Building blocks

## Investigating cubics

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?