2.16 Modulus graphs and transformations questions for IB Maths AA

Solve the inequality.

Hence solve ${\displaystyle \frac{3}{\mathrm{\mid }x\mathrm{\mid }-2}}>\mathrm{\mid }x\mathrm{\mid }$, where $x\in \mathbb{R}$, $\mathrm{\mid }x\mathrm{\mid }\mathrm{\ne }2$.

Find the value of $t$ when A and B are at the same position.

Find the values of $t$ for which the distance between A and B is less than $1$ metre.

Find an expression for the velocity of A and an expression for the velocity of B.

Find the values of $t$ for which A is moving faster than B.

Find the total distance travelled by A in the first $6$ seconds.

Find the greatest distance between A and B during the first $6$ seconds.

Find the exact value of $f\left(\frac{\pi }{6}\right)$.

Find the exact value of $f\left(\frac{5\pi }{6}\right)$.

Find the exact value of $f\left(\frac{7\pi }{6}\right)$.

Use your GDC to sketch the graph of $y=f(x)$ for $-\pi \le x\le 2\pi$. Clearly indicate the coordinates of any maximum and minimum points.

Show that $f(x)=\pi -x$ for $\frac{\pi }{2}\le x\le \frac{3\pi }{2}$.

Write down the period of $f$.

Hence write down an expression for $f(x)$ for $\frac{3\pi }{2}\le x\le \frac{5\pi }{2}$.

Show that $f$ is an odd function.

Write down an expression for $g(x)$ for $0\le x\le \pi$.

Show that $g(x)=2\pi -x$ for $\pi \le x\le 2\pi$.

Let $h(x)=f(x)+g(x)$, for $0\le x\le 2\pi$.

Find an expression for $h(x)$ on each of the intervals $0\le x\le \frac{\pi }{2}$, $\frac{\pi }{2}\le x\le \pi$, $\pi \le x\le \frac{3\pi }{2}$ and $\frac{3\pi }{2}\le x\le 2\pi$.

Find all the solutions of the equation $f(x)={\displaystyle \frac{x}{2}}$, for $x\ge 0$.

Show that, for $\cos x\mathrm{\ne }0$, $f^{\mathrm{\prime }}(x)={\displaystyle \frac{\cos x}{\mathrm{\mid }\cos x\mathrm{\mid }}}$.

Hence explain why $f$ is not differentiable at $x=\frac{\pi }{2}$.

Find the exact value of ${\displaystyle {\int }_0^{10\pi }\mathrm{\mid }f(x)\mathrm{\mid }dx}$.

Find the zeros of $f$.

Sketch the graph of $y=\mathrm{\mid }f(x)\mathrm{\mid }$ for $-1\le x\le 3.5$, showing the $x$-intercepts and the coordinates of the local maximum points.

Find the values of $k$ for which the equation $\mathrm{\mid }f(x)\mathrm{\mid }=k$ has exactly four solutions.

Consider the case $n=2$. Sketch the graph of $y=S_2(x)$ for $0\le x\le 3$, showing the coordinates of the end points and of the points where the gradient changes.

Write down the minimum value of $S_2(x)$ and the values of $x$ for which it occurs.

Consider the case $n=3$. Show that $S_3(x)=4-x$ for $1\le x\le 2$.

Hence or otherwise, find the minimum value of $S_3(x)$ and the value of $x$ at which it occurs.

Let $m_n$ be the minimum value of $S_n(x)$. Use technology to find $m_4$, $m_5$ and $m_6$.

Let $j$ be an integer with $1\le j\le n-1$. Show that, for $j<x<j+1$, the gradient of the graph of $y=S_n(x)$ is $2j-n$.

Hence, explain why the minimum value of $S_n$ occurs only at $x={\displaystyle \frac{n+1}{2}}$ when $n$ is odd, and for every $x$ with ${\displaystyle \frac{n}{2}}\le x\le {\displaystyle \frac{n}{2}}+1$ when $n$ is even.

Let $n=2p$, where $p\in {\mathbb{Z}}^{+}$. Show that $m_n=p^2$.

Let $n=2p+1$, where $p\in {\mathbb{Z}}^{+}$. Find $m_n$ in terms of $p$, and hence show that $m_n={\displaystyle \frac{n^2-1}{4}}$.

Five houses stand beside a straight road, at distances $0$, $2$, $3$, $7$ and $11$ km from a junction. A delivery depot is to be built beside the road, $x$ km from the junction, so that the total distance from the depot to the five houses is as small as possible.

By considering a function similar to $S_n$, find the value of $x$ and the minimum total distance.

A sixth house is built at a distance of $12$ km from the junction. Find the possible positions of the depot that minimise the total distance to the six houses.