How many pieces of information do we need to determine a straight line? How many pieces of information are needed to determine a quadratic? How many pieces of information are we looking for here in order to determine a cubic?

From previous knowledge about curves and their intersections with the \(x\)-axis we can say quite a lot about the form their equations might take.

From looking at curve \(A\), the orange curve, we see it passes through the points \((-2,0)\), \((-1,0)\) and \((1,0)\). Just as we saw in Can you find… cubic edition, this means we have an equation for this cubic of the form,

\[y=a(x+2)(x+1)(x-1).\]

We can determine \(a\) by looking at any point on the curve that is not on the \(x\)-axis. In this case we can choose the \(y\)-intercept (when \(x=0\)). This gives \(a=1\).

Here we needed to use four pieces of information to determine an equation of the cubic curve. Do we ever need more than four pieces of information about any cubic to determine an equation for it?

What would the derivative of this function look like and what does that tell us about the shape of the curve?

From our knowledge of roots and transformations of cubic curves, curve \(D\), the black curve, looks like a translation of \(y=x^3\) to \(y=(x-1)^3\).

The equation \(y=(x-1)^3\) looks right but we can test to make sure by substituting \(x=0\) into the equations and locating the \(y\)-intercept of curve \(D\).

What would the derivative of this function look like and what does that tell us about the shape of the curve?

From looking at curve \(B\), the blue curve, it passes through the point \((0,0)\) and touches the \(x\)-axis at \((-1,0)\). Just as we saw in Can you find… cubic edition, this means we have an equation for this cubic of the form,

\[y=bx(x+1)^2.\]

We can determine \(b\) by looking at a point on the curve. When \(x=1\), \(y=4\) and so \(b=1\).

What would the derivative of this function look like and what does that tell us about the shape of the curve?