EDEXCEL GCSE FOUNDATION MATHS - What are Cubic Graphs

Edexcel GCSE Foundation Maths - What are Cubic Graphs?

Introduction:

Cubic graphs are graphs of cubic functions, which are functions where the highest power of the variable is 3. They have a distinctive shape with one or two turning points.

General Form of a Cubic Function:

The general form of a cubic function is:

y = ax^3 + bx^2 + cx + d

Where:

• a, b, c, and d are constants
• x is the independent variable
• y is the dependent variable

Key Features of Cubic Graphs:

• Shape: Cubic graphs typically have an "S" shape, with one or two turning points.
• Turning Points: These are the points where the graph changes direction from increasing to decreasing or vice versa.
• Symmetry: Cubic graphs can be symmetrical about the y-axis if the coefficient of the x term is 0.
• Asymptotes: Cubic graphs do not have asymptotes.

Example:

Let's consider the cubic function:

y = x^3 - 3x^2 + 2x

To graph this function, we can follow these steps:

1. Find the y-intercept: Set x = 0 and solve for y.

   y = (0)^3 - 3(0)^2 + 2(0) = 0

   Therefore, the y-intercept is (0, 0). 2. Find the x-intercepts: Set y = 0 and solve for x.

   0 = x^3 - 3x^2 + 2x 0 = x(x^2 - 3x + 2) 0 = x(x - 1)(x - 2)

   Therefore, the x-intercepts are (0, 0), (1, 0), and (2, 0). 3. Find the turning points: We can find the turning points by finding where the derivative of the function is equal to 0.

   y' = 3x^2 - 6x + 2 0 = 3x^2 - 6x + 2

   Solving for x using the quadratic formula, we find that the turning points are approximately at x = 0.42 and x = 1.58. To find the corresponding y values, we plug these x values back into the original function. 4. Plot the points and draw the graph:

   Plot the y-intercept, x-intercepts, and turning points. Connect these points with a smooth curve to obtain the cubic graph.

Applications of Cubic Graphs:

Cubic graphs are used in various applications, including:

• Modeling real-world phenomena: They can be used to model the growth of populations, the motion of objects, and other natural phenomena.
• Engineering: They are used in designing structures, machines, and systems.
• Economics: They are used to model the relationship between price and demand, and other economic factors.

Tips for Graphing Cubic Functions:

• Use a graphing calculator or online graphing tool to help you visualize the graphs.
• Identify the y-intercept, x-intercepts, and turning points.
• Consider the symmetry of the graph.
• Use the information about the leading coefficient (a) to determine the end behavior of the graph. If a is positive, the graph will rise to the right and fall to the left. If a is negative, the graph will rise to the left and fall to the right.

In Summary:

Cubic graphs are an important type of graph in mathematics with various applications. By understanding their key features and properties, you can effectively graph and analyze them.