17 A – Cubic Polynomials

1: Graphing Basic Cubic Polynomials

Cubic Polynomials

• A cubic polynomial is a degree 3 polynomial in the form f(x) = ax3 + bx2 + cx + d.

Volume

• A 40 cm by 30 cm sheet of tinplate is to be used to make a cake tin. Squares are cut from its corners and the metal is then folded upwards along the dashed lines. Edges are fixed together to form the open rectangular tin.

• Consequently the capacity of the cake tin V, is given by V(x) = x(40 – 2x)(30 – 2x). How does the capacity change as x changes? What are the restrictions on x? What sized squares must be cut

out for the cake tin to have maximum capacity?

Forms of Cubics

• The function V(x) = x(40 – 2x)(30 – 2x) is the factored form of a cubic polynomial.

• The expanded form (or standard form) can be found by multiplying the factored form. V(x) = 4x3 – 140x2 + 1200x This form allows you to see why this

function is considered a cubic polynomial.

Expanding Cubics

• Write y = 2(x – 1)3 + 4 in general form (expand).

Write f(x) = 2(x – 3)3 + 7 in general form (expand).

Graphing Cubics

• Use technology to assist you to draw sketch graphs of:

f(x) = x3

f(x) = -x3

f(x) = 2x3

f(x) = ½x3

What effect does a have in f(x) = ax3?

Graphing Cubics

• Use technology to assist you to draw sketch graphs of:

f(x) = x3

f(x) = x3 + 2f(x) = x3 – 3

What effect does k have in f(x) = x3 + k?

Graphing Cubics

• Use technology to assist you to draw sketch graphs of:

f(x) = x3

f(x) = (x + 2)3

f(x) = (x – 3)3

What effect does h have in f(x) = (x – h)3?

Graphing Cubics

• Use technology to assist you to draw sketch graphs of:

f(x) = (x – 1)3 + 2f(x) = (x + 2)3 + 1

What is important about (h, k) in f(x) = (x – h)3 + k?