Advanced Algebra and Functions: Transformations of graphs, roots of polynomial equations.
Transformations of Graphs
- Understanding the transformation of graphs involves visualising how a parent graph is shifted, reflected or stretched to create a new graph.
- Five basic types of transformations include: translations, reflections, stretches, compressions, and combinations.
- A translation of a graph involves shifting the graph horizontally or vertically. This does not change the shape or size of the graph.
- A reflection flips a graph over a line, either horizontally (y-axis) or vertically (x-axis).
- Stretch and compression transformations involve changing the shape of the graph by either stretching it away from or compressing it towards a fixed line (usually the x or y axis).
- Combination transformations involve a sequence of two or more of the above transformations.
- The effect of any sequence of transformations can be described fully by a single transformation. This means that the order in which transformations are applied does not matter.
Roots of Polynomial Equations
- Polynomials are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients.
- A root of a polynomial is a solution of the polynomial equation set equal to zero.
- The terms 'zeros', 'roots' and 'solutions' of equations are synonymous.
- The Fundamental Theorem of Algebra states that every polynomial equation of degree n has exactly n roots in the complex number system.
- Real roots are real numbers that satisfy the equation.
- Complex roots are complex numbers that satisfy the equation.
- Polynomial equations of odd degree will always have at least one real root.
- Graphically, real roots are x-intercepts of the graph of the polynomial function.
- Techniques for finding roots include factoring, the Rational Root Theorem, synthetic division, and the use of the quadratic formula for polynomial equations of degree 2.
- Remember that roots of polynomial equations may have multiplicities, meaning they occur more than once. The power of the factor corresponding to a root in the factored form of the polynomial is called the multiplicity of the root.