Determine the time when the arrow should be released to hit the ball before the ball reaches its maximum height.

State the range of $g$.

Find the value of $b$.

Find the $y$-intercept of the graph of $f\left(x\right)+3$.

The tangent to $y=f\left(x\right)$ at P passes through the origin (0, 0).

Determine the value of $k$.

Find the $y$-intercept of the graph of $f\left(4x\right)$.

Determine the two positions where the path of the arrow intersects the path of the ball.

Find the equation of the axis of symmetry of the graph of $f$.

Find the $x$-intercept of the graph of $f\left(2x\right)$.

Find the value of $p$ and the value of $q$.

Find $g\left(0\right)$.

Write down the least value of $a$ such that $g$ has an inverse.

For $a=-\frac{\pi }{2}$, sketch the graph of $y=g\left(x\right)$. Indicate clearly the maximum and minimum values of the function.

Write down the maximum and minimum value of $v$.

For the value of $a$ found in part (b), write down the domain of $g^{-1}$.

Find the range of $g$.

Find the remainder when $p\left(x\right)$ is divided by $\left(x-2\right)$.

The range of $f$ is $k$ ≤ $f\left(x\right)$ ≤ $m$. Find $k$ and $m$.

On the same axes, sketch the graph of $f\left(-x\right)$.

Write down the value of $a$.

Find the coordinates of P.

The equation $g\left(x\right)=12$ has two solutions where $\pi$ ≤ $x$ ≤ $\frac{4\pi }{3}$. Find both solutions.

Find the value of $k$.

For the value of $a$ found in part (b), find an expression for $g^{-1}\left(x\right)$.

Sketch the graphs of $y=f\left(x\right)$ and $y=g\left(x\right)$ on the same axes, clearly stating the points of intersection with any axes.

Find the period of $g$.

Given that $p\left(t\right)$ can be written as $p\left(t\right)=a\text{sin}\left(b\left(t-c\right)\right)+d$ where $a$, $b$, $c$, $d$ > 0, use your graph to find the values of $a$, $b$, $c$ and $d$.

Sketch the graph of $y=p\left(t\right)$ for 0 ≤ $t$ ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.

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

Find the remainder when $p\left(x\right)$ is divided by $\left(x-3\right)$.

Find $p_{av}$(0.007).

Show that the equation of $L_1$ is $kx+p^{2}y-2pk=0$.

Prove that $p\left(x\right)$ has only one real zero.

Write down the transformation that will transform the graph of $y=p\left(x\right)$ onto the graph of $y=8x^3-12x^2+16x-24$.

The graph of the function $f\left(x\right)=\ln x$ is translated by $\left(\begin{array}{c}a\\ b\end{array}\right)$ so that it then passes through the points $(0, 1)$ and $({\text{e}}^3, 1+\ln 2)$ .

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

Find $f\text{'}\left(p\right)$ in terms of $k$ and $p$.

Describe the transformation $f\left(x+1\right)$.

Write down two transformations that will transform the graph of $y=v\left(t\right)$ onto the graph of $y=i\left(t\right)$.

Find the total time in the interval 0 ≤ $t$ ≤ 0.02 for which $p\left(t\right)$ ≥ 3.

The random variable $X$ follows a Poisson distribution with a mean of $\lambda$ and $6\text{P}\left(X=3\right)=3\text{P}\left(X=2\right)-2\text{P}\left(X=1\right)+3\text{P}\left(X=0\right)$.

Find the value of $\lambda$.

Consider the function $f\left(x\right)=x^4-6x^2-2x+4$, $x\in \mathbb{R}$.

The graph of $f$ is translated two units to the left to form the function $g\left(x\right)$.

Express $g\left(x\right)$ in the form $a{x}^4+b{x}^3+c{x}^2+dx+e$ where $a$, $b$, $c$, $d$, $e\in \mathbb{Z}$.

Another function, $g$, can be written in the form $g\left(x\right)=a\times f\left(x+b\right)$. The following diagram shows the graph of $g$.

Write down the value of a and of b.

Describe the transformation by which $f\left(x\right)$ is transformed to $g\left(x\right)$.

The outer dome of a large cathedral has the shape of a hemisphere of diameter 32 m, supported by vertical walls of height 17 m. It is also supported by an inner dome which can be modelled by rotating the curve $y=33-0.08x^3$ through 360° about the $y$-axis between $y$ = 0 and $y$ = 33, as indicated in the diagram.

Find the volume of the space between the two domes.

Write down the value of $h$.

The graph of $f$ is translated by $\left(\begin{array}{c}4\\ 3\end{array}\right)$ to give the graph of $g$.
In the following diagram:

• point $\text{Q}$ lies on the graph of $g$
• points $\text{C}$, $\text{D}$ and $\text{E}$ lie on the vertical asymptote of $g$
• points $\text{D}$ and $\text{F}$ lie on the horizontal asymptote of $g$
• point $\text{G}$ lies on the $x$-axis such that $\text{FG}$ is parallel to $\text{DC}$.

Line $L_2$ is the tangent to the graph of $g$ at $\text{Q}$, and passes through $\text{E}$ and $\text{F}$.

Given that triangle $\text{EDF}$ and rectangle $\text{CDFG}$ have equal areas, find the gradient of $L_2$ in terms of $p$.

Find the area of triangle $\text{AOB}$ in terms of $k$.

With reference to your graph of $y=p\left(t\right)$ explain why $p_{av}\left(T\right)$ > 0 for all $T$ > 0.

The graph of a second function, $g$, is obtained by a reflection of the graph of $f$ in the $y$-axis, followed by a translation of $\left(\begin{array}{c}-3\\ 6\end{array}\right)$.

Find the coordinates of the vertex of the graph of $g$.

On the same set of axes draw the graph of $y=g\left(x\right)$, showing any intercepts and asymptotes.