Exam-Style Questions.

1.	GCSE Higher

Write down the coordinates of the turning point on the graph of \(y = 9 - (x - 5)^2\)

Worked Solution

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2.	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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3.	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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4.	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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5.	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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6.	GCSE Higher

The diagram below is a sketch of a curve, a parabola, which is not drawn to scale.

The quadratic graph intersects the x-axis at (-5, 0) and at another point.

It also intersects the y-axis at (0, –10).

Work out the coordinates of the turning point of the graph if its equation is in the form \(y = x^2 + bx + c \).

Worked Solution

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7.	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.

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

Consider the function \(f(x)=\frac{1}{2}\left(2x-3\right)\left(x+5\right)\) for \(x \in \mathbb R\). The following diagram shows part of the graph of \(f\).

For the graph of \(f\)

(a) find the coordinates of the x-intercepts.

(b) find the coordinates of the vertex.

The function \(f\) can be written in the form \(f(x) = (x+h)^2 + k\)

(c) Write down the value of \(h\) and the value of \(k\).

Worked Solution

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10.	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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11.	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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12.	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 \(g(x)=Ax^2+Bx+C\).

Worked Solution

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13.	IB Analysis and Approaches

The functions \( f \) and \( g \) are defined for \( x \in \mathbb{R} \) by \( f(x) = 3 + 5x - 2x^2 \) and \( g(x) = x + k \), where \( k \in \mathbb{R} \).

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

(b) Given that \( (f \circ g)(x) \) has two distinct roots that sum to zero, determine the value of \( k \).

Worked Solution

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14.	IB Analysis and Approaches

The graphs of the functions \(f(x)\), a parabola, and \(g(x)\), a straight line, meet at exactly one point.

$$f(x) = px^2 - px $$ $$g(x) = px-5 $$

where \( x \in \mathbf R \text{ and } p \in \mathbf R \)

(a) Show that \(p = 5\)

The function \(f\) can be expressed in the form \(f(x) = 5(x-m)(x-n) \text{, where } m,n \in \mathbf R\)

(b) Find the value of \(m\) and the value of \(n\).

The function \(f\) can also be expressed in the form \(f(x) = 5(x-h)^2 + k, \text{ where } h,k \in \mathbf R\)

(c) Find the value of \(h\) and the value of \(k\).

(d) Hence find the values of \(x\) where the graph of \(f\) is both negative and decreasing.

Worked Solution

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