Exam-Style Questions.

1.	GCSE Higher

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

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2.	GCSE Higher

Match the equation with the letter of its graph

Equation	Graph
$$y=3-\frac{10}{x}$$
$$y=2^x$$
$$y=\sin x$$
$$y=x^2+7x$$
$$y=x^2-8$$
$$y=2-x$$

Worked Solution

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3.	GCSE Higher

The graph of the curve A with equation \(y=f(x)\) is transformed to give the graph of the curve B with equation \(y=5-f(x)\).

The point on A with coordinates (3, 9) is mapped to the point W on B.

Find the coordinates of W.

Worked Solution

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4.	GCSE Higher

The graph of y = f(x) is drawn accurately on the grid.

(a) Write down the coordinates of the turning point of the graph.

(b) Write down estimates for the roots of f(x) = 0

(c) Use the graph to find an estimate for f(-5.5).

Worked Solution

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5.	GCSE Higher

(a) By completing the square, solve \(x^2+8x+13=0\) giving your answer to three significant figures.

(b) From the completed square you found in part (a) find the minimum value of the curve \(y=x^2+8x+13\).

Worked Solution

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6.	IB Studies

Consider the graph of the function \(f(x)=7-3x^2-x^3\)

(a) Label the local maximum as A on the graph.

(b) Label the local minimum as B on the graph.

(c) Write down the interval where \(f(x)>5\).

(d) Draw the tangent to the curve at \(x=-3\) on the graph.

(e) Write down the equation of the tangent at \(x=-3\).

Worked Solution

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7.	GCSE Higher

The graph of the following equation is drawn and then reflected in the x-axis

$$y = 2x^2 - 3x + 2$$

(a) What is the equation of the reflected curve?

The original curve is reflected in the y-axis.

(b) What is the equation of this second reflected curve?

Worked Solution

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8.	GCSE Higher

(a) Write \(2x^2+8x+27\) in the form \(a(x+b)^2+c\) where \(a\), \(b\), and \(c\) are integers, by 'completing the square'

(b) Hence, or otherwise, find the line of symmetry of the graph of \(y = 2x^2+8x+27\)

(c) Hence, or otherwise, find the turning point of the graph of \(y = 2x^2+8x+27\)

Worked Solution

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9.	IB Analysis and Approaches

The function \(f\) is defined by:

$$f(x) = \frac{4x+2}{x+1}, \quad \text{ where } x \in \mathbb{R}, x \neq -1 $$

(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

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10.	IB Analysis and Approaches

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

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11.	IB Standard

A function is defined as \(f(x) = 2{(x - 3)^2} - 5\) .

(a) Show that \(f(x) = 2{x^2} - 12x + 13\).

(b) Write down the equation of the axis of symmetry of this graph.

(c) Find the coordinates of the vertex of the graph of \(f(x)\).

(d) Write down the y-intercept.

(e) Make a sketch the graph of \(f(x)\).

Let \(g(x) = {x^2}\). The graph of \(f(x)\) may be obtained from the graph of \(g(x)\) by the two transformations:

• a stretch of scale factor \(s\) in the y-direction;
• followed by a translation of \(\left( {\begin{array}{*{20}{c}} j\\ k \end{array}} \right)\) .

(f) Find the values of \(j\), \(k\) and \(s\).

Worked Solution

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12.	IB Standard

\(f\) and \(g\) are two functions such that \(g(x)=3f(x+2)+7\).

The graph of \(f\) is mapped to the graph of \(g\) under the following transformations:

A vertical stretch by a factor of \(a\) , followed by a translation \(\begin{pmatrix}b \\c \\ \end{pmatrix}\)

Find the values of

(a) \(a\);

(b) \(b\);

(c) \(c\).

(d) Consider two other functions \(h\) and \(j\). Let \(h(x)=-j(2x)\). The point A(8, 7) on the graph of \(j\) is mapped to the point B on the graph of \(h\). Find the coordinates of B.

Worked Solution

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13.	IB Standard

The diagram shows part of the graph of \(f(x)=Ae^{kx}+2\).

The y-intercept is at \((0,10)\).

(a) Show that \(A=8\).

(b) Given that \(f(8)=3.62\) (correct to 3 significant figures), find the value of \(k\).

(c) (i) Using your value of \(k\), find \(f'(x)\).

    (ii) Hence, explain why \(f\) is a decreasing function.

    (iii) Find the equation of the horizontal asymptote of the graph \(f\).

Let \(g(x)=-x^2+7x+5\)

(d) Find the area enclosed by the graphs of \(f\) and \(g\).

Worked Solution

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14.	IB Standard

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

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