Building blocks

## Things you might have noticed

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? (You might want to start with very simple values for $a$, $b$ and $c$.)

• What happens to the gradients as you change $c$?

• What happens to the gradients as you change $a$?

• What happens to the gradients as you change $b$?
• As we change $c$, the curve just translates up or down, so the gradient at a given $x$-value does not change.

• The graph of the gradients is a straight line, and its gradient changes as we change $a$.

The gradient function of $y=x^2$ is $2x$, as we discovered in Zooming in. If we stretch this function in the $y$-direction by a factor of $2$, by setting $a=2$, we double the gradients, giving a gradient function of $4x$.

Does the same sort of thing happen for other values of $a$?

Does changing $b$ affect the gradient of the gradient function line?

• Changing $b$ translates the gradient function. If we add $1$ to $b$, then the gradient function increases by $1$. This makes some sense: $b=1$ (with $a=c=0$) gives the function $y=x$, which has gradient $1$ everywhere. So if we add $x$ to our function, it seems reasonable that the gradient should increase by $1$. For example, when we go from $y=3x$ to $y=3x+x=4x$, the gradient increases from $3$ to $4$.

Putting these together, to find the gradient of the quadratic function $f(x)=ax^2+bx+c$, we can start with $x^2$, stretch it by a factor of $a$ to get $ax^2$ (which has what gradient function?), then add on $bx$ (which does what to the gradient function?) and then finally add $c$. This will give us the final gradient function.

Following on from the above ideas, we can predict what will happen for a cubic: starting with $x^3$, which appears to have a quadratic gradient function (which quadratic exactly?), we stretch it to obtain $ax^3$, and this stretches the gradient function. We then add on extra terms, whose behaviour we now understand from exploring quadratics: first we add on $bx^2$, then $cx$ and finally $d$.
Using this idea, we should be able to predict the gradient function of any cubic $f(x)=ax^3+bx^2+cx+d$.
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?
Some patterns definitely seem to be emerging here! What would you expect the gradient function of $x^4$ to be?