Overview
This lecture reviews key concepts from the Grade 11 Functions math course, focusing on recognizing, analyzing, and solving problems involving functions, quadratics, rational expressions, and systems.
Functions and Relations
- A function requires each x-value (domain) to correspond to only one y-value (range).
- The vertical line test determines if a relation is a function; failing it means the relation is not a function.
- State the domain (possible x-values) and range (possible y-values) by analyzing the graph's horizontal and vertical extents.
Function Notation
- f(x) replaces y to write equations, representing the y-value at a specific x.
- To find f(a), substitute a for x in the function and solve.
Quadratic Functions
- Quadratic graphs (parabolas) open up if a > 0, down if a < 0; vertex is the maximum or minimum point.
- Vertex form: y = a(x-h)² + k, where (h, k) is the vertex.
- Find the vertex by completing the square or using symmetry and intercepts (partial factoring).
Solving Quadratics
- Factoring method: for ax²+bx+c, find two numbers with product c and sum b (when a=1), otherwise use grouping.
- Quadratic formula: x = [-b ± √(b²-4ac)]/(2a); always works, even if not factorable.
- Discriminant (b²-4ac): positive = 2 solutions, zero = 1 solution, negative = no real solutions.
Linear-Quadratic Systems
- Solving means finding intersection points of a line and a parabola (can be 0, 1, or 2 points).
- Set y-values equal (substitution), rearrange to quadratic, then solve by factoring or quadratic formula.
Rational Expressions
- When dividing rational expressions, invert (reciprocal) the second fraction and multiply, then factor completely before reducing.
- For subtraction, get a common denominator, factor, combine, and simplify.
- Always state domain restrictions: exclude x-values that make any denominator zero at any step.
Key Terms & Definitions
- Function — A relation where each x-value matches only one y-value.
- Vertical Line Test — A method to determine if a graph represents a function.
- Domain — Set of all possible x-values in a relation/function.
- Range — Set of all possible y-values in a relation/function.
- Vertex — The highest or lowest point on a parabola.
- Quadratic Formula — Formula for solving any quadratic equation.
- Discriminant — The expression under the square root in the quadratic formula (b²–4ac).
- Rational Expression — A fraction with polynomials in numerator and/or denominator.
- Restriction — Values excluded from the domain to prevent division by zero.
Action Items / Next Steps
- Review homework problems on factoring, quadratic formula, and rational expressions.
- Practice identifying functions using the vertical line test.
- Prepare for exam questions on all discussed topics.