Worksheet 05 - Complete Function Analysis

Section 05

Instructions for Complete Function Analysis

For each function, perform a complete function analysis including:

  1. Domain: Identify all values of \(x\) for which the function is defined
  2. Zeros: Find where \(f(x) = 0\) (if they exist)
  3. Y-intercept: Find \(f(0)\) (if it exists)
  4. Symmetry: Check for even symmetry (\(f(-x) = f(x)\)) or odd symmetry (\(f(-x) = -f(x)\))
  5. Asymptotes:
    • Vertical asymptotes (for rational functions)
    • Horizontal asymptotes (behavior as \(x \to \pm\infty\))
    • Oblique/slant asymptotes (if applicable)
  6. First derivative \(f'(x)\):
    • Critical points (where \(f'(x) = 0\) or undefined)
    • Intervals of increase/decrease
    • Local extrema (maxima and minima)
  7. Second derivative \(f''(x)\):
    • Inflection points (where \(f''(x) = 0\) and changes sign)
    • Intervals of concavity (concave up where \(f''(x) > 0\), concave down where \(f''(x) < 0\))
  8. Sketch: Draw the graph incorporating all the above information

Problem Set 1: Polynomial Functions

Perform a complete function analysis for each of the following polynomial functions:

1. \(f(x) = -\frac{1}{20}x^{3} + 15x\)

  • This is an odd-degree polynomial with a negative leading coefficient
  • Factor out \(x\) to find zeros
  • Check for odd symmetry
  • The function will have no asymptotes

2. \(f(x) = \frac{1}{9}x^{3} - \frac{1}{6}x^{2} - 2x\)

  • Factor out \(x\) first
  • Use the quadratic formula for the remaining factor
  • No symmetry (has both even and odd powers)

3. \(f(x) = 1.5x^{4} + x^{3} - 9x^{2}\)

  • Factor out \(x^{2}\)
  • The remaining quadratic can be solved with the quadratic formula
  • Check the y-intercept carefully

4. \(f(x) = x^{3} - 6x^{2} + 9x\)

  • Factor out \(x\)
  • The quadratic factor is a perfect square
  • One zero has multiplicity 2

5. \(f(x) = -\frac{1}{20}x^{4} + \frac{6}{5}x^{2} - 4\)

  • This has even symmetry (only even powers)
  • Use substitution \(u = x^{2}\) to find zeros
  • Leading coefficient is negative, so the function opens downward

6. \(f(x) = -\frac{1}{36}\left(3x^{5} - 50x^{3} + 135x\right)\)

  • Simplify: \(f(x) = -\frac{1}{12}x^{5} + \frac{25}{18}x^{3} - \frac{15}{4}x\)
  • Factor out \(x\) first
  • This function has odd symmetry
  • The remaining quartic can be solved by substitution

7. \(f(x) = x^{3} + 4x^{2} - 11x - 30\)

  • Try rational root theorem: possible rational zeros are divisors of 30
  • Test \(x = -2, -3, 2, 3, 5, -5\), etc.
  • Once you find one zero, use polynomial division

8. \(f(x) = \frac{1}{9}x^{5} - \frac{20}{27}x^{4} + \frac{10}{9}x^{3}\)

  • Factor out \(\frac{1}{9}x^{3}\)
  • The remaining quadratic is straightforward
  • Note that \(x = 0\) is a zero with multiplicity 3

9. \(f(x) = x^{4} + x^{3} - 11x^{2} + 20\)

  • Use the rational root theorem
  • Try small integers as possible zeros
  • This one is more challenging - be systematic

10. \(f(x) = \frac{1}{32}\left(5x^{4} - x^{5}\right)\)

  • Simplify: \(f(x) = -\frac{1}{32}x^{5} + \frac{5}{32}x^{4}\)
  • Factor out \(\frac{1}{32}x^{4}\)
  • Note the behavior at the zero with multiplicity 4

Problem Set 2: Rational Functions

Perform a complete function analysis for each of the following rational functions:

1. \(f(x) = \frac{8}{4 - x^{2}}\)

  • Rewrite denominator: \(4 - x^{2} = (2-x)(2+x)\)
  • Vertical asymptotes at \(x = 2\) and \(x = -2\)
  • Has even symmetry
  • Horizontal asymptote at \(y = 0\)

2. \(f(x) = \frac{2 - x^{2}}{x^{2} - 9}\)

  • Factor numerator: \(2 - x^{2} = -1(x^{2} - 2) = -(x - \sqrt{2})(x + \sqrt{2})\)
  • Factor denominator: \(x^{2} - 9 = (x - 3)(x + 3)\)
  • Vertical asymptotes at \(x = \pm 3\)
  • Horizontal asymptote at \(y = -1\)
  • Has even symmetry

3. \(f(x) = \frac{x^{2} - 4}{x^{2} + 2}\)

  • Numerator: \((x-2)(x+2)\)
  • Denominator is always positive (no vertical asymptotes!)
  • Domain is all real numbers
  • Horizontal asymptote at \(y = 1\)

4. \(f(x) = \frac{x^{2}}{(x - 2)^{2}}\)

  • Vertical asymptote at \(x = 2\)
  • Zero at \(x = 0\)
  • Horizontal asymptote at \(y = 1\)
  • Always non-negative

5. \(f(x) = \frac{x}{x^{2} + 1}\)

  • No vertical asymptotes (denominator always positive)
  • Odd symmetry
  • Horizontal asymptote at \(y = 0\)
  • Zero at \(x = 0\)

6. \(f(x) = \frac{3x^{2} - 3x}{(x - 2)^{2}}\)

  • Factor numerator: \(3x(x - 1)\)
  • Zeros at \(x = 0\) and \(x = 1\)
  • Vertical asymptote at \(x = 2\)
  • For horizontal asymptote, divide leading terms

7. \(f(x) = \frac{3}{x} - \frac{12}{x^{2}}\)

  • Combine fractions: \(f(x) = \frac{3x - 12}{x^{2}}\)
  • Factor numerator: \(3(x - 4)\)
  • Vertical asymptote at \(x = 0\)
  • Zero at \(x = 4\)

8. \(f(x) = \frac{1}{2}x + \frac{1}{2} + \frac{2}{x^{2}}\)

  • Combine: \(f(x) = \frac{\frac{1}{2}x^{3} + \frac{1}{2}x^{2} + 2}{x^{2}}\)
  • Vertical asymptote at \(x = 0\)
  • Oblique asymptote: \(y = \frac{1}{2}x + \frac{1}{2}\)
  • Finding zeros requires solving a cubic equation

Problem Set 3: Function Families

For each function family (with parameter \(t > 0\)), perform a complete function analysis. Then sketch the graphs for \(t = 1, 2, 3\) in the same coordinate system.

1. \(f_{t}(x) = x - \frac{t^{3}}{x^{2}}\)

  • Combine: \(f_{t}(x) = \frac{x^{3} - t^{3}}{x^{2}}\)
  • Vertical asymptote at \(x = 0\)
  • Zero at \(x = t\) (since \(x^{3} = t^{3}\) when \(x = t\))
  • Oblique asymptote: \(y = x\)
  • As \(t\) increases, the zero moves to the right

2. \(f_{t}(x) = \frac{10}{x} - \frac{10t}{x^{2}}\)

  • Combine: \(f_{t}(x) = \frac{10x - 10t}{x^{2}} = \frac{10(x - t)}{x^{2}}\)
  • Vertical asymptote at \(x = 0\)
  • Zero at \(x = t\)
  • Horizontal asymptote at \(y = 0\)

3. \(f_{t}(x) = \frac{6x}{x^{2} + t^{2}}\)

  • No vertical asymptotes (denominator always positive)
  • Odd symmetry
  • Zero at \(x = 0\)
  • Horizontal asymptote at \(y = 0\)
  • As \(t\) increases, the function becomes “flatter”

4. \(f_{t}(x) = \frac{10x}{\left(x^{2} + t\right)^{2}}\)

  • No vertical asymptotes (denominator always positive)
  • Odd symmetry
  • Zero at \(x = 0\)
  • Horizontal asymptote at \(y = 0\)
  • More complex than previous - use quotient rule carefully

Additional Notes

NoteWorking with Function Families

When analyzing function families: 1. First analyze the general case with parameter \(t\) 2. Identify how \(t\) affects key features (zeros, extrema, asymptotes) 3. Then substitute specific values (\(t = 1, 2, 3\)) to sketch 4. Use different colors or line styles to distinguish the three graphs

WarningCommon Mistakes to Avoid
  • Forgetting to check the domain first (especially for rational functions)
  • Missing vertical asymptotes where the denominator equals zero
  • Confusing “no solution” with “zero” when finding zeros
  • Not checking if critical points are actually extrema (use first or second derivative test)
  • Forgetting to simplify before differentiating