Algebra II Final Exam Review Notes

Section 5.1: Polynomial Functions

• Identifying Polynomial Functions:

  • A polynomial function must have non-negative integer exponents.

  • Example: $5x^3 + 4x^2 - 3x$ is a polynomial.

• Standard Form:

  • Polynomials are written in descending order of exponents.

  • Example: $5x^3 + 4x^2 - 3x$ is in standard form.

• Key Characteristics:

  • Degree: The highest exponent of the variable.

  • Type: Determined by the degree (e.g., cubic, quartic).

  • Leading Coefficient: The coefficient of the term with the highest degree.

  • End Behavior: Describes what happens to the function as $x$ approaches positive or negative infinity.

    • Determined by the degree and leading coefficient.

      • Even degree with positive leading coefficient: both ends go up.

      • Even degree with negative leading coefficient: both ends go down.

      • Odd degree with positive leading coefficient: left goes down, right goes up.

      • Odd degree with negative leading coefficient: left goes up, right goes down.

• Examples:

  • $h(x) = 5x^3 - 7x^2 + x - 1$

    • Degree: 3

    • Type: Cubic

  • $g(x) = 6x^4 - 3x^3 + 12x^2 + 8x + 2$

    • Degree: 4

    • Type: Quartic

• Evaluating Functions:

  • Substitute the given value of $x$ into the function.

  • Example: $f(x) = -2x^4 + x^3 + 5x^2 - 3x - 7$; find $f(-1)$
    $f(-1) = -2(-1)^4 + (-1)^3 + 5(-1)^2 - 3(-1) - 7 = -2 - 1 + 5 + 3 - 7 = -2$

Analyzing Polynomial Graphs

• Intervals of Increase and Decrease:

  • Increasing: As $x$ increases, $f(x)$ increases.

  • Decreasing: As $x$ increases, $f(x)$ decreases.

• Sign of $f(x)$:

  • $f(x) > 0$: The function is above the x-axis.

  • $f(x) < 0$: The function is below the x-axis.

• Example:

  • Increasing on $(-\infty, -1)$ and $(1, \infty)$; Decreasing on $(-1, 1)$

  • $f(x) > 0$ on $(-2, 0)$ and $(2, \infty)$; $f(x) < 0$ on $(-\infty, -2)$ and $(0, 2)$

Sketching Graphs from Characteristics

• Example:

  • Function is increasing on $(-\infty, 1)$ and decreasing on $(1, \infty)$. $f(x) > 0$ on $(-1, 3)$; $f(x) < 0$ on $(-\infty, -1)$ and $(3, \infty)$.

  • This indicates a graph with a maximum at $x = 1$, crossing the x-axis at $-1$ and $3$.

• Degree and Leading Coefficient:

  • The shape of the graph helps determine the degree (even or odd) and the sign of the leading coefficient (positive or negative).

  • Even degree: Both ends go in the same direction.

  • Odd degree: Ends go in opposite directions.

Real-World Application

• Baseball Trajectory:

  • The height $h(t)$ of a baseball after $t$ seconds is modeled by $h(t) = -4.9t^2 + 28.62t + 2.4$.

  • Maximum height: Occurs at the vertex of the parabola.

  • Time to hit the ground: Find $t$ when $h(t) = 0$.

Section 5.2: Operations with Polynomials

• Addition and Subtraction:

  • Combine like terms.

  • Example:
    $(8x^7 - 6x^5 + 4x^3 - 6x) + (15x + 4x^5 - 3x^3 + 2) = 8x^7 - 2x^5 + x^3 + 9x + 2$

• Multiplication:

  • Use the distributive property.

  • Example:
    $(x^2 - 7x - 2)(x^2 - 3x - 6) = x^4 - 10x^3 + 13x^2 + 48x + 12$

• Special Products:

  • $(a + b)^2 = a^2 + 2ab + b^2$

  • Example: $(6t + 7)^2 = 36t^2 + 84t + 49$

Volume Application

• Rectangular Pool:

  • Length: $(3x - 1)$ feet

  • Width: $(x + 6)$ feet

  • Depth: $(x + 6)$ feet

  • Volume: $V = (3x - 1)(x + 6)(x + 6) = 3x^3 + 35x^2 + 96x - 36$ cubic feet

Section 5.3: Polynomial Division

• Long Division:

  • Used to divide polynomials when the divisor has a degree greater than 1.

  • Example:
    $(4x^4 + 2x^3 - 9x^2 - 36) \div (x^2 + x - 4) = 4x^2 - 2x + 9 - \frac{17x}{x^2 + x - 4}$

• Synthetic Division:

  • Used when dividing by a linear factor $(x - k)$.

  • Example:
    $(2x^3 - 5x^2 + 3) \div (x + 1) = 2x^2 - 7x + 7 - \frac{4}{x + 1}$

• Remainder Theorem:

  • If a polynomial $f(x)$ is divided by $(x - k)$, the remainder is $f(k)$.

  • Example: Evaluate $f(x) = -x^4 + 6x^2 + 6$ at $x = -2$
    $f(-2) = -10$

Section 5.4: Factoring Polynomials

• Factoring Techniques:

  • Greatest Common Factor (GCF)

  • Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$

  • Factoring by Grouping

  • Factoring Trinomials

• Examples:

  • $5t^5 - 320t^3 = 5t^3(t^2 - 64) = 5t^3(t - 8)(t + 8)$

  • $2p^6 - 26p^5 + 84p^4 = 2p^4(p^2 - 13p + 42) = 2p^4(p - 7)(p - 6)$

  • $x^3 - 7x^2 + 5x - 35 = (x - 7)(x^2 + 5)$

  • $81g^4 - 625 = (9g^2 - 25)(9g^2 + 25) = (3g - 5)(3g + 5)(9g^2 + 25)$

• Factor Theorem:

  • A binomial $(x - k)$ is a factor of $f(x)$ if and only if $f(k) = 0$.

Sections 5.5, 5.6, 5.8: Polynomial Equations and Zeros

• Solving Polynomial Equations:

  • Factor and set each factor equal to zero.

  • Examples:

    • $4x^4 + 12x^3 + 9x^2 = 0 \implies x^2(2x + 3)^2 = 0 \implies x = 0, -\frac{3}{2}$

    • $6h^5 = 12h^3 \implies 6h^3(h^2 - 2) = 0 \implies h = 0, \pm \sqrt{2}$

    • $16q^4 - 8q^2 + 1 = 0 \implies (4q^2 - 1)^2 = 0 \implies q = \pm \frac{1}{2}$

    • $2x^3 - 3x^2 + 18x - 27 = 0 \implies (2x - 3)(x^2 + 9) = 0 \implies x = \frac{3}{2}, \pm 3i$

• Finding Zeros of Functions:

  • Set $f(x) = 0$ and solve for $x$.

  • The zeros are the x-intercepts of the graph.

• Fundamental Theorem of Algebra:

  • A polynomial of degree $n$ has exactly $n$ complex roots (counting multiplicity).

• Complex Conjugate Theorem:

  • If $a + bi$ is a zero of a polynomial with real coefficients, then $a - bi$ is also a zero.

Sketching Graphs and Analyzing Intervals

• Example:

  • $f(x) = -5x^4 + 20x^3 + 60x^2 = -5x^2(x - 6)(x + 2)$

    • Zeros: $0, 6, -2$

  • $g(x) = -x^3 - x^2 + 30x = -x(x + 6)(x - 5)$

    • Zeros: $0, -6, 5$

Using Calculator for Approximations

• Finding zeros and local maxima/minima.

Identifying Number of Solutions/Zeros

• The maximum number of zeros is equal to the degree of the polynomial.

Writing Polynomial Functions from Zeros

• Example:

  • Zeros: $2, 3 + i$
    $f(x) = (x - 2)(x - (3 + i))(x - (3 - i)) = (x - 2)(x^2 - 6x + 10) = x^3 - 8x^2 + 22x - 20$

  • Zeros: $3, -\sqrt{7}$
    $f(x) = (x - 3)(x + \sqrt{7})(x - \sqrt{7}) = (x - 3)(x^2 - 7) = x^3 - 3x^2 - 7x + 21$

Section 5.7: Transformations of Polynomial Functions

• Horizontal Stretch/Compression:

  • $f(ax)$ stretches (if $0 < a < 1$) or compresses (if $a > 1$) horizontally.

• Reflection:

  • Over x-axis: $-f(x)$

  • Over y-axis: $f(-x)$

• Translation:

  • Vertical: $f(x) + k$

  • Horizontal: $f(x - h)$

Examples:

• $f(x) = x^3 - 3x^2 + 2$
  $g(x) = -\frac{1}{27}x^3 + \frac{1}{3x^2} + 3$

Describing Transformations:

• Reflection in the y-axis
  Vertical shrink by a factor of $1/3$
  Translation 1 units up

$\frac{1}{3}f(-x) + 1$

Section 5.9: Writing Polynomial Functions for Sets of Points

• Using Given Points:

  • Write a cubic function that passes through given points.

  • Example: $(2, 0), (0, 5), (-1, 0), (5, 0)$
    $y = a(x + 1)(x - 2)(x - 5)$
    Using $(0, 5)$, $5 = a(1)(-2)(-5) \implies a = \frac{1}{2}$
    $y = \frac{1}{2}(x + 1)(x - 2)(x - 5)$

Finite Differences

• Used to determine the degree of the polynomial function.

Section 6.1: nth Roots and Rational Exponents

• nth Root:

  • $x^n = a \implies x = \sqrt[n]{a}$

• Rational Exponents:

  • $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$

Simplify

• $(-128)^{\frac{1}{7}} = ((-2)^7)^{\frac{1}{7}} = -2$

Section 6.2: Properties of Rational Exponents and Radicals

• Simplifying Radicals:

  • $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$

Section 6.3: Graphing Radical Functions

• Describe the Transformation:
  $g(x) = -(2x-2)^2 + 3$

Section 6.4: Solving Radical Equations and Inequalities

• Solve the equation: $(\sqrt{x - 1} + 2)^2=(3x + 1) ^(\frac{1}{2})^2$

  • Isolate the Radical then Square both sides

Section 6.5: Operations with Functions

• Addition: $(f + g)(x) = f(x) + g(x)$

• Subtraction: $(f - g)(x) = f(x) - g(x)$

• Multiplication: $(f \cdot g)(x) = f(x) \cdot g(x)$

• Division: (f/g)(x) = \frac{f(x)}{g(x)

Section 6.6: Inverse of Nonlinear Functions

• Inverse Functions:

  • switch the x and y.

Section 7.1: Exponential Growth and Decay

• Formula:$A = p(1 + \frac{r}{n})^{nt}$
  Where:
  A = final amount
  p = principal amount
  r = rate of interest
  n = number of times interest is compounded per year
  t = time in years

Section 7.2: The Natural Base e

• The Natural Base e: An irrational number approximately equal to 2.71828…

Section 7.3: Logarithmic Functions

• Definition:

  • If $x = b^y$, then $log_b(x) = y$

Section 7.4: Transformations of Logarithmic Functions

• f(x) = −2 log₂(−x + 1) + 3

Section 7.5: Properties of Logarithms

• Properties:

Section 7.6: Exponential and Logarithmic Equations

Section 7.7: Classify Data Sets

Find the function that fits the data

Section 8.1: Classify Variations

*Write equation
Then find y when x = 4.

Section 8.2: Graphing Rational Function

Section 8.3: Rational Expression

Section 8.4: Add and Subtract Rational Expression

Section 8.5: Solve Rational Equation

Solve for Inverse when given a function