Let $g\left(x\right)=p^x+q$, for $x\text{, }p\text{, }q\in <!--\; \in \; -->\mathbb{R}\text{, }p>1$. The point $\text{A}\left(0\text{, }a\right)$ lies on the graph of $g$.

Let $f\left(x\right)=g^{-<!--\; -\; -->1}\left(x\right)$. The point $\text{B}$ lies on the graph of $f$ and is the reflection of point $\text{A}$ in the line $y=x$.

The line $L_1$ is tangent to the graph of $f$ at $\text{B}$.

Consider the binomial expansion ${(x+1)}^7=x^7+a{x}^6+b{x}^5+35x^4+\dots +1$ where $x\ne 0$ and $a, b\in {\mathrm{\mathbb{Z}}}^{+}$.

The following diagram shows part of the graph of a quadratic function $f$.

The graph of $f$ has its vertex at $(3, 4)$, and it passes through point $\text{Q}$ as shown.

The function can be written in the form $f\left(x\right)=a{(x-h)}^2+k$.

The line $L$ is tangent to the graph of $f$ at $\text{Q}$.

Now consider another function $y=g\left(x\right)$. The derivative of $g$ is given by $g\prime \left(x\right)=f\left(x\right)-d$, where $d\in \mathrm{ℝ}$.

Consider the functions $f\left(x\right)=\frac{1}{x-4}+1$, for $x\ne 4$, and $g\left(x\right)=x-3$ for $x\in \mathrm{ℝ}$.

The following diagram shows the graphs of $f$ and $g$.

The graphs of $f$ and $g$ intersect at points $\text{A}$ and $\text{B}$. The coordinates of $\text{A}$ are $(3, 0)$.

In the following diagram, the shaded region is enclosed by the graph of $f$, the graph of $g$, the $x$-axis, and the line $x=k$, where $k\in \mathrm{\mathbb{Z}}$.

The area of the shaded region can be written as $\ln \left(p\right)+8$, where $p\in \mathrm{\mathbb{Z}}$.

The following diagram shows the graph of a function $f$, with domain $-<!--\; -\; -->2⩽<!--\; ⩽\; -->x⩽<!--\; ⩽\; -->4$.

The points $(-<!--\; -\; -->2,0)$ and $(4,7)$ lie on the graph of $f$.

Olava’s Pizza Company supplies and delivers large cheese pizzas.

The total cost to the customer, $C$, in Papua New Guinean Kina ($\text{PGK}$), is modelled by the function

$C\left(n\right)=34.50n+8.50 , n\ge 2 , n\in \mathrm{\mathbb{Z}},$

where $n$, is the number of large cheese pizzas ordered. This total cost includes a fixed cost for delivery.

The graph of the quadratic function $f(x)=c+bx-<!--\; -\; -->x^2$ intersects the $x$-axis at the point $\text{A}(-<!--\; -\; -->1,0)$ and has its vertex at the point $\text{B}(3,16)$.

The following table shows the probability distribution of a discrete random variable $A$, in terms of an angle $\mathrm{\theta <!--\; \theta \; -->}$.

The function $f$ is defined for all $x\in \mathrm{ℝ}$. The line with equation $y=6x-1$ is the tangent to the graph of $f$ at $x=4$.

The function $g$ is defined for all $x\in \mathrm{ℝ}$ where $g\left(x\right)=x^2-3x$ and $h\left(x\right)=f\left(g\left(x\right)\right)$.

A function $f$ is defined by $f\left(x\right)=\frac{2x-1}{x+1}$, where $x\in \mathrm{ℝ}, x\ne -1$.

The graph of $y=f\left(x\right)$ has a vertical asymptote and a horizontal asymptote.

Consider the function $f\left(x\right)=-2\left(x-1\right)\left(x+3\right)$, for $x\in \mathrm{ℝ}$. The following diagram shows part of the graph of $f$.

For the graph of $f$

Consider the function $f\left(x\right)=a^x$ where $x, a\in \mathrm{ℝ}$ and $x>0, a>1$.

The graph of $f$ contains the point $\left(\frac{2}{3}, 4\right)$.

Consider the arithmetic sequence ${\log }_8 27 , {\log }_8 p , {\log }_8 q , {\log }_8 125 ,$ where $p>1$ and $q>1$.

The diagram shows the graph of the quadratic function $f\left(x\right)=a{x}^2+bx+c$ , with vertex $\left(-2, 10\right)$.

The equation $f\left(x\right)=k$ has two solutions. One of these solutions is $x=2$.

A quadratic function $f$ can be written in the form $f(x)=a(x-<!--\; -\; -->p)(x-<!--\; -\; -->3)$. The graph of $f$ has axis of symmetry $x=2.5$ and $y$-intercept at $(0,-<!--\; -\; -->6)$

The following diagram shows part of the graph of $f\left(x\right)=\frac{k}{x}$, for $x>0, k>0$.

Let $\text{P}\left(p, \frac{k}{p}\right)$ be any point on the graph of $f$. Line $L_1$ is the tangent to the graph of $f$ at $\text{P}$.

Line $L_1$ intersects the $x$-axis at point $\text{A}\left(2p, 0\right)$ and the $y$-axis at point B.

A particle $P$ moves along the $x$-axis. The velocity of $P$ is $v \mathrm{m} {\mathrm{s}}^{-1}$ at time $t$ seconds, where $v\left(t\right)=4+4t-3t^2$ for $0\le t\le 3$. When $t=0, P$ is at the origin $\mathrm{O}$.

Let $f\left(x\right)=m{x}^2-2mx$, where $x\in \mathrm{ℝ}$ and $m\in \mathrm{ℝ}$. The line $y=mx-9$ meets the graph of $f$ at exactly one point.

The function $f$ can be expressed in the form $f\left(x\right)=4\left(x-p\right)\left(x-q\right)$, where $p, q\in \mathrm{ℝ}$.

The function $f$ can also be expressed in the form $f\left(x\right)=4{\left(x-h\right)}^2+k$, where $h, k\in \mathrm{ℝ}$.

A function, $f$, has its derivative given by $f\prime \left(x\right)=3x^2-12x+p$, where $p\in \mathrm{ℝ}$. The following diagram shows part of the graph of $f\prime$.

The graph of $f\prime$ has an axis of symmetry $x=q$.

The vertex of the graph of $f\prime$ lies on the $x$-axis.

The graph of $f$ has a point of inflexion at $x=a$.

Jean-Pierre jumps out of an airplane that is flying at constant altitude. Before opening his parachute, he goes through a period of freefall.

Jean-Pierre’s vertical speed during the time of freefall, $S$, in ${\text{m\hspace{0.05em}s}}^{-1}$, is modelled by the following function.

$S\left(t\right)=K-60\left(1.2^{-t}\right) , t\ge 0$

where $t$, is the number of seconds after he jumps out of the airplane, and $K$ is a constant. A sketch of Jean-Pierre’s vertical speed against time is shown below.

Jean-Pierre’s initial vertical speed is $0 {\text{m\hspace{0.05em}s}}^{-1}$.

Let $f(x)=1+{\text{e}}^{-<!--\; -\; -->x}$ and $g(x)=2x+b$, for $x\in <!--\; \in \; -->\mathbb{R}$, where $b$ is a constant.

The following diagram shows the graph of $y=-1-\sqrt{x+3}$ for $x\ge -3$.

A function $f$ is defined by $f\left(x\right)=-1-\sqrt{x+3}$ for $x\ge -3$.

Line $L$ intersects the $x$-axis at point A and the $y$-axis at point B, as shown on the diagram.

The length of line segment OB is three times the length of line segment OA, where O is the origin.

Point $\text{(2, 6)}$ lies on $L$.

Consider the functions $f\left(x\right)=x^4-<!--\; -\; -->2$ and $g\left(x\right)=x^3-<!--\; -\; -->4x^2+2x+6$

The functions intersect at points P and Q. Part of the graph of $y=f\left(x\right)$ and part of the graph of $y=g\left(x\right)$ are shown on the diagram.

Let $g\left(x\right)=x^2+bx+11$. The point $\left(-<!--\; -\; -->1\text{, }8\right)$ lies on the graph of $g$.

Let $y=\frac{\ln x}{x^4}$ for $x>0$.

Consider the function defined by $f\left(x\right)\frac{\ln x}{x^4}$ for $x>0$ and its graph $y=f\left(x\right)$.

The following table shows values of $f\left(x\right)$ and $g\left(x\right)$ for different values of $x$.

Both $f$ and $g$ are one-to-one functions.

The function $f$ is defined by $f\left(x\right)=\frac{2x+4}{3-x}$, where $x\in \mathrm{ℝ}, x\ne 3$.

Write down the equation of

Find the coordinates where the graph of $f$ crosses

Consider the series $\ln x+p \ln x+\frac{1}{3}\ln x+\dots$, where $x\in \mathrm{ℝ}, x>1$ and $p\in \mathrm{ℝ}, p\ne 0$.

Consider the case where the series is geometric.

Now consider the case where the series is arithmetic with common difference $d$.

The functions $f$ and $g$ are defined such that $f\left(x\right)=\frac{x+3}{4}$ and $g\left(x\right)=8x+5$.

Let $f\left(x\right)=a {\log }_3\left(x-4\right)$, for $x>4$, where $a>0$.

Point $\text{A}\left(13, 7\right)$ lies on the graph of $f$.

Consider the graph of the function $f\left(x\right)=x+\frac{12}{x^2}, x\ne 0$.

The points $\text{A}$ and $\text{B}$ have position vectors $\left(\begin{array}{c}-<!--\; -\; -->2\\ 4\\ -<!--\; -\; -->4\end{array}\right)$ and $\left(\begin{array}{c}6\\ 8\\ 0\end{array}\right)$ respectively.

Point $\text{C}$ has position vector $\left(\begin{array}{c}-<!--\; -\; -->1\\ k\\ 0\end{array}\right)$. Let $\text{O}$ be the origin.

Find, in terms of $k$,

The graph of $y=f\left(x\right)$ for $-4\le x\le 6$ is shown in the following diagram.

Consider the points $\text{A}(-2, 20)$, $\text{B}(4, 6)$ and $\text{C}(-14, 12)$. The line $L$ passes through the point $\text{A}$ and is perpendicular to $\text{[BC]}$.

Consider the functions $f\left(x\right)=\sqrt{3}\sin x+\cos x$ where $0\le x\le \pi$ and $g\left(x\right)=2x$ where $x\in \mathrm{ℝ}$.

Let $f(x)=x^2-<!--\; -\; -->x$, for $x\in <!--\; \in \; -->\mathbb{R}$. The following diagram shows part of the graph of $f$.

The graph of $f$ crosses the $x$-axis at the origin and at the point $\text{P}(1,0)$.

The line $L$ intersects the graph of $f$ at another point Q, as shown in the following diagram.

The function $f$ is of the form $f(x)=ax+b+\frac{c}{x}$, where $a$ , $b$ and $c$ are positive integers.

Part of the graph of $y=f(x)$ is shown on the axes below. The graph of the function has its local maximum at $(-<!--\; -\; -->2,-<!--\; -\; -->2)$ and its local minimum at $(2,6)$.

Let $f(x)=x^2-<!--\; -\; -->4x+5$.

The function can also be expressed in the form $f(x)=(x-<!--\; -\; -->h{)}^2+k$.

Consider the function $f$, with derivative $f^{\prime }\left(x\right)=2x^2+5kx+3k^2+2$ where $x\text{, }k\in <!--\; \in \; -->\mathbb{R}$.

Consider the function $f$ defined by $f\left(x\right)=\ln (x^2-16)$ for $x>4$.

The following diagram shows part of the graph of $f$ which crosses the $x$-axis at point $\text{A}$, with coordinates $(a, 0)$. The line $L$ is the tangent to the graph of $f$ at the point $\text{B}$.