17 a – cubic polynomials 1: graphing basic cubic polynomials
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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?
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