Study Notes
Graphs and transformations involve sketching and analyzing different types of graphs and applying transformations to them. You will learn about cubic and reciprocal graphs, points of intersection, and various transformations like translations, stretches, reflections, and combinations.
- Cubic graphs — A cubic function has the form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are real numbers and a is non-zero. Example: The graph of y = x^3 - 3x^2 + 2x crosses the x-axis at its roots.
- Reciprocal graphs — These graphs have asymptotes at x = 0 and y = 0, where the graph approaches but never touches these lines. Example: The graph of y = 1/x has vertical and horizontal asymptotes at x = 0 and y = 0.
- Points of intersection — The x-coordinates at the points where two graphs intersect are the solutions to the equation f(x) = g(x). Example: The intersection points of y = x(x - 3) and y = x^2(1 - x) are solutions to the equation.
- Translating graphs — Moving a graph along the x-axis or y-axis without changing its shape. Example: y = f(x) + a translates the graph vertically by a units.
- Stretching graphs — Changing the size of a graph by multiplying by a constant. Example: y = af(x) stretches the graph vertically by a factor of a.
- Reflections — Flipping a graph over a line, such as the x-axis or y-axis. Example: y = -f(x) reflects the graph in the x-axis.
- Combined transformations — Applying more than one transformation in sequence. Example: Translate a graph 2 units right, then reflect in the x-axis.
Exam Tips
Key Definitions to Remember
- Cubic function: f(x) = ax^3 + bx^2 + cx + d
- Reciprocal function: y = 1/x
- Asymptote: A line that a graph approaches but never touches
Common Confusions
- Confusing translations with stretches
- Misidentifying the direction of translations
Typical Exam Questions
- What is the form of a cubic function? Answer: f(x) = ax^3 + bx^2 + cx + d
- How do you find the points of intersection of two graphs? Answer: Solve the equation f(x) = g(x) for x.
- What happens when you reflect a graph in the x-axis? Answer: The graph of y = f(x) becomes y = -f(x).
What Examiners Usually Test
- Ability to sketch and identify key features of cubic and reciprocal graphs
- Understanding of how to apply and combine transformations
- Solving equations using points of intersection