4.1 Cubic Graphs
Loading...
...
Learning Objectives
- Understand the general shape and form of cubic graphs
- Identify key features such as turning points and points of inflection
- Sketch cubic graphs from equations
- Interpret the roots and behaviour of cubic functions
- Apply knowledge of cubic graphs to solve equations and inequalities
Key Concepts
What Are Cubic Graphs?
A **cubic graph** is the graph of a function of the form *y = ax³ + bx² + cx + d*.
Cubic functions can have up to **three real roots**, and their graphs can have up to **two turning points**.
The general shape resembles an 'S' curve, though this varies based on the sign and values of the coefficients.
Sketching Cubic Graphs
To sketch a cubic graph, identify:
- The **roots** (solutions to y = 0)
- The **y-intercept** (when x = 0)
- The **shape** and **direction** of the curve (based on the leading coefficient)
Plot key points and smoothly join them to reflect the graph's curvature.
Example 1
Sketch the graph of y = (x - 1)(x + 2)(x - 3)
Solution:
Roots at x = 1, -2, and 3. Positive coefficient, so graph rises to the right and falls to the left. Y-intercept: (0, (0 - 1)(0 + 2)(0 - 3)) = -6.
Turning Points and Shape
Cubic graphs may have **two turning points** (a local maximum and minimum) or **one point of inflection**.
If the graph has a point of inflection, it doesn’t change direction but still changes curvature.
The number of turning points depends on the nature of the equation and its derivative.
Solving Cubic Equations Graphically
Cubic equations can be solved by finding where the graph crosses the x-axis (i.e., the roots).
You can also solve equations like *f(x) = g(x)* by plotting both functions and finding points of intersection.
Inequalities such as *f(x) > 0* relate to the regions where the curve is above the x-axis.
Applications and Interpretation
Cubic graphs can model real-world situations involving acceleration, population growth, and economics.
Understanding the shape and behaviour helps in analysing trends, especially where growth turns into decline (or vice versa).
Test Your Knowledge
Ready to test your understanding of 4.1 Cubic Graphs?