Overview
This lecture reviewed key concepts of transformations and polynomials, including transformation rules, polynomial graph characteristics, and division theorems, in preparation for exams.
Transformations Overview
- The general transformation formula is y = a·f(bx - h) + k.
- "a" controls vertical stretch/compression; negative "a" reflects over the x-axis.
- "b" controls horizontal stretch/compression; negative "b" reflects over the y-axis.
- "h" represents horizontal translation (left/right).
- "k" represents vertical translation (up/down).
- Elements inside brackets affect horizontal (x), outside affect vertical (y).
Truth vs. Math in Transformations
- "Truth" includes verbal descriptions and mapping notation (coordinate changes).
- Mapping notation examples: (x, y) → (bx + h, ay + k).
- "Math" includes replacement notation and formula manipulation.
- Replacement notation: show variable changes needed in equations (e.g., y → (1/a)y).
Practice & Examples
- Vertical stretch by 4: mapping (x, y) → (x, 4y), replacement y → (1/4)y, formula y = 4f(x).
- Horizontal translation of 3 right: mapping (x, y) → (x + 3, y), replacement x → x - 3, formula y = f(x - 3).
Order of Transformations
- When vertical and horizontal transformations are both present, order does not matter.
- If both affect the same direction (both vertical or both horizontal), order matters.
- Order: stretch/reflection before translation (RST/SRT), translation always last.
Inverse Functions
- Inverse swaps x and y: (x, y) → (y, x).
- x- and y-intercepts, and domain and range, are swapped in the inverse.
- To keep inverse as a function, restrict original domain (e.g., x ≥ 0 for y = x² + 3).
Polynomial Graphs Overview
- Polynomials split into algebra and graphing components.
- Graphs are analyzed by vocabulary, expanded form, and factored form.
Polynomial Vocabulary
- Degree: highest power of x.
- Constant term: value without x (also y-intercept).
- Leading coefficient: coefficient of term with highest degree.
- Odd degree & negative leading coefficient: right arm down, left arm up (opposite directions).
- End behavior is determined by degree and leading coefficient.
Expanded and Factored Forms
- Expanded form makes degree and y-intercept obvious; not all x-intercepts visible.
- Factored form reveals x-intercepts and multiplicities.
- Multiplicity: exponent on each factor indicates how graph touches/crosses x-axis (even —touch/tangent, odd—cross/point of inflection).
Types of Polynomials by Degree
- x¹: linear
- x²: quadratic
- x³: cubic
- x⁴: quartic
- x⁵: quintic
- x⁶: hexic
Algebra with Polynomials
- Synthetic division is preferred over long division for polynomials on exams.
- Always check for missing terms (write as zero coefficients).
- Remainder Theorem: plug k into f(x) to get remainder when divided by x - k.
- Factor Theorem: x - k is a factor if remainder is zero.
Key Terms & Definitions
- Vertical Stretch/Compression — Changes the y-values by factor a.
- Horizontal Stretch/Compression — Changes x-values by factor b.
- Reflection (x-axis) — Flips graph over x-axis (negative a).
- Reflection (y-axis) — Flips graph over y-axis (negative b).
- Translation — Moves graph horizontally (h) or vertically (k).
- Multiplicity — The exponent on a factor in factored form; influences graph shape at intercept.
- Synthetic Division — Shortcut for dividing polynomials when dividing by linear terms.
- Remainder Theorem — The remainder of f(x) divided by x - k is f(k).
- Factor Theorem — x - k is a factor if f(k) = 0.
Action Items / Next Steps
- Complete assigned transformation mapping and formula exercises.
- Review vocabulary and polynomial graph forms.
- Be prepared for new lesson on radicals next class.
- Check posted notes on D2L or pick up printed materials at school.