Cubic Function Explorer
A cubic function is of the form y = ax3 + bx2 + 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

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

## Things to try

Assuming you already have a knowledge of cubic equations, the following activities can help you get a more intuitive feel for the action of the four coefficients a, b, c , d.

## The simplest case. Y = constant. (y = d)

1. Click 'zero' on all four sliders

a, b, c, d are all set to zero, so this is the graph of the equation y = 0x3+0x2+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.

1. Now move the d slider and let it settle on, say, 12.
This is the graph of the equation y = 0x3+0x2+0x+12. This simplifies to y = 12 and so the function has the value 12 for all values of x. It is therefore a straight horizontal line through 12 on the y axis. Play with different values of d and observe the result.

## Linear equations. (y = cx)

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

This is the graph of the equation y = 0x3+0x2+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.

1. Now move both sliders c and d to some value.
This is the equation of y = cx+d and combines the effects of the c and d coefficients. Play with various values to get a feel for the effects of their values on the graph.

## The squared term. (y = bx2)

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

This is the graph of the equation y = 0x3+2x2+0x+0. This simplifies to y = 2x2. 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.

## The cubed term. (y = ax3)

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

This is the graph of the equation 2x3+0x2+0x+0. This simplifies to y = 2x3. 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.

## The full cubic. (y = ax3+bx2+cx+d)

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

This is the graph of the equation y = 4x3+5x2-25x+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.

## Try it yourself

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