Sketching graphs: notes and practice questions

A pharmaceutical company is testing a new drug. The concentration of the drug in a patient's bloodstream, $C$, in mg/L, $t$ hours after administration, is modeled by the function $C(t) = 10t \text{e}^{-0.5t}$, for $0 \le t \le 10$.

Sketch the graph of $C(t)$ on the grid below.

(b) Find the time, in hours, at which the concentration of the drug in the bloodstream is at its maximum.

A biologist is modelling the growth of two different bacterial colonies. The first colony, A, grows such that its population at time $x$ is given by $P_A(x) = a^x$, where $a$ is a growth factor and $x \ge 0$. The second colony, B, grows linearly such that its population at time $x$ is $P_B(x) = x$.

Consider the cases where the growth factor $a = 2$ and $a = 10$. On the same set of axes, sketch the following three graphs for $x \ge 0$:

$y = 2^x$

$y = 10^x$

$y = x$

Clearly label each graph with its equation and state the coordinates of any non-zero $y$-axis intercepts.

In parts (b) and (c), consider the case where the growth factor $a = e$.

Use calculus to find the minimum value of the expression $e^x - x$, justifying that this value is a minimum.

Hence deduce that $e^x > x$ for all $x \in \mathbb{R}$.

There exist values of $a$ for which the graph of $y = a^x$ and the line $y = x$ have different numbers of intersection points. The following table gives three intervals for the value of $a$.

By investigating the graph of $y = a^x$ for different values of $a$, write down the values of $p, q$ and $r$.

In parts (e) and (f), consider $a \in \mathbb{R}^+, a \neq 1$.

For $1.4 \leq a \leq 1.5$, a value of $a$ exists such that the line $y = x$ is a tangent to the graph of $y = a^x$ at a point P.

Find the exact coordinates of P and the exact value of $a$.

Write down the exact set of values for $a$ such that the graphs of $y = a^x$ and $y = x$ have

(i) two intersection points;

(ii) no intersection points.

The amplitude of a sound wave, $A(t)$, is modeled by a composite function $(f \circ g)(t)$, where $t$ is time in seconds. The initial signal is given by $g(t) = \sin t$, and the amplification stage is defined by $f(x) = x^3 - 3x$.

(a) Find $(f \circ g)(t)$.

On the following grid, sketch the graph of $y = (f \circ g)(t)$ for $-2 \le t \le 2$. Write down and clearly label the coordinates of any local maximum or minimum points.

A civil engineer is analyzing the structural integrity of a new bridge design. The deflection of a certain point on the bridge, $D(x)$, in millimeters, is modeled by the function $D(x) = \frac{2x+1}{x^2-4}$, where $x$ represents the horizontal distance in meters from a central support. The model is valid for $x \in \mathbb{R}$, $x\neq p$, $x\neq q$.

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

Find an expression for $D'(x)$.

The graph of $y = D(x)$ has exactly one point of inflexion.

Find the x-coordinate of the point of inflexion.

Sketch the graph of $y = D(x)$ for $-4 \leq x \leq 4$, showing the values of any axes intercepts, the coordinates of any local maxima and local minima (if they exist), and giving the equations of any asymptotes.

Consider a related model for stress distribution, $S(x) = \frac{x^2-4}{2x+1}$ for $x \in \mathbb{R}$, $x \neq -\frac{1}{2}$.

Find the equations of all the asymptotes on the graph of $y = S(x)$.

The engineer needs to identify the regions where the bridge deflection $D(x)$ is less than $1$ mm. Solve $D(x) < 1$ for $x \in \mathbb{R}$.

Consider the function $f(x) = e^x - 3x - 6$.

On the following axes, sketch the graph of $f$ for $-3 \le x \le 3$.

The function $g$ is defined by $g(x) = e^{2x} - 6x - 10$.

The graph of $g$ is obtained from the graph of $f$ (from part a) by a horizontal stretch with scale factor $k$, followed by a vertical translation of $c$ units.

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

A team of engineers is designing a new roller coaster ride. The path of a certain section of the ride can be modelled by the function $g(x)=\frac{x^2 + 3x - 10}{x-4}$, where $x$ is the horizontal distance in metres from the starting point and $g(x)$ is the vertical height in metres. The domain of the function is $x \in \mathbb{R}, x\neq 4$.

Find the coordinates where the path of the roller coaster crosses the horizontal ground (x-axis).

Find the coordinates where the path of the roller coaster crosses the vertical axis (y-axis).

Write down the equation of the vertical asymptote of the graph of $g$.

The oblique asymptote of the graph of $g$ can be written as $y = ax + b$ where $a, b \in \mathbb{Z}$.

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

Sketch the graph of $g$ for $-20 \le x \le 20$, clearly indicating the points of intersection with each axis and any asymptotes.

Engineers want to analyse the inverse of the roller coaster's height function, $h(x) = \frac{1}{g(x)}$.

Express $h(x)$ in partial fractions.

Hence find the exact value of $\int_{0}^{1} h(x) dx$, expressing your answer as a single logarithm.

A scientist is studying the growth of a certain bacterial culture. The population, in thousands, at time $t$ hours is modeled by the function $P(t) = e^t - 4t - 5$.

On the following axes, sketch the graph of $P(t)$ for $-2 \leq t \leq 4$.

Another bacterial culture, observed under different conditions, has its population modeled by the function $Q(t) = e^{2t} - 8t - 12$.

The graph of $Q(t)$ is obtained from the graph of $P(t)$ by a horizontal stretch with scale factor $k$, followed by a vertical translation of $c$ units.

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

A designer is creating a decorative glass container shaped like a dome. The outer profile of the container can be modelled by the function $f(x) = \sqrt{9-x^2}$, where $0 \le x \le 3$ and $x$ and $y$ are measured in metres.

Sketch the curve $y = f(x)$, clearly indicating the coordinates of the endpoints.

Show that the inverse function of $f$ is given by $f^{-1}(x) = \sqrt{9-x^2}$.

State the domain and range of $f^{-1}$.

The container is formed by rotating the curve $y = f(x)$ by $2\pi$ about the y-axis. Show that the volume, $V \text{ m}^3$, of liquid in the container when it is filled to a height of $h$ metres is given by $V = \pi \left( 9h - \frac{1}{3}h^3 \right)$.

Hence, determine the maximum volume of the container.

At $t = 0$, the container is empty. Liquid is then added to the container at a constant rate of $0.5 \text{ m}^3\text{s}^{-1}$.

Find the time it takes to fill the container to its maximum volume.

Find the rate of change of the height of the liquid when the container is filled to half its maximum volume.

The functions $f$ and $g$ are defined by $f(x) = 2x - x^3$ and $g(x) = \cos x$.

(a) Find $(f \circ g)(x)$.

On the following grid, sketch the graph of $y = (f \circ g)(x)$ for $0 \leq x \leq \pi$. Write down and clearly label the coordinates of any local maximum or minimum points.

A pharmaceutical company is testing a new drug. The concentration of the drug, $C$, in the bloodstream of a patient, in micrograms per millilitre ($\mu\text{g/mL}$), $t$ hours after administration, is modelled by the function $C(t) = 5te^{-0.2t}$, for $0 \le t \le 15$.

Sketch the graph of $C(t)$ for $0 \le t \le 15$, clearly indicating the coordinates of the initial concentration point $A$, the maximum concentration point $M$, and the concentration point $B$ at $t=15$ hours.

State the range of the concentration $C(t)$ during the observed period.

Find the equation of the straight line connecting the initial concentration point $A$ and the concentration point $B$ at $t=15$ hours.

Show that the rate of change of the drug concentration is given by $C'(t) = (5-t)e^{-0.2t}$.

At a certain time, the rate of change of the drug concentration is parallel to the line AB. Find the equation of the tangent line to the graph of $C(t)$ at this time. Give all coefficients in your equation correct to $3$ significant figures.

Calculate the area of the region enclosed by the graph of $C(t)$ and the line AB.

A scientist is modeling the growth of a certain bacterial colony. The population size, $P(t)$, at time $t$ hours, is given by the function $f(t) = 2^t - 2t - 5$ for $-3 \leq t \leq 4$.

(a) On the following axes, sketch the graph of $f$ for $-3 \leq t \leq 4$.

Another bacterial colony, under different conditions, has its population size modeled by $g(t) = 4^t - 4t - 8$. It is observed that the growth curve of this second colony can be obtained from the graph of $f$ by a horizontal stretch with scale factor $k$, followed by a vertical translation of $c$ units.

(b) Find the value of $k$ and the value of $c$.

The rate of change of the concentration, $C$, of a reactant in a chemical process at time $t$ is modelled by the differential equation $\frac{dC}{dt} = 2t - \frac{C}{t}$, for $t > 0$. Given that the initial concentration is $C(1) = 2$, use Euler's method with a step size of $h = 0.2$ to find an approximate value of $C$ when $t = 1.6$.

Solve the differential equation $\frac{dC}{dt} = 2t - \frac{C}{t}$ using an appropriate analytical method, and find the exact value of $C$ when $t = 1.6$.

Sketch the approximate values you found in part (a) and the graph of the function you found in part (b) on the same coordinate axes. Use your sketch to explain why your answer in part (a) is greater than or less than the value in part (b).

(a) Using your GDC, sketch the curve of $y = -0.8x^2 + 3.2x + 4.5$ for $-1 \le x \le 5$. Clearly label any intercepts and the vertex.

(b) Write down the coordinates of the points where the curve intersects the x-axis and the y-axis. Give your answers correct to three significant figures.

(c) Write down the range of $y$.

A company is designing a new power generator. The power output, $P$, in megawatts (MW), of a prototype generator at time $t$ hours after startup is modelled by the function $P(t) = t^3 - 3ct + K$, where $t \in \mathbb{R}$, $c$ is a control parameter, and $K$ is a constant representing the initial power. For parts (a) to (e), assume $K = 4$.

On separate axes, sketch the graph of $P = P(t)$ showing the value of the $P$-intercept and the coordinates of any points with zero gradient, for

(i) $c = 1$;

(ii) $c = 2$.

Write down an expression for $P'(t)$.

Hence, or otherwise, find the set of values of $c$ such that the graph of $P = P(t)$ has

(i) a point of inflexion with zero gradient;

(ii) one local maximum point and one local minimum point;

(iii) no points where the gradient is equal to zero.

Given that the graph of $P = P(t)$ has one local maximum point and one local minimum point, show that

(i) the $P$-coordinate of the local maximum point is $2c^{\frac{3}{2}} + 4$;

(ii) the $P$-coordinate of the local minimum point is $-2c^{\frac{3}{2}} + 4$.

Hence, for $c > 0$, find the set of values of $c$ such that the graph of $P = P(t)$ has

(i) exactly one $t$-axis intercept;

(ii) exactly two $t$-axis intercepts;

(iii) exactly three $t$-axis intercepts.

Consider a modified power output function $Q(t) = t^3 - 3ct + d$ for $t \in \mathbb{R}$ and where $c, d \in \mathbb{R}$. Find all conditions on $c$ and $d$ such that the graph of $P = Q(t)$ has exactly one $t$-axis intercept, explaining your reasoning.

This question explores the characteristics of a company's profit function, $P_n(x) = x^n(L-x)^n$, where $x$ represents the number of units produced (in thousands), $L$ is the maximum production capacity (in thousands of units), with $L \in \mathbb{R}^+$ and $n \in \mathbb{Z}^+$.

For parts (a) and (b), consider the case where $L = 4$.

Consider $P_1(x) = x(4-x)$.

Sketch the graph of $y = P_1(x)$, clearly indicating the values of any axes intercepts and the coordinates of any local maximum or minimum points.

Consider $P_n(x) = x^n(4-x)^n$, where $n \in \mathbb{Z}^+$, $n > 1$.

Use your graphic display calculator to explore the graph of $y = P_n(x)$ for:

• the odd values $n = 3$ and $n = 5$;
• the even values $n = 2$ and $n = 4$.

Hence, copy and complete the following table:

Number of local maximum points | Number of local minimum points | Number of points of inflexion with zero gradient

---|---|---

$n=3$ and $n = 5$ | |

$n=2$ and $n = 4$ | |

Now consider $P_n(x) = x^n(L-x)^n$ where $L \in \mathbb{R}^+$ and $n \in \mathbb{Z}^+$, $n > 1$.

Show that $P_n'(x) = nx^{n-1}(L-2x)(L-x)^{n-1}$.

State the three solutions to the equation $P_n'(x) = 0$.

Show that the point $(\frac{L}{2}, P_n(\frac{L}{2}) )$ on the graph of $y = P_n(x)$ is always above the horizontal axis.

Hence, or otherwise, show that $P_n'(\frac{L}{4}) > 0$, for $n \in \mathbb{Z}^+$.

By using the result from part (f) and considering the sign of $P_n'(-1)$, show that the point $(0, 0)$ on the graph of $y = P_n(x)$ is

(i) a local minimum point for even values of $n$, where $n > 1$ and $L \in \mathbb{R}^+$;

(ii) a point of inflexion with zero gradient for odd values of $n$, where $n > 1$ and $L \in \mathbb{R}^+$.

Consider the graph of $y = x^n(L-x)^n - k$, where $n \in \mathbb{Z}^+$, $L \in \mathbb{R}^+$ and $k \in \mathbb{R}$.

State the conditions on $n$ and $k$ such that the equation $x^n(L-x)^n = k$ has four distinct solutions for $x$.

A manufacturing company models its production efficiency, $E(t)$, at time $t$ (in hours) after a new process is implemented, using the function $E(t) = \frac{2t^2 + 3t}{t}$, for $t > 0$.

(a) Sketch the graph of $E(t)$ for $t > 0$. Clearly indicate any discontinuities.

(b) The company wants to understand the theoretical efficiency rate as time approaches zero, even though the model is strictly for $t > 0$. Find $\lim_{t \to 0} E(t)$ numerically.

(a) A cubic equation with real coefficients has roots $z = -2$ and $z = 3 + 2i$.

(i) Write down the third root.

(ii) Verify that the real part of the complex roots is 3.

(b) Let $f(x) = (x+2)(x^2 - 6x + 13)$. Show that the line $y = 4x + 8$ is tangent to the curve $y = f(x)$ at the point $A(3, 20)$.

(c) Sketch the curve $y = f(x)$ and the tangent to the curve at point $A$, clearly showing where the tangent crosses the $x$-axis.

(d) Let a general cubic function be given by $g(x) = (x-r)((x-a)^2 + b^2)$, where $a, b, r \in \mathbb{R}$ and $b > 0$.

(i) Show that $g'(x) = (x-a)^2 + b^2 + 2(x-r)(x-a)$.

(ii) Hence, or otherwise, prove that the tangent to the curve $y = g(x)$ at the point $A(a, g(a) )$ intersects the $x$-axis at the point $R(r, 0)$.

(e) Deduce from part (d)(ii) that the complex roots of the equation $(z - r)((z-a)^2 + b^2) = 0$ can be expressed as $a \pm i\sqrt{g'(a)}$.

(f) Consider a cubic function of the form $g(z) = (z-r)((z-a)^2 + b^2) = 0$. Given that $r = -1$ and $b = 3$, and the $y$-coordinate of the point of tangency $A(a, g(a) )$ is $45$.

(i) Use this information to determine the roots of the corresponding equation for $z \in \mathbb{C}$.

(ii) State the coordinates of the complex conjugate root with negative imaginary part, $C_2$, in the Argand diagram.

(g) The point of inflection for the curve $y=g(x)$ is denoted by $P$.

(i) Show that the $x$-coordinate of $P$ is $\frac{1}{3}(2a+r)$. You are not required to demonstrate a change in concavity.

(ii) Hence describe numerically the horizontal position of point $P$ relative to the horizontal positions of the points $R$ and $A$.

(h) Consider the special case where $a=r$.

(i) Sketch the curve $y = (x-r)((x-a)^2 + b^2)$ for $a=r=1$ and $b=2$.

(ii) For $a=r$ and $b > 0$, state in terms of $r$, the coordinates of points $P$ and $A$.

A hot air balloon takes off from a launch platform. Its height above the ground, $h$ metres, $t$ minutes after takeoff, is modelled by the function $h(t) = 24t - 0.8t^2$.

Determine the height of the hot air balloon after 5 minutes and after 10 minutes.

Find an expression for the vertical velocity of the hot air balloon, $v(t)$, in terms of $t$.

Calculate the time at which the hot air balloon reaches its maximum height.

Determine the total duration of the balloon's flight before it lands.

Sketch the graph of $h = h(t)$ for the duration of the flight, clearly indicating the intercepts with the $t$-axis and the coordinates of the maximum point. State the domain of the model.

Find the vertical velocity of the hot air balloon at the instant it lands.

Show that the vertical acceleration of the hot air balloon is constant, stating its value.

An architect is designing a decorative archway for a garden entrance. The shape of the archway's inner curve is modelled by the equation $y^2 = x^3 + ax + b$, where $x$ and $y$ are in meters.

(a.i) On the same set of axes, sketch the curve $C_1: y^2 = x^3$ for $x \ge 0$, clearly indicating any points of intersection with the coordinate axes. Assume a suitable range for $x$ and $y$ that shows the key features.

(a.ii) On the same set of axes, sketch the curve $C_2: y^2 = x^3 + 2x^2$ for $x \ge -2$, clearly indicating any points of intersection with the coordinate axes. Assume a suitable range for $x$ and $y$ that shows the key features.

(a.iii) By considering each curve from part (a), identify two key features that would distinguish $C_1$ from $C_2$.

(b.i) For the curve $C_2: y^2 = x^3 + 2x^2$, show that $\frac{dy}{dx} = \pm \frac{3x^2 + 4x}{2\sqrt{x^3 + 2x^2}}$ for $x > -2, x \ne 0$.

(b.ii) Find the $x$-coordinates of any local maximum or minimum points on $C_2: y^2 = x^3 + 2x^2$.

(c) The curve $C_2: y^2 = x^3 + 2x^2$ has points of inflexion. Find the $x$-coordinate of these points, giving your answer in the form $x = \frac{p \pm \sqrt{q}}{r}$ where $p, q, r \in \mathbb{Z}$.

Consider a different archway design modelled by the curve $C_3: y^2 = x^3 + 2$, for $x \ge -\sqrt[3]{2}$.

(d.i) The point P(-1, -1) is a rational point on $C_3$. Find the equation of the tangent to $C_3$ at P.

(d.ii) This tangent intersects $C_3$ at another rational point Q. Find the coordinates of Q, expressing each coordinate as a fraction.

(e) The point S(-1, 1) also lies on $C_3$. The line [QS] intersects $C_3$ at a further point R. Determine the coordinates of R.