03Jcubicpowerform

= Power Functions : Cubics =


 * A power function is any function in the form .. y= x n .. {where n is any rational number}


 * You already know about the quadratic power function y = x 2, (the parabola)
 * A basic cubic like y = x 3 is also a power function

The graph of y = x 3.

= =
 * The basic cubic graph could be described as almost like a parabola (steeper)
 * Except that the negative portion of the graph has been flipped down below the x-axis


 * Instead of a turning point, the basic cubic has a ** stationary point of inflection ** at (0, 0),
 * A ** stationary point of inflection ** is a point on the graph where the gradient (or slope) is briefly zero
 * Remember that a zero gradient means the line is horizontal.
 * Each side of the point of inflection, the gradient has the same sign
 * Either both positive or both negative.


 * The basic power from of the cubic, y = x 3, can have transformations (like the parabola)
 * It can be
 * dilated
 * inverted
 * translated
 * The rules for these transformations are exactly the same as for the parabola.


 * y = ax 3 **


 * If a > 1 the cubic will be thinner (steeper)
 * If a < 1 the cubic will be wider (less steep)
 * If a is negative, the cubic will be inverted (trending down from left to right)


 * y = x 3 + k **


 * The cubic will shift __up__ by a distance of k if k is positive
 * The cubic will shift __down__ by a distance of k if k is negative


 * The stationary point of inflection will have coordinates (0, k)


 * y = (x – h) 3 **


 * The cubic will shift sideways in the opposite direction to the sign


 * If y = (x + 4) 3, the cubic will shift left by 4 places
 * If y = (x – 5) 3, the cubic will shift right by 5 places


 * The stationary point of inflection will have coordinates (h, 0)


 * y = a(x – h) ** 3 ** + k **


 * If a is positive, the cubic will trend up from left to right
 * If a is negative, the cubic will trend down from left to right


 * The cubic will shift sideways by a distance of h (opposite to sign)
 * The cubic will shift up by a distance of k (same direction as sign)


 * The coordinates of the stationary point of inflection will be (h, k)


 * Example 1 **

... ... Sketch y = 2(x – 3) 3 + 2


 * Solution:**


 * Cubic is positive (trends up)
 * Stationary point of inflection at (3, 2)
 * y-intercept (let x = 0) at (0, –52)
 * x-intercept (let y = 0) at (2 ,0)


 * Example 2 **

... ... Sketch y = (1 – 2x) 3 + 1


 * Solution:**


 * Manipulate to get in standard power form

math . \qquad y = \Big( \big(1 - 2x \big) \Big)^3 + 1 \qquad \{ \textit{take out -2 as common factor} \}\\. \\ . \qquad y = \left( -2 \Big( x - \dfrac{1}{2} \Big) \right)^3 + 1 \\. \\ . \qquad y = -8 \left( x - \dfrac{1}{2} \right)^3 + 1 math


 * Cubic is negative (trends down)
 * Stationary point of inflection at (½, 1)
 * y-intercept (let x = 0) at (0, 2)
 * x-intercept (let y = 0) at (1 ,0)

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