2.3 The graph of a function questions for IB Maths AA

Write down the period of $h$.

Write down the coordinates of the maximum point on the graph of $h$.

Sketch the graph of $h$, showing the coordinates of any points where the graph meets the $x$-axis and the coordinates of the minimum point.

Find the range of $f$.

Find the values of $x$ for which $f(x)=-5$.

Hence explain why $f$ does not have an inverse.

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.

Write down the range of $f$.

Write down the value of $f^{-1}(1)$.

Sketch the graph of $y=f^{-1}(x)$ on the same axes, showing the coordinates of its endpoints.

Find the coordinates of the local maximum point and the two local minimum points on the graph of $y=p(x)$.

Find the real roots of the equation $p(z)=0$, where $z\in \mathbb{C}$.

Hence, by considering the sum and the product of the roots, find the two non-real roots of $p(z)=0$.

Find the set of values of $k$ for which all the roots of the equation $p(z)=k$ are real.

Find the value of $k$ for which $p(z)=k$ has a root of the form $z=b\mathrm{i}$, where $b\in \mathbb{R}$, $b>0$.

For this value of $k$, find the other roots of $p(z)=k$.

Find $(f\circ f)(-2)$.

Find $(g\circ f)(4)$.

Find the values of $x$ for which $(f\circ g)(x)=0$, where $-1\le x\le 3$.