2.11 Transformations of graphs questions for IB Maths AA

Show that $\cos (x-{\displaystyle \frac{\pi }{6}})\equiv \sin (x+{\displaystyle \frac{\pi }{3}})$.

Hence show that $f(x)=1-\cos (2x+{\displaystyle \frac{2\pi }{3}})$.

Write down the range of $f$.

Describe a sequence of transformations that maps the graph of $y=\cos x$ onto the graph of $y=f(x)$.

Solve the equation $f(x)={\displaystyle \frac{1}{2}}$.

Find the exact area between the graph of $f$ and the $x$-axis, from $x=0$ to $x=\frac{\pi }{2}$.

Find the value of $a$ and the value of $d$.

Find the value of $b$.

Write down the smallest possible value of $c$.

The model can also be written as $D(t)=a\sin (b(t-q))+d$. Find the smallest positive value of $q$.

The model can also be written as $D(t)=-a\cos (b(t-r))+d$. Find the smallest positive value of $r$.

Find the first time after midnight at which the depth is $7$ metres.

A ship can only enter the harbour when the depth is at least $7$ metres. By using the symmetry of the graph of $D$ about each high tide, find the total length of time between midnight and 24:00 during which the ship can enter the harbour.

Show that $D(t)+D(t+6.2)=12$ for all $t$.

Consider the equation $x^3-3x+c=0$, where $c\in \mathbb{R}$.

By sketching the graph of $y=x^3-3x+c$, or otherwise, write down the number of distinct real roots of the equation when $c=1$.

Write down the number of distinct real roots of the equation when $c=2$.

Write down the number of distinct real roots of the equation when $c=3$.

Factorise $x^3-3x+2$ completely.

Let $p(x)$ be a polynomial with a repeated root $x=a$, so that $p(x)=(x-a{)}^{2}g(x)$, where $g(x)$ is a polynomial.

Show that $p(a)=0$ and $p^{\mathrm{\prime }}(a)=0$.

Consider now the equation $x^3+px+q=0$, where $p,q\in \mathbb{R}$ and $p\mathrm{\ne }0$. Suppose that this equation has a repeated root $x=a$.

Use part (c) to show that $3a^2+p=0$.

Show that $a=-{\displaystyle \frac{3q}{2p}}$.

Hence show that $4p^3+27q^2=0$.

Verify that the equation in part (b) satisfies this condition.

Find the values of $k$ for which the equation $x^3-12x+k=0$ has a repeated root. For each of these values of $k$, find all the roots of the equation.

By using the substitution $x=y-2$, find the values of $c$ for which the equation $x^3+6x^2+3x+c=0$ has a repeated root.

Give your answers in the form $m\pm n\sqrt{3}$, where $m,n\in \mathbb{Z}$.

Suppose now that $p<0$. The graph of $y=x^3+px+q$ has stationary points where $x=\pm s$, where $s=\sqrt{-{\displaystyle \frac{p}{3}}}$.

Show that the product of the $y$-coordinates of the two stationary points is $q^2+{\displaystyle \frac{4p^3}{27}}$.

Hence state the condition on $p$ and $q$ for the equation $x^3+px+q=0$ to have three distinct real roots. Justify your answer.

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

The graph of $y=f(x)$ is translated by the vector $\left(\begin{array}{c}k\\ 0\end{array}\right)$, where $k>0$, onto the graph of $y=g(x)$. Write down the smallest possible value of $k$.

Let $h(x)=\tan (x+{\displaystyle \frac{\pi }{2}})$, where $x\in \mathbb{R}$, $\sin x\mathrm{\ne }0$.

Show that $h(x)=-\cot x$.

Hence solve $h(x)=\sqrt{3}$ for $0<x<\pi$.

Show that $f(\pi -x)=f(x)$.

Hence write down the equation of the line of symmetry of the graph of $f$.

Find $f^{\mathrm{\prime }}(x)$.

Find the coordinates of the maximum point and of the two minimum points of the graph of $f$.

Solve the equation $f(x)=2$.

Find the area of the region bounded by the curve $y=f(x)$ and the line $y=2$.

Find the set of values of $k$ for which the equation $f(x)=k$ has exactly two solutions.

Let $g(x)=e^{\cos x}$, where $-\pi \le x\le \pi$. The graph of $g$ can be obtained from the graph of $f$ by a translation. Find the translation vector.

Write down the area of the region bounded by the curve $y=g(x)$ and the line $y=2$.

Explain why the graph of $y=f(x)$ is symmetric about the origin if and only if $f$ is an odd function.

Consider $f(x)=x^3-6x^2+9x+1$, for $x\in \mathbb{R}$.

Find the coordinates of the point of inflexion on the graph of $f$.

Show that $g(t)=f(2+t)-3$ is an odd function.

Hence state the coordinates of a point about which the graph of $f$ is symmetric.

Now consider the general cubic $f(x)=x^3+p{x}^2+qx+r$, where $p,q,r\in \mathbb{R}$.

Show that $f(t-{\displaystyle \frac{p}{3}})=t^3+(q-{\displaystyle \frac{p^2}{3}})t+S$, where $S$ is a constant that does not depend on $t$.

Hence show that the graph of every such cubic is symmetric about its point of inflexion.

Suppose that the graph of $y=f(x)$ is symmetric about the point $(a,b)$, and that $f$ is continuous on $[a-h,a+h]$, where $h>0$.

By using the substitution $x=a+t$, show that ${\displaystyle {\int }_{a-h}^{a+h}f(x)dx=2hb}$.

Hence find the value of ${\displaystyle {\int }_{-1}^5(x^3-6x^2+9x+1)dx}$.

Let $u(x)={\displaystyle \frac{3x+1}{x-1}}$, for $x\mathrm{\ne }1$.

Show that the graph of $u$ is symmetric about the point $(1,3)$.

Let $k(x)={\displaystyle \frac{1}{1+2^{x^3}}}$, for $x\in \mathbb{R}$.

Show that the graph of $k$ is symmetric about the point $(0,{\displaystyle \frac{1}{2}})$.

Hence write down the value of ${\displaystyle {\int }_{-2}^{2}k(x)dx}$.

Use your GDC to verify your answer to part (g)(ii).