2.6 Quadratic functions questions for IB Maths AA

Write $f(x)$ in the form $(x-h{)}^2+k$, where $h,k\in \mathbb{Z}$.

Hence find the exact value of ${\displaystyle {\int }_3^5\frac{1}{x^2-6x+13}\mathrm{d}x}$.

Find $d$.

Show that $z_n=(2n+2)+(10-3n)\text{i}$.

Show that ${\mid z_n\mid }^2=13n^2-52n+104$.

Hence find the least value of $\mid z_n\mid$, and the value of $n$ for which it occurs.

Let $S_n=z_1+z_2+\cdots +z_n$.

Show that $S_n=n(n+3)+{\displaystyle \frac{n(17-3n)}{2}}\text{i}$.

Show that $S_n$ is never a real number.

Find the least value of $n$ for which $\mid S_n\mid >1000$.

The graph of the quadratic function $y=a{x}^2+bx+c$ passes through the points $(-1,2)$, $(0,1)$ and $(1,4)$.

Find the value of $a$, the value of $b$ and the value of $c$.

Find ${\displaystyle {\int }_{-1}^1(a{x}^2+bx+c)\mathrm{d}x}$ for these values of $a$, $b$ and $c$.

Now consider the quadratic $y=a{x}^2+bx+c$ whose graph passes through the points $(-h,p)$, $(0,q)$ and $(h,r)$, where $h>0$ and $p,q,r\in \mathbb{R}$.

Show that, for all values of $p$, $q$ and $r$, this system has the unique solution

Hence show that ${\displaystyle {\int }_{-h}^h(a{x}^2+bx+c)\mathrm{d}x=\frac{h}{3}(p+4q+r)}$.

It can be shown that the result of part (d) holds for any three equally spaced points $(m-h,p)$, $(m,q)$ and $(m+h,r)$. The rule ${\displaystyle {\int }_{m-h}^{m+h}f(x)\mathrm{d}x\approx \frac{h}{3}(f(m-h)+4f(m)+f(m+h))}$ is used to approximate integrals.

Use this rule with $m=0.5$ and $h=0.5$ to find an approximate value for ${\displaystyle {\int }_0^1{\mathrm{e}}^{x^2}\mathrm{d}x}$, and find the percentage error of this approximation.

Show that the rule in part (d) gives the exact value of ${\displaystyle {\int }_{-h}^{h}f(x)\mathrm{d}x}$ for any cubic function $f(x)=k{x}^3+a{x}^2+bx+c$.

A rule of the form ${\displaystyle {\int }_{-1}^{1}f(x)\mathrm{d}x\approx \alpha f(-1)+\beta f(0)+\gamma f(1)}$ is required to give the exact value when $f(x)=1$, $f(x)=x$ and $f(x)=x^2$.

Write down a system of three equations in $\alpha$, $\beta$ and $\gamma$, and solve it. Show that your answer agrees with part (d).

A different rule uses unequally spaced points: ${\displaystyle {\int }_0^{3}f(x)\mathrm{d}x\approx Af(0)+Bf(1)+Cf(3)}$. It is required to give the exact value when $f(x)=1$, $f(x)=x$ and $f(x)=x^2$.

Find the value of $A$, the value of $B$ and the value of $C$.

Determine whether the rule in part (h) gives the exact value of ${\displaystyle {\int }_0^3{x}^3\mathrm{d}x}$.

Explain briefly the significance of the value of $A$ found in part (h).

Write $5+4x-x^2$ in the form $a-(x-b{)}^2$, where $a,b\in \mathbb{Z}$.

Hence find an expression for $y$ in terms of $x$.

Find the value of $y$ when $x={\displaystyle \frac{7}{2}}$.

Find the time at which the ball hits the ground.

Hence write down the domain of $h$ in this context.

Find the range of $h$ in this context.

Write $x^2+4x+13$ in the form $(x+p{)}^2+q$, where $p,q\in \mathbb{Z}$.

Hence find ${\displaystyle \int \frac{1}{x^2+4x+13}\mathrm{d}x}$.

By writing $2x+7=(2x+4)+3$, find ${\displaystyle \int h(x)\mathrm{d}x}$.

Hence find the exact value of ${\displaystyle {\int }_{-2}^{1}h(x)\mathrm{d}x}$, giving your answer in the form $\ln a+{\displaystyle \frac{\pi }{b}}$, where $a,b\in {\mathbb{Z}}^{+}$.

The region bounded by the curve $y={\displaystyle \frac{3}{\sqrt{x^2+4x+13}}}$, the $x$-axis and the lines $x=-2$ and $x=1$ is rotated through $2\pi$ radians about the $x$-axis.

Find the exact volume of the solid formed.