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

Consider the function $f(x) = \frac{7x + 6}{5x - 3}$, for $x \neq 3$.

$a$ Find the $x$ and $y$ intercepts.

$b$ Find the equations of the vertical and horizontal asymptotes.

$c$ Find the inverse function $f^{- 1}(x)$.

Consider the functions $f(x) = -3\cos x + 5$ and $g(x) = -3\cos\left(x+\frac{\pi}{2}\right) + 5 - k$, where $x \in \mathbb{R}$ and $k > 0$.

The graph of $g$ is obtained by two transformations of the graph of $f$.

Describe these two transformations.

The $y$-intercept of the graph of $g$ is at $(0, p)$.

Given that the maximum value of $g(x)$ is less than or equal to 1, find the largest possible value of $p$.

A function $f$ is defined by $f(x) = \frac{4x-1}{2x+3}$, where $x \in \mathbb{R}, x \neq -\frac{3}{2}$.

The graph of $y = f(x)$ is shown below.

(a) Write down the equation of the horizontal asymptote.

Consider the function $g(x) = mx - \frac{1}{3}$, where $m \in \mathbb{R}, m \neq 0$.

(i) Write down the number of solutions to $f(x) = g(x)$ for $m < 0$.

(ii) Determine the value of $m$ such that $f(x) = g(x)$ has only one solution for $x$.

(iii) Determine the range of values for $m$ for which $f(x) = g(x)$ has two distinct solutions for $x \le 0$.

Consider the function $h(x) = \frac{8-2x}{x+1}$.

(a) State the largest possible domain for the function $h$.

(b) State the largest possible range for the function $h$.

(c) Find the coordinates of any points where the curve $y = h(x)$ intersects the $x$ and $y$-axes.

The function $g$ is defined by $g(x)=\frac{5x-10}{3x+6}$ for $x \in \mathbb{R}$, $x \neq -2$.

(a) Find the zero of $g(x)$.

(b) For the graph of $y = g(x)$, write down the equation of

(i) the vertical asymptote;

(ii) the horizontal asymptote.

(c) Find $g^{-1}(x)$, the inverse function of $g(x)$.

Consider the functions $f(x) = -2\cos(x) + 5$ and $g(x) = -2\cos(x - \frac{\pi}{4}) + 5 - k$, where $x \in \mathbb{R}$ and $k > 0$.

The graph of $g$ is obtained by two transformations of the graph of $f$.

(a) Describe these two transformations.

The $y$-intercept of the graph of $g$ is at $(0, s)$.

(b) Given that the maximum value of $g(x)$ is 1, find the value of $s$.

Let $g(x) = \frac{3x+6}{x-2}$.

(a) Find the coordinates of the $x$-intercept.

(b) Write down the equation of the vertical asymptote.

(c) Find the equation of the horizontal asymptote.

The function $g$ is defined by $g(x)=\frac{5x-10}{3x+6}$ for $x \in \mathbb{R}, x \ne -2$.

(a) Find the zero of $g(x)$.

(b) For the graph of $y = g(x)$, write down the equation of

(i) the vertical asymptote;

(ii) the horizontal asymptote.

(c) Find $g^{-1}(x)$, the inverse function of $g(x)$.

A function is defined by $f(x) = \frac{1}{2}x^2 + x + 4$. The following diagram shows part of the graph of $f$.

The graph has a vertex at V and intersects the y-axis at point P.

(a) Find the coordinates of the vertex V.

(b) Write down the coordinates of the y-intercept, P.

(c) The line L is the normal to the graph of $f$ at point P. Find the equation of L, giving your answer in the form $y=mx+c$.

(d) The line L intersects the graph of $f$ at a second point, Q. Calculate the distance between P and Q.

A function is defined by $f(x) = \frac{ax+b}{x+d}$.

(a) The asymptotes of the graph of $y=f(x)$ are a vertical line and a horizontal line which intersect at the point $(-2, 4)$. Find the value of $a$ and the value of $d$.

(b) The graph of $y=f(x)$ passes through the point $(0, -3)$. Find the value of $b$.

A function $f$ is defined by $f(x) = \frac{\ln(x)}{x}$, for $x > 0$.

The following diagram shows part of the graph of $f$.

(a) Find the coordinates of the x-intercept of the graph of $f$.

(b) Find $f'(x)$.

The graph of $f$ has a local maximum at point M.

(c) Hence, find the exact coordinates of M.

(d) (i) Show that $f''(x) = \frac{2\ln(x) - 3}{x^3}$.

The graph of $f$ has a point of inflection at point P.

(d) (ii) Hence, find the exact coordinates of P.

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 $x \in \mathbb{R}$.

A function $f$ is defined by $f(x) = \frac{a}{x-2} + k$, for $x \neq 2$.

(a) The graph of $y=f(x)$ has a horizontal asymptote with equation $y=1$. Write down the value of $k$.

(b) The graph of $y=f(x)$ has an $x$-intercept at $5$. Find the value of $a$.

(c) Find the coordinates of the $y$-intercept of the graph of $y=f(x)$.

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 company models the efficiency of a new production line by the function $f(x) = \frac{2x+3}{x+2}$, where $f(x)$ represents the output rate (in units per hour) after $x$ hours of operation. For mathematical analysis, we consider the function over its natural domain $x \in \mathbb{R}$, $x \ne -2$.

For the graph of f,

write down the equation of the vertical asymptote;

find the equation of the horizontal asymptote.

Find $f^{-1}(x)$.

Using an algebraic approach, show that the graph of $f^{-1}$ is obtained by a reflection of the graph of f in the y-axis followed by a reflection in the x-axis.

The graphs of f and $f^{-1}$ intersect at $x = p$ and $x = q$, where $p < q$.

Find the value of p and the value of q.

Hence, find the area enclosed by the graph of f and the graph of $f^{-1}$.

Consider the function $f(x) = 5 - \frac{1}{x+b}$, where $b$ is a constant and $x \neq -b$.

(a) Write down the equation of the horizontal asymptote of the graph of $f$.

(b) The vertical and horizontal asymptotes of the graph of $f$ intersect at the point $(3, 5)$. Find the value of $b$.

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 $c$.

The growth of a bacterial colony, $B$, in a petri dish can be modelled by the logistic differential equation

$\frac{\text{d}B}{\text{d}t} = k B \left(1 - \frac{B}{N}\right)$

where $t$ is the time measured in hours and $k, N$ are positive constants.

The constant $N$ represents the maximum number of bacteria the petri dish can sustain indefinitely due to limited nutrients.

In the context of this bacterial growth model, interpret the meaning of $\frac{\text{d}B}{\text{d}t}$.

Show that $\frac{\text{d}^2B}{\text{d}t^2} = k^2B\left(1-\frac{B}{N}\right)\left(1-\frac{2B}{N}\right)$.

Hence show that the bacterial colony will grow at its maximum rate when $B = \frac{N}{2}$. Justify your answer.

Hence determine the maximum value of $\frac{\text{d}B}{\text{d}t}$ in terms of $k$ and $N$.

Let $B_0$ be the initial number of bacteria.

By solving the logistic differential equation, show that its solution can be expressed in the form

$kt = \ln\left(\frac{B(N-B_0)}{B_0(N-B)}\right)$.

After 5 hours, the number of bacteria is $2B_0$. It is known that $N = 3B_0$.

Find the value of $k$ for this bacterial growth model.

A freshly baked cake is taken out of an oven and placed on a wire rack to cool in a kitchen.

The temperature of the cake, $T^\circ\text{C}$, after $t$ minutes is modelled by the function $T(t) = 21 + 144\text{e}^{-0.035t}$, for $t \ge 0$.

(a) Find the initial temperature of the cake.

(b) Find the temperature of the cake after 20 minutes.

(c) Write down the temperature of the kitchen.

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 $c$.