Sketch the graph of $y=f\left(x\right)$ showing clearly the position of the points A and B.

Sketch the graph of $C$ against $x$, labelling the maximum point and the $x$-intercepts with their coordinates.

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

Sketch the graph of $y=g\left(t\right)$ for t ≤ 0. Give the coordinates of any intercepts and the equations of any asymptotes.

Sketch the graphs $y=\text{si}{\text{n}}^{3}x+\text{ln}x$ and $y=1+\text{cos}x$ on the following axes for 0 < $x$ ≤ 9.

Use the model to calculate the total amount of time when fishing should be stopped.

Write down one feature of this graph which suggests a cubic function might be appropriate to model this scenario.

Write down the value of $c$.

For the first year of the model, find the length of time when there are more than $10$ hours and $30$ minutes of daylight per day.

Sketch the graphs on the same set of axes.

Sketch the graph of y = f (x), for −4 ≤ x ≤ 3 and −50 ≤ y ≤ 100.

Write down the second $x$-intercept of the function.

This region is now rotated through $2\pi$ radians about the $x$-axis. Find the volume of revolution.

Sketch the graph of $y=g\left(x\right)$. State the equations of any asymptotes and the coordinates of any intercepts with the axes.

Show that $\alpha$ + β < −2.

Use your graphic display calculator to find how long it will take for Jashanti to have saved enough money to buy the car.

Solve $f\left(x\right)=f^{-1}\left(x\right)$.

Find the coordinates of $\text{X}$.

By considering the graph of $h\left(t\right)$, determine the length of time during one revolution of the Ferris wheel for which the chair is higher than the cowboy statue.

Determine the maximum amount of coffee the cafe can make that will not result in a loss of money for the morning.

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

Use your graphic display calculator to find the equation of the tangent to the graph of y = f (x) at the point (–2, 38.75).

Give your answer in the form y = mx + c.

The wind speed increases. The blades rotate at twice the speed, but still at a constant rate.

At any given instant, find the probability that Tim can see point $\text{C}$ from his window. Justify your answer.

Find the time, in seconds, that point $\text{C}$ is above a height of $100 \text{m}$, during each complete rotation.

Determine an equation for the axis of symmetry of the parabola that models the archway.

Determine whether the crate will fit through the archway. Justify your answer.

Write down the amplitude of the function.

Find the height of $\text{C}$ above the ground when $t=2$.

Sketch the function $h\left(t\right)$ for $0\le t\le 5$, clearly labelling the coordinates of the maximum and minimum points.

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

Sketch the graphs of $y=\frac{x}{2}+1$ and $y=\left|x-2\right|$ on the following axes.

Express this volume in $\text{c}{\text{m}}^3$.

Given that the graphs enclose a region of area 18 square units, find the value of b.

Use your graphic display calculator to find the zero of f (x).

Write down the amount of money Jashanti saves per month.

Write down the number of possible solutions to the equation $f(x)=5$.

State the domain of $P$.

Sketch the curve $y=f\left(x\right)$, 0 ≤ $x$ ≤ 5 indicating clearly the coordinates of the maximum and minimum points and any intercepts with the coordinate axes.

With reference to your graph, explain why $f$ is a function on the given domain.

Find the inverse function $g^{-1}$ and state its domain.

Factorize $x^2+3x+2$.

Sketch the graph of $y=f\left(x\right)$ for $-\frac{5\pi }{8}⩽x⩽\frac{\pi }{8}$.

Write down the $x$-coordinates of these two points;

Write down the $y$-intercept of the graph.

Find $f(2)$.

Express $g\left(x\right)$ in the form $A+\frac{B}{x-2}$ where A, B are constants.

Explain why $f$ has no inverse on the given domain.

Sketch the graph of $y=f\left(\left|x\right|\right)$.

Explain why $f$ is an even function.

find the value of $b$.

The endpoints of the minute hand and hour hand meet when $\theta =k$.

Find the smallest possible value of $k$.

Use your graphic display calculator to find the coordinates of the local minimum point.

State the range of $f$.

Show that $a=8$.

Hence find the value of $p$ if ${\int }_0^{1}f(x)\text{d}x=\ln (p)$.

Show that the $x$-coordinate of the minimum point on the curve $y=f(x)$ satisfies the equation $\tan x=2x$.

Write down the $x$-coordinate of the point of inflexion on the graph of $y=f\left(x\right)$.

Calculate how much extra money Jashanti needs.

This straight road crosses the highway and then carries on due north.

State whether the straight road will ever cross the river. Justify your answer.

Hence, show that there are no solutions to $(g^{-1}{)}^{\prime }(x)=0$.

The function $h$ is defined by $h\left(x\right)=\sqrt{x}$, for $x$ ≥ 0.

State the domain and range of $h\circ g$.

Sketch the graph of $y=\frac{1-3x}{x-2}$, showing clearly any asymptotes and stating the coordinates of any points of intersection with the axes.

Show that there is exactly one point of inflexion, B, on the graph of $y=f\left(x\right)$.

Solve the equation $\frac{x}{2}+1=\left|x-2\right|$.

The graph of $y=f\left(x\right)$ has a local maximum at A. Find the coordinates of A.

Express $x^2+3x+2$ in the form $(x+h{)}^2+k$.

Write down the intervals where the gradient of the graph of $y=f(x)$ is positive.

Using your answer to part (e), find the value of $r$ which minimizes $A$.

state the value of $a$ and the value of $c$;

Find the coordinates of the point on the graph of $f$ where the normal to the graph is parallel to the line $y=-x$.

Sketch the graph of $y=f(x)$ showing clearly the minimum point and any asymptotic behaviour.

Find $f\left(1\right)$.

State, in the context of the question, what the value of $34.50$ represents.

State, in the context of the question, what the value of $8.50$ represents.

Kaelani has $450 \text{PGK}$.

Find the maximum number of large cheese pizzas that Kaelani can order from Olava’s Pizza Company.

Write down the coordinates of the local minimum point.

Determine the values of $x$ for which $f(x)$ is a decreasing function.

Show that $\frac{1}{x+1}-\frac{1}{x+2}=\frac{1}{x^2+3x+2}$.

Solve $f\left(x\right)=0$.

Write down the minimum number of pizzas that can be ordered.

The equation $f(x)=m$, where $m\in \mathbb{R}$, has four solutions. Find the possible values of $m$.

Solve the inequality $\left|3x\arccos (x)\right|>1$.

Find $g^{\prime }(x)$.

Find $\frac{\text{d}A}{\text{d}r}$.

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

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

The coordinates of B can be expressed in the form B$\left(2^a,b\times 2^{-3a}\right)$ where a, b$\in \mathbb{Q}$. Find the value of a and the value of b.

Explain why the inverse function $f^{-1}$ does not exist.

Hence, show that there are no solutions to $g^{\prime }(x)=0$;

Hence or otherwise, solve the inequality .

Find $\alpha$ and β in terms of $k$.

Write down the range of $f(x)$.

The graph of $f$ has a local minimum at point $\text{A}$.

Find the coordinates of $\text{A}$.

Find the distance from the centre of Orangeton to the point at which the road meets the highway.

find the value of $f\left(1\right)$.

Consider the function $g\left(x\right)=3-x$.

Solve $f\left(x\right)=g\left(x\right)$.

The price of gas at Erica’s gas station is $\$1.30$ per litre. A customer must buy a minimum of $10$ litres of gas. The total cost at Erica’s gas station is cheaper than Leon’s gas station when $x>k$.

Find the minimum value of $k$.

Hence solve  in the range 0 < $x$ ≤ 9.

Find the inverse function $f^{-1}$, stating its domain.

Write down, in terms of $r$ and $h$, an equation for the volume of this water container.

Write down a formula for $A$, the surface area to be coated.

Find the least number of cans of water-resistant material that will coat the area in part (g).

On day $n$ of the fitness programmes Daniella runs more than Charlie for the first time.

Find the value of $n$.

Explain why $f$ is not a function for $-\frac{3\pi }{4}⩽x⩽\frac{\pi }{4}$.

Sketch the graph of $f$ indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.

Show that $A=\pi r^2+\frac{1000000}{r}$.

Sketch the graph of $y=f(x)$ showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.

Estimate the value of Roger’s laptop after $5$ years.

Comment on the validity of your answer to part (b).

Find the largest possible domain $D$ for $f$ to be a function.

Find the value of this minimum area.

find the value of $f\left(4\right)$.

Show that $g\left(t\right)={\left(\frac{1+t}{1-t}\right)}^2$.

Determine the area of the region enclosed between the graph of $y=f\left(\left|x\right|\right)$, the $x$-axis and the lines with equations $x=-1$ and $x=1$.

Sketch the graph of $f(x)$, indicating on it the equations of the asymptotes, the coordinates of the $y$-intercept and the local maximum.

Determine the first year in which this model predicts the average number of visitors per concert will exceed the total seating capacity of the concert hall.

Whilst swimming, Scarlett can hear the siren only if the sound intensity at her location is greater than $1.5\times {10}^{-6}$ ${\text{W\hspace{0.05em}m}}^{-2}$.

Find the values of $d$ where Scarlett cannot hear the siren.

Find the value of $k$.

minimum value of $h$.

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

Find the time, in seconds, it takes for the blade $\text{[BC]}$ to make one complete rotation under these conditions.

maximum value of $h$.

minimum value of $h$.

Find the time, in seconds, it takes for the blade $\text{[BC]}$ to make one complete rotation under these conditions.

Calculate the angle, in degrees, that the blade $\text{[BC]}$ turns through in one second.

Find the period of the function.

Sketch the function $h\left(t\right)$ for $0\le t\le 5$, clearly labelling the coordinates of the maximum and minimum points.

At any given instant, find the probability that point $\text{C}$ is visible from Tim’s window.

maximum value of $h$.

Calculate the angle, in degrees, that the blade $\text{[BC]}$ turns through in one second.

Write down the amplitude of the function.

Find the period of the function.

Find the height of $\text{C}$ above the ground when $t=2$.

Find the time, in seconds, that point $\text{C}$ is above a height of $100 \text{m}$, during each complete rotation.

The wind speed increases and the blades rotate faster, but still at a constant rate.

Given that point $\text{C}$ is now higher than $110 \text{m}$ for $1$ second during each complete rotation, find the time for one complete rotation.