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Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
1. | 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.
2. | 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 \(y=x^2+8x+13\).
3. | 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?
4. | GCSE Higher |
(a) Find the interval for which \(x^2 - 9x + 18 \le 0\)
(b) The point (-4, -4) is the turning point of the graph of \(y = x^2 + ax + b\), where a and b are integers. Find the values of a and b.
5. | 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 \(y = 2x^2+8x+27\)
6. | IB Standard |
Let \(f (x)=a(x-b)^2+c\). The vertex of the graph of \(f\) is at (4, -3) and the graph passes through (3, 2).
(a) Find the value of \(c\).
(b) Find the value of \(b\).
(c) Find the value of \(a\).
7. | 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:
(f) Find the values of \(j\), \(k\) and \(s\).
8. | 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.
9. | IB Standard |
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 \(g(x)=Ax^2+Bx+C\).
10. | IB Standard |
Let \(f(x) = {x^2}\) and \(g(x) = 3{(x+2)^2}\) .
The graph of \(g\) can be obtained from the graph of \(f\) using two transformations.
(a) Give a full description of each of the two transformations.
(b) The graph of \(g\) is translated by the vector \( \begin{pmatrix}-4\\5\\ \end{pmatrix}\) to give the graph of \(h\).
The point \(( 2{\text{, }}-1)\) on the graph of \(f\) is translated to the point \(P\) on the graph of \(h\).
Find the coordinates of \(P\).
11. | IB Analysis and Approaches |
Part of the graph of a function, \(f\) , is shown in the following diagram. The graph of \(y = f(x)\) has a y-intercept at \((0, 1.5)\) , an x-intercept at \((a , 0)\) and horizontal asymptotes \(y = 5\) and \(y = -2\).
Consider the function \(g(x) = |f(|x|)| \)
(a) Sketch the graph of \(y = g(x)\), labelling any axis intercepts and giving the equation of the asymptotes.
(b) Find the possible values of \(k\) such that \( (g(x))^2 = k \) has exactly four solutions.
12. | IB Standard |
Let \(f\) and \(g\) be functions such that \(g(x) = 3f(x - 2) + 7\) .
The graph of \(f\) is mapped to the graph of \(g\) under the following transformations: vertical stretch by a factor of \(k\) , followed by a translation \(\left( \begin{array}{l} p\\ q \end{array} \right)\) .
Write down the value of:
(a) \(k\)
(b) \(p\)
(c) \(q\)
(d) Let \(h(x) = - g(2x)\) . The point A(\(8\), \(7\)) on the graph of \(g\) is mapped to the point \({\rm{A}}'\) on the graph of \(h\) . Find \({\rm{A}}'\)
13. | IB Analysis and Approaches |
The graph of \( y = f(|x|) \) for \( -2 \leq x \leq 2 \) is shown in the following diagram.
(a) On the following axes, sketch the graph of \( y = |f(|x|)| \) for \( -2 \leq x \leq 2 \).
It is given that \( f \) is an odd function.
(b) On the following axes, sketch the graph of \( y = f(x) \) for \( -2 \leq x \leq 2 \).
It is also given that
$$ \int_{0}^{2} f(|x|) \, dx = \dfrac{2}{3} $$(c) Write down the value of
$$ \int_{-2}^{2} f(x) \, dx; $$(d) Evaluate
$$ \int_{-2}^{2} \left( f(|x|) + f(x) \right) \, dx. $$14. | IB Standard |
Two functions are defined as follows: \(f(x) = 2\ln x\) and \(g(x) = \ln \frac{x^2}{3}\).
(a) Express \(g(x)\) in the form \(f(x) - \ln a\) , where \(a \in {{\mathbb{Z}}^ + }\) .
(b) The graph of \(g(x)\) is a transformation of the graph of \(f(x)\) . Give a full geometric description of this transformation.
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