Binomial Expansion and Cubic Functions

Cubic Functions

Learning Intentions

  • Recognize and determine features of cubic graphs:
    • y=x3y = x^3
    • y=a(x−h)3+ky = a(x - h)^3 + k
    • y=a(x−x<em>1)(x−x</em>2)(x−x3)y = a(x - x<em>1)(x - x</em>2)(x - x_3)
    • Including shape, intercepts, and behavior as x→∞x \rightarrow \infty and x→−∞x \rightarrow -\infty.
  • Sketch cubic function graphs (with and without technology).

Key Concepts

  • General Cubic Polynomial:
    • y=a(x−h)3+ky = a(x - h)^3 + k where (h,k)(h, k) is a stationary point of inflection.

Transformations

  • Dilation:
    • aa affects dilation; if a < 0, reflection in the x-axis.
  • Translations:
    • hh: Sideways shift; xx-coordinate of the turning point (h,k)(h, k). Positive hh moves right, negative moves left.
    • kk: Vertical shift; yy-coordinate of the turning point (h,k)(h, k). Positive kk translates up, negative translates down.

Graphing Cubics in Turning Point Form: y=a(x−h)3+ky = a(x - h)^3 + k

  • Shape.
  • Key feature: stationary (flat) point of inflection.
  • Intercepts: let x=0x = 0 to find the yy-intercept; let y=0y = 0 to find the xx-intercept(s).

Cubic Graphs with Three x-intercepts

  • Equation form: y=a(x−x<em>1)(x−x</em>2)(x−x3)y = a(x - x<em>1)(x - x</em>2)(x - x_3)
  • Turning points do not lie halfway between x-intercepts; use technology to find them.

Cubic Graphs with Two or One x-intercept

  • Form y=(x−a)2(x−b)y = (x - a)^2(x - b) has a turning point on the x-axis at (a,0)(a, 0) and another x-intercept at (b,0)(b, 0). The graph touches the x-axis at x=ax = a and cuts at x=bx = b.
  • Some cubic graphs have one x-intercept without a stationary point of inflection and cannot be expressed in the form y=a(x−b)3+cy = a(x - b)^3 + c.

Finding the Equation of Cubic Functions

  1. Identify the equation type.
  2. Substitute the key feature into the equation.
  3. Substitute another point (x,y)(x, y) to solve for aa.
  4. Write the final equation by adding xx and yy back.

Solving Cubic Equations

  • With or without technology.
  • Use division of polynomials to determine linear factors.

Modeling with Cubic Functions

  • Create an equation for a situation and solve problems involving cubic functions (with/without technology).