How to Write a Cubic Function From Given Zeros or X-Intercepts

Key Concepts

• Zeros/X-Intercepts/Roots/Solutions: These terms are used interchangeably and refer to the points where the graph crosses the x-axis.
• Factors: Derived from the zeros, represented as (x - zero). For example, if a zero is at -3, the factor is (x - (-3)) or (x + 3).
• Coefficient (a): Adjusts the graph by stretching or compressing it vertically.

Steps to Write the Cubic Equation

1. Identify the Factors: Convert zeros into factors.

   • If zeros are at -3, 1, and 4:
     • Zero at -3: Factor is (x + 3)
     • Zero at 1: Factor is (x - 1)
     • Zero at 4: Factor is (x - 4)

2. Determine the Coefficient (a):

   • Use a point that is not on the x-axis (e.g., (0, 2)) to find a.
   • Substitute x and y values into the equation to solve for a:
     • Example: Given point (0, 2), the equation becomes: 2 = a(0 + 3)(0 - 1)(0 - 4)
     • Calculate: 2 = a(3)(-1)(-4)
     • Simplify: 2 = 12a
     • Solve for a: a = 2/12 or a = 1/6

3. Write the Cubic Equation:

   • Substitute a and factors back into the equation: y = (1/6)(x + 3)(x - 1)(x - 4)

4. Optional: Expand and simplify the equation.

   • Expanding the factored form and multiplying by a will give the simplified polynomial form.

Summary

• Identify x-intercepts from the graph
• Convert x-intercepts to factors
• Use another point to determine the coefficient (a)
• Write the cubic equation in factored form or simplified polynomial form

This method ensures that the cubic function you derive correctly represents the graph given the x-intercepts and an additional point. I hope this helps!