Cubic and polynomial graphsIB MYP Maths Extended: Revision notes

A cubic function has an $x^3$ term as its highest power, $y=ax^3+bx^2+cx+d$ with $a\ne0$. The sign of $a$ gives the overall shape (the end behaviour):

• $a>0$: the graph comes from bottom left and goes to top right.
• $a<0$: the graph comes from top left and goes to bottom right. A cubic crosses the $x$-axis at least once and at most three times. It has at most two turning points, a local maximum and a local minimum, or none at all. The $y$-intercept is $d$, the value when $x=0$.

If $y=a(x-p)(x-q)(x-r)$ the graph crosses the $x$-axis at $x=p,q,r$ (the roots). Set each bracket to zero, so the signs reverse. Example: $f(x)=(x+2)(x-1)(x-3)$ has roots $-2,1,3$. Its $y$-intercept is $f(0)=2\times(-1)\times(-3)=6$. Testing $f(2)=4\times1\times(-1)=-4$ shows the graph is below the axis between $1$ and $3$, and it is above the axis for $-2<x<1$ and for $x>3$. A repeated root, such as in $y=x(x+3)^2$, makes the graph touch the axis at $x=-3$ and turn round instead of crossing.

1. Use the sign of the $x^3$ term for the overall shape.
2. Mark the roots on the $x$-axis.
3. Find the $y$-intercept by putting $x=0$.
4. Join the points with a smooth curve, turning between the roots (a repeated root touches the axis). Example: $y=-(x+1)(x-2)(x-4)$. The shape is top left to bottom right because the coefficient is $-1$. The roots are $-1,2,4$ and the $y$-intercept is $-(1)(-2)(-4)=-8$. A sketch does not need an accurate scale, but it must show the roots, the $y$-intercept and the correct shape.

The highest power is the degree. A polynomial of degree $n$ has at most $n$ roots and at most $n-1$ turning points.

• Even degree (quadratic, quartic): both ends point the same way. $y=x^2$ is a U shape, and a quartic with positive coefficient is W-shaped or U-shaped.
• Odd degree (cubic, $y=x^5$): the ends point in opposite directions. Roots from factors work in the same way: $y=x(x-1)(x+2)(x-3)$ is a quartic with roots $0,1,-2,3$, positive $x^4$ coefficient, so both ends go up.

Graphing software or a graphing calculator can find features that are hard to get by hand.

• Turning points: use the maximum or minimum function. For $y=x^3-4x$ the local minimum is $(1.15,-3.08)$ and the local maximum is $(-1.15,3.08)$.
• Solving $f(x)=k$: graph $y=f(x)$ and $y=k$ and read the intersections. Solving $x^3-4x=2$ gives $x=-1.68,\ -0.539,\ 2.21$ (3 s.f.).
• Roots: use the zeros or solver function. Always give answers to the accuracy asked for, and check the answer lies inside any allowed domain.

Cubic functions model volumes. Cutting squares of side $x$ cm from a 30 cm square card and folding gives an open box of volume $V=x(30-2x)^2=4x^3-120x^2+900x$. The domain is limited by the situation: $0<x<15$, since you cannot cut a negative length and $x=15$ leaves no base. A graphing calculator shows the maximum volume of $2000$ cm$^3$ at $x=5$. To ask 'for which $x$ is $V\ge1800$?', graph $y=V$ and $y=1800$ and find where the curve is above the line: $3.27\le x\le6.96$. A real-life answer should always be checked against the context.