A cubic function is of the form *y = ax*^{3} + bx^{2} + cx + d

In the applet below, move the sliders on the right to change the values of a, b, c and d and note the effects it has on the graph. See also Linear Explorer, Quadratic Explorer and General Function Explorer

In the applet below, move the sliders on the right to change the values of a, b, c and d and note the effects it has on the graph. See also Linear Explorer, Quadratic Explorer and General Function Explorer

See also General Function Explorer where you can graph up to three functions of your choice simultaneously using sliders for independent variables as above.

- Click 'zero' on all four sliders

a, b, c, d are all set to zero, so this is the graph of the equation
y = 0x^{3}+0x^{2}+0x+0.
This simplifies to y = 0 and is of course zero for all values of x.
Its graph is therefore a horizontal straight line through the origin.

- Now move the d slider and let it settle on, say, 12.

- Click 'zero' on all four sliders
- Move the c slider to get different values of c. Let it settle on, say, 2.

This is the graph of the equation y = 0x^{3}+0x^{2}+2x+0 which simplifies to
y = 2x.
This is a simple linear equation and so is a straight line whose slope is 2.
That is, y increases by 2 every time x increases by one.
Since the slope is positive, the line slopes up and to the right.
Play with the c slider and observe the results, including negative values.

- Now move both sliders c and d to some value.

- Click 'zero' on all four sliders
- Move the b slider to get different values of b. Let it settle on, say, 2.

This is the graph of the equation y = 0x^{3}+2x^{2}+0x+0.
This simplifies to y = 2x^{2}.
Equations of this form and are in the shape of a parabola, and since b is positive,
it goes upwards on each side of the vertex.
Play with various values of b. As b gets larger the parabola gets steeper and 'narrower'.
When b is negative it slopes downwards each side of the vertex.

- Click 'zero' on all four sliders
- Move the a slider to get different values of a. Let it settle on, say, 2.

This is the graph of the equation 2x^{3}+0x^{2}+0x+0.
This simplifies to y = 2x^{3}.
Equations of this form and are in the cubic "s" shape, and since a is positive, it goes up and to the right.
Play with various values of a. As a gets larger the curve gets steeper and 'narrower'.
When a is negative it slopes downwards to the right.

- Click 'zero' on all four sliders
- Set d to 25, the line moves up
- Set c to -10, the line slopes
- Set b to 5, The parabola shape is added in.
- Set a to 4. The cubic "s" shape is added in.

This is the graph of the equation y = 4x^{3}+5x^{2}-25x+25.
Note how it combines the effects of the four coefficients.
Play with various values of a, b, c, d.
Changing d moves it up and down,
changing c changes the slope.
Changing b alters the curvature of the parabolic element,
and changing a changes the steepness of the cubic "s" curve.

- Press "reset", then "hide details"
- Adjust the sliders until you see a curve that appeals to you
- Estimate the values of a,b,c,d and write the equation for the curve.
- Click on "show details" and see how close you got

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