2nd Semester Final Exam Review

Big Idea 11

• Conjugate of complex numbers:

  • $4 + 3i$: $-4 - 3i$

  • $2 - 6i$: $2 + 6i$

• Simplify complex numbers to $a + bi$ form:

  • $-i\sqrt{-4} + \sqrt{-9} = -2i^2 + 3i = 2 + 3i$

• $(1 + i) - (3 - 2i) = -2 + 3i$

• $(2 + 5i)(2 - i) = 4 - 2i + 10i - 5i^2 = 9 + 8i$

• $(1 - 3i)^2 = 1 - 6i + 9i^2 = -8 - 6i$

Big Idea 12

• Solving quadratic equations:

  • $4x^2 = -5 \implies x = \pm \frac{i\sqrt{5}}{2}$

  • $12x^2 = 2(-2x) \implies 6x^2 + 2x = 0 \implies x(6x + 2) = 0 \implies x = 0, -\frac{1}{3}$

  • $x^2 + 3 = 15 \implies x^2 = 12 \implies x = \pm 2\sqrt{3}$

• Completing the square: $y = x^2 - 8x + 1 = 0 \implies (x - 4)^2 - 15 = 0 \implies x = 4 \pm \sqrt{15}$

• Factoring: $x^2 + 16 = (x + 4i)(x - 4i)$

• Quadratic regression:

  • Regression equation: $y = 0.0419x^2 + 0.0492x - 3.6513$

  • If $x = 73$, then $y \approx 223.2$

• Quadratic inequality: x^2 - 3x - 4 > 0 \implies (x - 4)(x + 1) > 0 \implies x < -1 \text{ or } x > 4

• Solutions to a system (graphical): $x = -1, 4$

• Type of roots: Imaginary (no real solutions).

Big Idea 13

• Zeros and multiplicities:

  • Zeros: $-5$ (multiplicity 2), $-3$ (multiplicity 2), $0$ (multiplicity 1), $2$ (multiplicity 1)

  • Degree: Even

  • Leading Coefficient (LC): Positive

• Factored form equation: $y = x^2(x-2)$

• Total roots for $P(x) = 9x^3 + x^2 - 8.5x - 6.5$: 3

• Total roots for $k(x) = x^2(x - 3)(x^2 - 5)$: 5; Roots: $0$ (multiplicity 2), $3$, $\pm\sqrt{5}$

• Graph properties using Desmos:

  • For $g(x) = x^4 - 2x^3 - 3x^2 + 5x - 3$:

    • Max: $(0.612, -1.382)$

    • Mins: $(-1.053, -8.027)$, $(1.941, -5.029)$

    • Intervals of INC: $(-1.053, 0.612) \cup (1.941, \infty)$

    • Intervals of DEC: $(-\infty, -1.053) \cup (0.612, 1.941)$

    • Positive: $(-\infty, -1.783) \cup (2.586, \infty)$

    • Negative: $(-1.783, 2.586)$

    • Domain: $(-\infty, \infty)$

    • Range: $[-8.027, \infty)$

    • As $x \to -\infty, y \to \infty$

    • As $x \to \infty, y \to \infty$

• Polynomial function identification from a graph: $p(x) = (x+1)(x-2)(x-5)$

Big Idea 14

• Linear programming problem:

  • Variables:

    • $x$: # of amberjacks

    • $y$: # of flounder

  • Constraints:

    • $x + y \geq 50$

    • $x \leq 30$

    • $y \leq 35$

  • Objective function: $P = 4x + 3y$

• Solutions to a system of inequalities: Check if the point lies within the feasible region.

Big Idea 15

• Simplifying rational expressions:

  • $\frac{x^2 + 11x + 10}{x + 4} \div \frac{x^2 + 4x + 3}{x + 4} = \frac{(x + 10)(x + 1)}{x + 4} \cdot \frac{x + 4}{(x + 3)(x + 1)} = \frac{x + 10}{x + 3}$

  • $\frac{b^2 - 10b + 16}{7b} \div \frac{b^2 - 14b}{x + 4x} = \frac{(b - 8)(b - 2)}{7b} \cdot \frac{b(b - 2)}{b - 8} = \frac{b-2}{7b(b-2)} = \frac{x(x+4)}{7b^2-14b}$

  • $\frac{x + 5}{(x + 4)(x + 4)} - \frac{x - 1}{(x + 4)(x + 4)} = \frac{6}{(x + 4)(x + 4)}$

  • $\frac{k + 2}{3k - 5} - \frac{4k - 2}{9k - 15} = \frac{3(k + 2) - (4k - 2)}{3(3k - 5)} = \frac{-k + 8}{3(3k - 5)}$

• Solving rational equations:

  • $\frac{4x + 5}{x + 5} = \frac{2}{9} \implies x = -\frac{35}{18}$

  • $\frac{x + 1}{x^2 + 8x + 15} + \frac{18}{x + 3} = \frac{9}{x + 5} \implies x = 8$

Big Idea 16

• Finding asymptotes, holes, and x-intercepts:

  • $f(x) = \frac{x - 4}{x^2 - 2x - 8} = \frac{x - 4}{(x - 4)(x + 2)} = \frac{1}{x+2}$

    • HA: $y = 0$

    • VA: $x = -2$

    • Hole: $(4, 1/6)$

    • X-intercepts: None

  • $f(x) = \frac{2x^2 - 8}{x^2 + x - 6} = \frac{2(x - 2)(x + 2)}{(x + 3)(x - 2)} = \frac{2(x+2)}{x + 3}$

    • HA: $y = 2$

    • VA: $x = -3$

    • Hole: $(2, 8/5)$

    • X-intercepts: $x = -2$

• Properties of rational functions from graph:

  • Domain: $(-\infty, -2) \cup (-2, \infty)$

  • Range: $(-\infty, 3) \cup (3, \infty)$

  • Interval of INC: N/A

  • Interval of DEC: $(-\infty, -2) \cup (-2, \infty)$

  • As $x \to -\infty, y \to 3$

  • As $x \to \infty, y \to 3$

Big Idea 17

• Matrix operations:

  • Possible operations:

    • 2A - 3B: Yes (same dimensions)

    • C + A: No (different dimensions)

    • CB: No (# of columns in C != # of rows in B)

• Matrix multiplication:

• System of equations:

  • $2x + y = 37$

  • $4x + 3y = 81$

  • Solution: Paint costs $15, brush costs $7.

Big Idea 18

• Trigonometric values:

  • $\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}$

  • $\sin(210^\circ) = -\frac{1}{2}$

  • $\tan(210^\circ) = \frac{\sqrt{3}}{3}$

  • $\tan\left(\frac{3\pi}{2}\right) = \text{undefined}$

  • $\cos\left(\frac{3\pi}{2}\right) = 0$

• Solving trigonometric equations:

  • $\cos(x) = -\frac{1}{2} \implies x = \frac{2\pi}{3}, \frac{4\pi}{3}$

  • $2\sin(x) + \sqrt{3} = 0 \implies \sin(x) = -\frac{\sqrt{3}}{2} \implies x = \frac{4\pi}{3}, \frac{5\pi}{3}$

  • $3\tan(x) = 0 \implies \tan(x) = 0 \implies x = 0, \pi, 2\pi$

• Modeling with trigonometric functions:

  • Daylight hours: $h(t) = -3.5\cos(\frac{\pi}{6}t) + 12.25$. $10.5$ hours of daylight in February and October.

  • Average daily temperature: $y = -22\cos(\frac{\pi}{6}t) + 57$. Average temperature in May is 68°F.