Exam-Style Questions.

At depths below 900 metres, the temperature of the water in the sea is given by the formula:

Where \(x\) is the depth in metres and the temperature of the water is T °C.

(a) A fish swims downwards from a depth of 1000 metres to a depth of 1500 metres. How many degrees colder is the sea water at 1500m than 1000m?

(b) Roughly sketch the shape of a graph that shows that T is inversely proportional to \(x\).

Worked Solution

The images below show a graphic display calculator screen with different functions displayed as graphs.

a) Which function is trigonometric?

b) Which function is inversely proportional to \(x\)?

c) Which function is exponential?

d) Which function is proportional to \(x^3\)?

Worked Solution

Match the equation with the letter of its graph

Worked Solution

The following table shows corresponding values for two variables \(x\) and \(y\).

(a) If \(y\) is inversely proportional to the square of \(x\) find an equation for \(y\) in terms of \(x\).

(b) Find the positive value for \(x\) when \(y = 20\).

Worked Solution

(a) Sketch a graph on the axes below left that shows that \(y\) is directly proportional to \(x\).

(b) Sketch the graph of \(y = -x^3\) on the axes above right.

(c) It is possible to draw many rectangles that have area \(48 cm^2\).

Draw a graph to show the relationship between length and width for rectangles with area \(48 cm^2\) and sides less than \(50cm\).

Worked Solution

At a constant temperature, the volume of a gas \(V\) is inversely proportional to its pressure \(p\). By what percentage will the pressure of a gas change if its volume increases by 15% ?

Worked Solution

(a) Sketch the graph of \( y = f(x) \) for values of \( x \) between \(-5\) and \(5\) given that:

$$ f(x) = \frac{1}{x-1} - x $$

(b) Write down the equations of any asymptotes parallel to the \( y \)-axis.

(c) Find the coordinates of the y-axis intercept.

(d) Find the coordinates of the x-axis intercepts.

(e) Find a value of \( k \) for which \( g(x) = kx \) does not intercept \( f(x)\).

(f) Draw the graph of \(h(x)=x-3\) on the same set of axes.

(g) Solve the equation \( f(x) = h(x) \).

(h) Find the solutions to the inequality \( f(x) > h(x) \).

Worked Solution

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

(a) Find the zero of \( f(x) \).

(b)For the graph of \( y = f(x) \), write down the equation of the asymptotes;

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

Worked Solution

The function \(f\) is defined by:

(a) Write down the equation of the vertical asymptote of the graph of \(f\).

(b) Write down the equation of the horizontal asymptote of the graph of \(f\).

(c) Find the coordinates of the \(x\)-axis and \(y\)-axis intercepts.

(d) Sketch the graph of \(f\).

Worked Solution

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

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

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

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

Find the coordinates of the point where the graph of \(y\) intersects:

(c) the y-axis;

(d) the x-axis.

Worked Solution

Let \(f(x)=5x^2-20x+k\). The equation \(f(x)=0\) has two equal roots.

(a) Write down the value of the discriminant.

(b) Hence, show that \(k=20\).

The graph of \(f\) has its vertex on the x-axis.

(c) Write down the solution of \(f(x)=0\).

(d) Find the coordinates of the vertex of the graph of \(f\).

The function can be written in the form \(f(x)=a(x-h)^2+j\).

(e) Find the value of \(a\).

(f) Find the value of \(h\).

(g) Find the value of \(j\).

(h) The graph of a function \(g\) is obtained from the graph of \(f\) by a reflection in the x-axis, followed by a translation by the vector \(\begin{pmatrix} 0 \\ 3 \\ \end{pmatrix} \). Find \(g\), giving your answer in the form \(g(x)=Ax^2+Bx+C\).

Worked Solution

The diagram on the right shows a sketch of part of the graph:

(a) State the range of \(f\).

(b) Solve the equation \(f(x) = \frac{x}{3} + 20 \).

(c) Given that the equation \(f(x) = k\), where \(k\) is a constant, has two distinct roots, state the set of possible values for k.

Worked Solution

Let \(f(x) = \frac{9x-3}{bx+9}\) for \(x \neq -\frac9b, b \neq 0\).

(a) The line \(x = 3\) is a vertical asymptote to the graph of \(f\). Find the value of b.

(b) Write down the equation of the horizontal asymptote to the graph of \(f\).

(c) The line \(y = c\) , where \(c\in \mathbb R\) intersects the graph of \( \begin{vmatrix}f(x) \end{vmatrix} \) at exactly one point. Find the possible values of \(c\).

Worked Solution