Sketching graphs: notes and practice questions

(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.

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 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 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.

[Maximum mark: 9]

Consider the function $f(x) = 2x^2 - \frac{8}{x}$, $x \neq 0$. The graph of $f$ for $0 < x \le 4$ is shown on the following axes.

(a) (i) Sketch the graph of $f$, for $-4 \le x < 0$, on the same axes.

(ii) Write down the x-coordinate of the local minimum point.

(b) The company wants to find the control parameter values $x$ for which the efficiency $f(x)$ is 10 units per hour. Use your graphic display calculator to find the solutions to the equation $f(x) = 10$.

(c) Write down the equation of the vertical asymptote for the graph of $f$.

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 slope field for the differential equation $\frac{dy}{dx} = y - x^2$ is shown.

Some of the solutions to the differential equation have a local maximum point and a local minimum point.

Write down the equation of the curve on which all these maximum and minimum points lie.

Sketch this curve on the slope field.

The solution to the differential equation that passes through the point (0, 1) has both a local maximum point and a local minimum point.

On the slope field, sketch the solution to the differential equation that passes through (0, 1).

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.

A metal rod is being heated in an industrial furnace. The temperature, $T$ (in $^{\circ}\text{C}$ ), of the rod $t$ minutes after it is placed in the furnace is modelled by the function $T = 200 - 120 \times b^{-t}$, where $t$ is the time in minutes.

(a) Find the initial temperature of the metal rod when it is placed in the furnace.

The metal rod reaches a temperature of $140^{\circ}\text{C}$ after $1$ minute.

(b) Find the value of $b$.

Draw the graph of $T = 200 - 120 \times 2^{-t}$ on the following set of axes.

State a mathematical reason why the model predicts the temperature of the metal rod will never reach $200^{\circ}\text{C}$.

(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.

The graph of the function $f$ is shown in the following diagram, representing the concentration of a certain chemical in a solution over time $x$ (in minutes).

(a) Write down the concentration of the chemical at time $x=1$ minute.

(b) On the same axes, sketch the graph of $y = f^{-1}(x)$, representing the time required to reach a certain concentration.

A different chemical's concentration, $h(x)$, is modeled by the function $h(x) = 2x - 3$, where $x$ is time in minutes.

(c) Find an expression for $h^{-1}(x)$.

(d) Find a value of $x$ where $f^{-1}(x) = h^{-1}(x)$.

(a) When the profit is zero, find the possible number of units produced, $x$.

(b) Determine the positive values of profit, $P$, for which there is only one positive value of $x$ (units produced).

A printing company has received a large order for flyers. The number of flyers remaining to be printed, $N$, is modelled by the equation $N(t) = 25000 - 500t$, where $t$ is the time in hours since printing began.

Calculate the time it will take to complete the entire order of flyers.

Determine a reasonable domain and range for this model in the context of the problem.

Sketch a graph of $N(t)$ for the domain found in part (b).

Find $N(5)$ and interpret its meaning in the context of the problem.

A chemical engineer is studying the concentration of an intermediate product, $C$, in a reaction vessel. The change in concentration over time $t$ (in minutes) is modelled by the second order differential equation:

$\frac{d^2C}{dt^2} + 4\frac{dC}{dt} + 3C = 0$

where $t \ge 0$. It is known that when $t = 0$, the initial concentration is $C = 0$ and the rate of change of concentration is $\frac{dC}{dt} = 2$.

Show that the system of coupled first order equations:

$\frac{dC}{dt} = y$

$\frac{dy}{dt} = -3C - 4y$

can be written as the given second order differential equation.

Find the eigenvalues of the system of coupled first order equations given in part (a).

Hence find the exact solution of the second order differential equation, given the initial conditions $C(0) = 0$ and $\frac{dC}{dt}(0) = 2$.

Sketch the graph of $C$ against $t$ for $t \ge 0$, labelling the maximum point of the graph with its coordinates.

If the concentration of the intermediate product $C$ exceeds $0.2$ arbitrary units, the reaction needs to be monitored closely. Use the model to calculate the total amount of time (in minutes) during which the reaction needs to be monitored closely.

The chemical engineer decides to monitor the reaction for 15% longer than the time found from the model in part (e).

Write down one reason, with reference to the context, to support this decision.

The two graphs illustrate relationships within a specialized coffee roasting company.

Write down which graph represents direct variation, with a reason.

(b) Find and interpret the gradient of each graph in context.

(c) Find an equation for the roasting time $T$ (in minutes) as a function of the batch size, $S$ (in number of bags), and use it to predict how long the roasting process will be for a batch of $1500$ bags.

A marine engineer is modelling the vertical displacement, $h$ meters, of a buoy from its equilibrium position at time $t$ minutes after being disturbed by a wave. The motion is described by the differential equation:

$\frac{\text{d}^2 h}{\text{d}t^2} + 6\frac{\text{d}h}{\text{d}t} + 8h = 0$.

This equation can be rewritten as a system of coupled first-order differential equations:

$\frac{\text{d}h}{\text{d}t} = y$

$\frac{\text{d}y}{\text{d}t} = -8h - 6y$.

Find the general solution for $h$.

Initially, the buoy is at its equilibrium position ($h = 0$) and has an initial upward velocity of $2$ m/min ($\frac{\text{d}h}{\text{d}t} = 2$).

Find an expression for $h$ in terms of $t$.

Sketch $h$ against $t$ in the interval $0 \le t \le 4$.

The engineer is interested in the maximum upward displacement of the buoy.

Find the time, in minutes, when this maximum displacement occurs.

Calculate the maximum upward displacement of the buoy.

An improved model for the buoy's motion, considering an external periodic force, is given by:

$\frac{\text{d}^2 h}{\text{d}t^2} + 6\frac{\text{d}h}{\text{d}t} + 8h = h \cos t$.

The same initial conditions as in part (b.i) apply ($h(0) = 0$ and $\frac{\text{d}h}{\text{d}t}(0) = 2$).

Use Euler's method with a $t$-interval of $0.1$ to predict the value of $h$ when $t = 1$.

(a) The height, $h$ metres, of a small rocket above the ground, $t$ seconds after launch, is modelled by the function $h(t) = t^2 - 4t + 3$. Sketch the graph of $h(t)$ for $0 \le t \le 4$. Clearly label any axis intercepts and the vertex.

(b) A company's daily profit, $P$ (in thousands of dollars), depends on the number of units, $x$, it produces, according to the function $P(x) = -x^2 + 2x + 3$. Sketch the graph of $P(x)$ for $0 \le x \le 4$. Clearly label any axis intercepts and the vertex.

A suspension bridge arch can be modelled by a parabolic curve. The arch starts at one end of the river bank at $x = 1$ meter and ends at the other bank at $x = 7$ meters, where $x$ represents the horizontal distance from a central reference point. The highest point of the arch is $9$ meters above the water level.

(a) Determine the horizontal position, in meters, of the highest point of the arch.

(b) Find the equation of the parabolic curve in the form $y = ax^2 + bx + c$.