator: (2x)^2 = 4x^2. Then 8x^3 / 4x^2 = 2x.
  • Subtract (13w³ + 2w² − 3) − (6w² + 5w − 3)
    Result: 13w³ + 2w² − 3 − 6w² − 5w + 3 = 13w³ − 4w² − 5w.
  • Multiply (b − 2)(b² − 4b + 4)
    Use distributive property: b³ − 4b² + 4b − 2b² + 8b − 8 = b³ − 6b² + 12b − 8.
  • Factor 16x² − 81
    Difference of squares: (4x − 9)(4x + 9).
  • Factor y² + 12y + 36
    Perfect square trinomial: (y + 6)².
  • Find the y-intercept for 2x − 5y = 6
    Set x=0: −5y=6 → y=−6/5.
  • Distance between (5, −3) and (−2, 8)
    Distance = √[(−2−5)² + (8+3)²] = √[49 + 121] = √170.
  • Midpoint between (5, −3) and (−2, 8)
    Midpoint = ((5−2)/2, (−3+8)/2) = (1.5, 2.5).
  • Interval notation for {x | x ≤ 10}
    Interval: (−∞, 10].
  • Solve |3x − 2| ≥ 4
    3x − 2 ≥ 4 or 3x − 2 ≤ −4 → x ≥ 2 or x ≤ −2/3.
  • Equation of line through (−4, 2) and (5, −3)
    Slope m = (−3−2)/(5+4) = −5/9; equation: y − 2 = (−5/9)(x + 4).
  • Add (3 − 2i) + (4 − i)
    Sum: (3+4) + (−2i − i) = 7 − 3i.
  • Multiply (14 − 3i)(8 + 5i)
    Product: 14×8 + 14×5i − 3i×8 − 3i×5i = 112 + 70i − 24i − 15i² = 112 + 46i + 15 = 127 + 46i.
  • Solve 4x² − 4x = 1
    Rewrite: 4x² − 4x − 1 = 0; use quadratic formula.
  • Find domain of g(x) = 8 − x
    Domain: all real numbers (−∞, ∞).
  • Compute (f ○ g)(4) for f(x) = 3x² − 10, g(x) = 2x + 5
    g(4) = 2(4) + 5 = 13; f(13) = 3(13)² − 10 = 3(169) − 10 = 497.
  • Find inverse of f(x) = 0.5√(x − 3)
    Swap x and y: x = 0.5√(y − 3); solve for y: y = 4x² + 3.
  • Determine if f(x) = −3x² + 4 is one-to-one
    No, because it is a quadratic (parabola) failing the horizontal line test.
  • Write quadratic function in standard form: y = x² + 2x − 8
    Complete the square: y = (x + 1)² − 9.
  • Use Rational Root Theorem on f(x) = 4x⁴ + 8x³ + 21x² + 17x + 4
    Possible rational roots are ± factors of 4 over factors of 4: ±1, ±2, ±4, ±1/2, ±3/4, etc.
  • Find vertical asymptote(s) of f(x) = (x + 4) / (x² − 2x − 3)
    Denominator zeroes: x² − 2x − 3 = (x − 3)(x + 1) = 0 → x = 3, −1 vertical asymptotes.
  • Find slant asymptote of g(x) = (x² + 3x − 3) / (x − 1)
    Divide numerator by denominator: quotient is x + 4; slant asymptote y = x + 4.