Graph features (max / min, axis intercepts, symmetry, vertex, asymptotes & intercepts (GDC)): notes and practice questions

The population, $P$, of a certain bacterial colony, in thousands, can be modelled by the function $P(t) = 80 \times 1.15^t + 20$, where $t$ is the time in hours.

Write down the equation of the horizontal asymptote of the graph of $P(t)$.

Write down the coordinates of the point where the graph of $P(t)$ cuts the $P$-axis.

(a) The intensity of light, I, from a lighthouse varies inversely with the square of the distance, d, from the lighthouse, where $d > 0$.

It is known that at a distance of 2 metres from the lighthouse, the light intensity is 50 candelas per square metre (cd m$^{-2}$).

Show that $I = \frac{200}{d^2}$.

(b) Sketch the curve of I on the axes below, showing clearly the point (2, 50).

(c) A small boat needs a light intensity greater than $0.005 \text{ cd m}^{-2}$ to navigate safely.

Find the values of d where the boat cannot navigate safely.

A botanical garden features a winding path for visitors. The path can be modelled by the function $f(x) = x^3 - 3x^2 - 9x + 5$. All distances in the garden are in kilometres.

A new straight maintenance path needs to be constructed. This path will start at point P on the visitor path, which has coordinates $(2, -17)$.

(a) Using your graphic display calculator, find the value of $f'(2)$.

(b) Find the equation of the line normal to $f(x)$ at point P.

(c) The maintenance path connects point P to another point Q on the visitor path, where the normal line intersects the path again. For safety regulations, point Q must have a positive x-coordinate. Determine the length of this new maintenance path.

A scientist is studying the growth of a bacterial colony. The population of the colony, $P$, after $t$ hours is modelled by the function $P(t) = 150(1.08)^t + 50$.

(a) Write down the equation of the horizontal asymptote to the graph of $P(t)$.

(b) Calculate the initial number of bacteria in the colony.

A manufacturing company purchases a new specialized machine. The machine's value depreciates exponentially over time. The value of the machine, $V$, in thousands of dollars, $t$ years after its purchase, is modelled by the function $V(t) = A e^{-kt}$, for $t \ge 0$.

The initial value of the machine was 150 thousand dollars. After 3 years, its value had decreased by 40%.

(a) Find the value of $k$.

(b) Calculate the value of the machine after 5 years and 6 months, giving your answer to two decimal places.

(c) The company believes that, according to this model, the machine will always have some residual value, however small.

State a mathematical reason why the company might believe this.

(d) Write down one possible limitation of the domain of the model.

(a) The position of a reconnaissance drone, relative to a control tower, is given by the vector equation $r = \begin{pmatrix} 7 \\ 2 \end{pmatrix} + t \begin{pmatrix} -3 \\ 4 \end{pmatrix}$, where $r$ is the position vector in metres and $t$ is the time in minutes.

Write down the position vector of the drone when $t = 0$ and when $t = 1$.

(b) Calculate the speed of the drone.

(c) Find an expression for the distance of the drone from the origin at time $t$.

(d) Hence find the minimum distance of the drone from the origin and the time at which it occurs.

An architect is designing a parabolic arch for a new building. The shape of the arch can be modelled by the function $f(x) = ax^2 + bx + c$, where $x$ and $f(x)$ are measured in metres. The highest point of the arch (the vertex) is at $(2, 4)$. One end of the arch is at the origin $(0, 0)$, and the other end is at $(p, 0)$.

(a) Find the value of $p$.

(b) Find the value of

(i) $a$.

(ii) $b$.

(iii) $c$.

(c) Write down the equation of the axis of symmetry of the arch.

In a controlled chemical experiment, the initial temperature, $x$ (in degrees Celsius), of a reactant mixture is adjusted before a reaction begins. The adjusted temperature, $f(x)$, is then used to determine the reaction rate, $g(f(x) )$.

The adjustment function is given by $f(x) = x - 3$, for $x \in \mathbb{R}$.

State the range of $f(x)$.

The overall reaction rate, $gf(x)$, as a function of the initial temperature $x$, is modelled by $gf(x) = 4x^2 - 24x + 30$, for $x \in \mathbb{R}$.

State the range of $gf(x)$.

The experiment is designed to achieve a specific 'target state' when the adjusted reaction rate $f(gf(x) )$ equals 0. Solve the equation $fgf(x) = 0$.

Determine the function $g(x)$.

A new fertilizer is being tested on a specific plant species. The concentration of a key nutrient, $C$, in the plant (in mg/L) is modelled by the function $C(x) = 5 - \frac{30}{x-2}$, where $x$ is the amount of fertilizer added (in grams). The amount of fertilizer used is restricted to $-4 \le x \le 8$, where $x \neq 2$ (due to experimental constraints).

Find the range of the nutrient concentration $C(x)$.

Determine the amount of fertilizer $x$ that results in a nutrient concentration of $-1$ mg/L. Give your answer in the form $C^{-1}(-1)$.

The concentration of a reactant A, in mg/L, in a chemical reaction over time $t$ (in minutes) is modelled by the function:

$C(t) = -t^3 + 15t^2 - 48t + 100$, for $0 \le t \le 10$.

(a) Find the coordinates of the local minimum point of the concentration.

(b) Find the coordinates of the local maximum point of the concentration.

(c) Find the set of values of $t$ for which the concentration of reactant A is above $100$ mg/L.

(a) An architect is designing the entrance to a new tunnel. The shape of the entrance is modelled by a quadratic curve, with the base of the tunnel on the x-axis. The curve has end points (0, 6) and (10, 6), and its vertex is (5, 9). Distances are measured in metres.

The quadratic curve can be expressed in the form $y = ax^2 + bx + c$ for $0 \leq x \leq 10$.

(a.i) Write down the value of $c$.

(a.ii) Hence, form two equations in terms of $a$ and $b$.

(a.iii) Hence, find the equation of the quadratic curve.

(b) Calculate the area of the tunnel entrance.

The concentration of a certain chemical, $C$, in a solution over a period of time can be modelled using the function $C(t) = -0.005t^3 + 0.1t^2 - 0.2t + 5$, where $t$ is the time in hours after the experiment begins.

Sketch the graph of $C(t) = -0.005t^3 + 0.1t^2 - 0.2t + 5$ for $0 \le t \le 20$.

Find the concentration after $2$ hours.

Find the concentration after $15$ hours.

Find the maximum concentration and the time in hours at which this occurs.

Find the minimum concentration and the time in hours at which this occurs.

Find the times in hours when the concentration is $6$ mol/L.

The height of a diver above the water surface after jumping from a diving board is modelled by the function

$h(t) = -4.9t^2 + 5t + 10$

where $h(t)$ is the height in metres and $t$ is the time in seconds after the diver leaves the board.

(a) Write down the height of the diving board above the water surface.

(b) Find the value of $t$ when the diver enters the water. Give your answer to three significant figures.

(c) State an appropriate domain for $t$ in this model.

A drone launches a package, and its trajectory is modelled by the equation $h(x) = -0.015x^2 + 0.6x + 5$, where $h(x)$ is the height of the package in metres and $x$ is the horizontal distance in metres from the launch point.

On paper, sketch the graph of the path that the package flies for $x \ge 0$. Clearly indicate the initial height, the maximum height, and the horizontal distance when it lands.

Find the height of the package when it has travelled a horizontal distance of $15$ metres.

Find the maximum height of the package.

Find the horizontal distance at which the package lands on the ground, and explain what this value represents in the context of the problem.

A new experimental drug is administered to a patient. The concentration of the drug in the patient's bloodstream, $C$, in mg/L, $t$ hours after administration, is modelled by the function $C(t) = C_0 e^{-kt}$, where $C_0$ and $k$ are positive constants.

Initially, the concentration of the drug is $250$ mg/L. After $2$ hours, the concentration drops to $150$ mg/L.

Determine the value of $k$.

Using this model, calculate the concentration of the drug in the bloodstream $5$ hours after administration.

Based on this model, will the drug ever completely leave the patient's bloodstream? Justify your answer.

State one limitation of the domain of this model in a real-world context.

The number of visitors (in hundreds) to a new eco-tourism resort $t$ months after its opening is modelled by the function $N(t) = -0.001 t^3 + 0.045 t^2 - 0.375 t + 10$.

Sketch the graph of $N$ against $t$ for the first $40$ months, clearly indicating any intercepts and local extrema within this domain.

Find the maximum number of visitors (to the nearest whole number) during the first $40$ months.

Find the time(s) when the number of visitors is above $1200$. Give your answer in months, correct to two decimal places.

A civil engineer is designing a parabolic arch for a pedestrian bridge. The arch starts at ground level at the origin (0,0) and reaches its maximum height of 5 meters at a horizontal distance of 10 meters from the start. The shape of the arch can be modelled by the function $f(x) = ax^2 + bx + c$, where $f(x)$ is the height of the arch above the ground in meters and $x$ is the horizontal distance in meters from the start of the arch.

(a) Find the total horizontal span of the bridge, i.e., the x-coordinate where the arch meets the ground again.

(b) Determine the values of $a$, $b$, and $c$.

(c) Write down the equation of the axis of symmetry of the parabolic arch.

A company is designing an open-top storage container with a square base. The side length of the base is $x$ cm and the height is $h$ cm. The container needs to have a volume of $500$ cm$^3$.

Explain why $x^2 h = 500$.

Rearrange the equation in part (a) to make $h$ the subject.

Write down an expression for the surface area, $A$, of the open-top container.

Show that this can be written as

$A = x^2 + \frac{2000}{x}$

Plot the graph of $A = x^2 + \frac{2000}{x}$ for $x > 0$.

Find the minimum surface area and the value of $x$ when this occurs.

The graph of $y = f(x)$ represents the concentration of a certain chemical in a solution (in mol/L) at time $x$ (in hours). The graph passes through the points $(1, 10)$ and $(3, 4)$, and has a horizontal asymptote at $y = 2$.

Let $g(x) = 3f(x + 1) - 5$ represent the concentration in a different experiment.

Find $g(0)$.

On a new set of axes, sketch the graph of $y = g(x)$, clearly indicating its horizontal asymptote and the $y$-intercept. You do not need to show the graph of $f(x)$.

(a) A designer is creating a prototype for a decorative vase. The cross-section of the vase can be modelled by the function $f(x) = x\sqrt{4-x^2}$, for $-2 \le x \le 2$.

Sketch the graph of $y = f(x)$ on the following pair of axes.

(b) The region enclosed by the graph of $y = f(x)$ and the x-axis is rotated $360^\circ$ about the x-axis to form the body of the vase.

(i) Write down an integral that represents the volume of this vase.

(ii) Calculate the value of this integral.

(c) The designer decides to create a new, larger version of the vase, $y = g(x)$, by applying the following transformations to the original cross-section $y = f(x)$:

• A horizontal stretch by a scale factor of 3, parallel to the x-axis.

• A vertical stretch by a scale factor of 0.75, parallel to the y-axis.

Find the volume of this new vase.