03Hcubicgraphintform

= Cubic Graphs Intercept Form =


 * The graph of a cubic function has two basic shapes
 * the power function form which has a stationary point (See Ex 3J)
 * the general cubic form which has two changes in direction (a maximum and a minimum)
 * In this section, we look at the general cubic form.




 * A cubic graph can either be positive (trends up from left to right)
 * both examples above are positive cubics
 * or negative (trends down from left to right)
 * The direction of the cubic is controlled by the sign of the coefficient in front of the x 3 term.


 * The turning points are NOT half-way between the x-intercepts


 * x-Intercepts **


 * The x-intercepts of a cubic function can be found by setting y = 0
 * The x values can then be found from the factorised function using the Null Factor Law


 * Example 1 **

... ... Sketch y = 0.5(x + 3)(x – 1)(x – 4)


 * Solution:**

... ... ** X-Intercepts **

math . \qquad \text{Solve} \\.\\ . \qquad 0.5 \big( x + 3 \big) \big( x - 1 \big) \big( x - 4 \big) = 0 \\.\\ . \qquad \;\; \big( x + 3 \big) \big( x - 1 \big) \big( x - 4 \big) = 0 \\.\\ . \qquad \;\; x + 3 = 0 \;\; or \;\; x - 1 = 0 \;\; or \;\; x-4 = 0 \\.\\ . \qquad \quad x = -3 \;\;\; or \;\;\; x = 1 \;\;\; or \;\;\; x = 4 math

... ... ** Y-Intercept **

math . \qquad \text{Let } x = 0 \\. \\ . \qquad y = 0.5 \big( 3 \big) \big( -1 \big) \big( -4 \big) \\.\\ . \qquad y = 6 math

... ... ** Turning Points **

... ... Coordinates can be found by sketching on the calculator

... ... ( –1.36, 10.37 )

... ... ( 2.69, –6.30 )


 * Repeated Factors **


 * Where there is a factor squared, there will be a turning point on the x-axis.


 * Example 2 **

... ... Sketch y = –(x + 2)(x – 1) 2.

... ... ** X-Intercepts **
 * Solution:**

math . \qquad \text{Solve} \\. \\ . \qquad - \big( x + 2 \big) \big( x - 1 \big)^2 = 0 \\. \\ . \qquad \; x + 2 = 0 \;\; or \;\; x - 1 = 0 \\.\\ . \qquad \;\; x = -2 \;\;\; or \;\;\; x = 1 \\.\\ math

... ... ** Shape **


 * The negative sign out the front makes the graph a negative graph
 * It will trend downwards from left to right
 * The squared term (x - 1) 2 makes a turning point on the x-axis at x = 1

... ... ** Y-Intercept **

math . \qquad \text{ Let } x = 0 \\.\\ . \qquad y = - \big( 2 \big) \big( -1 \big)^2 \\.\\ . \qquad y = -2 math

... ... ** Turning Points **

... ... Turning points can be found by sketching on the calculator

... ... (–1, –4)

... ... (1, 0)

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