Here is an applet very similar to that in the first section, only this time it is for cubic graphs \(y=f(x)\) where \(f(x)=ax^3+bx^2+cx+d\).

Again, the big question: Can you find a relationship between the equation of the cubic and its gradient at \(x\)?

As with the quadratic case, for each of the following questions, describe what you think will happen before you try it out on the interactivity. Then reflect on your ideas: did the actual behaviour surprise you? And can you explain the observed behaviour?

What happens to the gradients as you change \(d\)?

What happens to the gradients as you change \(a\)?

What happens to the gradients as you change \(c\)?

What happens to the gradients as you change \(b\)?

Given what you have discovered, can you use it to predict what the gradients will be for some or all of the following cubics? Test your predictions!

- \(y=2x^3\)
- \(y=x^3+x\)
- \(y=x^3-x^2\)
- \(y=2x^3+x^2\)
- \(y=2x^3-x^2+3x\)
- \(y=-x^3+2x^2-3x+4\)

Now predict the gradients for some other cubics and test your results until you are confident that you have found a way of predicting the gradient function for any cubic.

Can you predict what the gradient will be for quartics (equations of the form \(y=ax^4+bx^3+\cdots\)) or polynomials of higher degree?