## 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 Analysis and Approaches |

Part of the function of \(y=f(x)\) is shown in the following diagram.

(a) Write down the value of \(f(1)\).

(b) Write down the value of \(ff(7)\).

(c) Sketch the graph of \(g(x)\) if \(g(x) = -f(x)-1\) on the same set of axes above.

## 7. | 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 \(a\).

## 8. | 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 \(s\).

## 9. | 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.

## 10. | 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 \(g(x)=Ax^2+Bx+C\).

## 11. | 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 \(P\).

## 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 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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