01Fcubicgraphs

toc = Cubic Graphs =

The general form of the cubic function is y = ax 3 + bx 2 + cx + d
 * the sign of a (the coefficient of x 3 ) controls the direction of the curve
 * a > 0, the graph trends __up__ from left to right (ie it starts in 3rd Quadrant, ends in 1st Quadrant)
 * a < 0, the graph trends __down__ from left to right (ie it starts in 2nd Quadrant, ends in 4th Quadrant)
 * The constant term (d) gives the y-intercept
 * Some cubics have a single stationary point of inflection, the rest have two turning points (a local minimum and a local maximum)
 * A cubic may have one or two or three x-intercepts
 * The turning points are NOT midway between the x-intercepts
 * The x-intercepts are given by the solutions of ax 3 + bx 2 + cx + d = 0
 * This can be solved by factorising, then using the Null Factor Law
 * Factors can be found using the Factor Theorem {If P(a) = 0, then x – a is a factor}
 * A CAS calculator can factorise {select **factor** from the ACTION menu, TRANSFORMATION submenu}
 * A CAS calculator can solve the equation P(x) = 0 to find the x-intercepts {select **solve** from the ACTION menu, EQUATION submenu}
 * The x-value of the stationary pointscan be found by
 * setting the derivative to zero and solving for x
 * graphing on a CAS calcuator {**Max** and **Min** are in the ANALYSIS menu, G-SOLVE submenu}

Basic Form (or Power Form)


__Some__ cubics are transformations of the basic cubic, y = x 3.

The basic cubic has a stationary point of inflection at the origin (no turning points).

The Power Form of the cubic can be written as: ... ... y = a(x – h) 3 + k

The usual transformations apply:
 * The sign of a controls the direction of the curve
 * a > 0, the graph trends __up__ from left to right
 * a < 0, the graph trends __down__ from left to right
 * The magnitude of a controls the dilation of the curve
 * a > 1, (or a < –1), the graph is thinner
 * a < 1, (or a > –1), the graph is wider
 * The stationary point of inflection is at (x = h, y = k)
 * To find the y-intercept, substitute x = 0
 * To find the x-intercept (there will be exactly one), substitute y = 0

**Example 1**
Sketch y = 2(x – 2) 3 – 1


 * Solution:**


 * Stationary point at (2, –1)
 * Dilation by a factor of 2 (thinner)
 * a > 0 so trend is up from left to right

Y-intercept

... ... y = 2(–2) 3 – 1

... ... y = –17

X-intercept

math \\ . \qquad 2 \big( x - 2 \big)^3 - 1 = 0 \\. \\ . \qquad 2 \big( x - 2 \big)^3 = 1 \\. \\ . \qquad (x - 2)^3 = \dfrac{1}{2} math . math \\ . \qquad x-2=\sqrt[3]{\dfrac{1}{2}} \\. \\ . \qquad x=2+\sqrt[3]{\dfrac{1}{2}} \\. \\ . \qquad x \approx 2.8 math

Factor Form
__Some__ cubic equations can be factorised into the form:

... y = a(x – b)(x – c)(x – d)
 * the sign of a controls the direction of the curve
 * +a trends up from left to right
 * –a trends down from left to right
 * the magnitude of a gives the dilation factor
 * b, c and d are the x-intercepts
 * a factor repeated twice indicates a turning point on the axis at that x-value
 * to find the y-intercept, substitute x = 0
 * the turning points are NOT midway between the x-intercepts

** Example 2 **
Sketch y = 0.5(x + 3)(x – 1)(x – 4)


 * Solution:**

X-intercepts

... ... From the equation, x = –3, 1, 4

Y-intercept

... ... substitute x = 0

... ... y = 0.5(3)(–1)(–4)

... ... y = 6

Dilation Factor

... ... a = 0.5 (wider)

... ... a > 0 so it is a positive cubic, trends up from left to right

Stationary Points

... ... Sketch graph on your calculator, then use it to get values for the turning points ... ... ... { ** Max ** and ** Min ** are in the ANALYSIS menu, G-SOLVE submenu}

... ... Local maximum at (–1.36, 10.37)

... ... Local minimum at (2.69, –6.30)

** Example 3 **
Sketch y = –(x + 2)(x – 1) 2


 * Solution:**

X-intercepts

... ... From the equation x = –2, 1

Y-intercept

... ... Substitute x = 0

... ... y = –(2)(–1) 2

... ... y = –2

Dilation Factor

... ... a = –1 (dilation factor = 1, standard width)

... ... a < 0 so negative cubic, trends down

Stationary Points

... ... Repeated factor means one turning point is on the axis at x = 1

... ... Use calculator for the other turning point

... ... Local maximum at (1, 0)

... ... Local minimum at (–1, –4)


 * Note: **
 * Stationary points can be located algebraically by
 * Finding the derivative of the function
 * Setting the derivative to equal zero

If the domain of a cubic is R, then the range is also R

If the domain is restricted, then care must be taken when stating the range
 * because the endpoints for the domain may not give the maximum or minimum values for the range.
 * The y-values of the turning points must also be considered.

** Example 4 **
math \\ \text{Sketch } y = \frac{1}{3}x^3 + x^2 -3x-3 \text{ in the domain } x \in \big( -5\frac{1}{4}, \,2 \big] \\ \text{and state the range} math


 * Solution:**


 * ** With the aid of a calculator, draw the entire graph (in pencil), **
 * ** then mark in the endpoints and draw the desired section more clearly. **
 * ** Erase the unwanted portions of the graph. **

X-intercepts

... ... x = –4.54, –0.83, 2.38

Y-intercepts

... ... y = –3

Turning Points

... ... Local maximum (–3, 6)

... ... Local minimum (1, –4.67)

Endpoints

... ... (–5.25, –8) ** {open circle} **

... ... (2, –2.33) ** {closed circle} **


 * Desired section is between the two endpoints (in dark blue)

Domain

... ... x Î (–5.25, 2]

Range


 * ** by observation of the graph **

... ... y Î (–8, 6] Go to top of page flat

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