Advanced Algebra Unit 6 Review

Linear Functions

• Linear functions have a constant rate of change, yielding straight-line graphs.
• Key equations:
  • Slope-intercept form: (y = mx + b) (m = slope, b = y-intercept)
  • Point-slope form: ((y - y_1) = m(x - x_1))
  • Standard form: (Ax + By = C)
  • Vertical line: (x = a)
  • Horizontal line: (y = b)
• Slope calculation: (\text{slope} = \frac{\Delta y}{\Delta x})

Absolute Value Functions

• Absolute value functions yield V-shaped graphs and measure distance from zero.
• Vertex form: (y = a|x - h| + k) (h, k is the vertex)

Quadratic Functions

• Quadratic functions create U-shaped graphs, known as parabolas.
• Standard form: (y = ax^2 + bx + c)
• Factored form: (y = a(x - r_1)(x - r_2)) where (r_1) and (r_2) are roots.
• Vertex form: (y = a(x - h)^2 + k) (h, k is the vertex)

Exponential Functions

• Exponential functions display a J-shaped curve on graphs.
• General equation: (y = a(b)^{x - h} + k) (b = constant ratio, h represents horizontal shift, k represents vertical shift)

Piecewise Functions

• Evaluate values of piecewise functions by examining defined intervals.
• Key elements include determining domain (all possible x-values) and range (all possible y-values).

Polynomials

• Area of a rectangle: ( \text{Area} = \text{Length} \times \text{Width} )
• Perimeter is the summation of all sides.
• Operations on polynomials must result in expressions in standard form.

Polynomial Operations

• To simplify expressions, ensure polynomials are in standard form after performing operations.
• Factoring out the greatest common factor is critical for simplifying polynomial expressions.

Classification of Polynomials

• Monomial: a single term (e.g., (5x^5))
• Binomial: two terms (e.g., (3x + 2))
• Trinomial: three terms (e.g., (x^2 + 3x + 4))
• Highest degree indicates polynomial classification (e.g., cubic, quadratic, linear).