[Maximum number: 7]

Let $f(x)=x^{2}+2 x+1$ and g(x)=x-5, for $x \in \mathbb{R}$ .

(a)

Find f(8).

[ 2 ]

(b)

Find $(g \circ f)(x)$ .

[ 2 ]

(c)

Solve $(g \circ f)(x)=0$ .

[ 3 ]

[Maximum number: 3]

Kiran and Logan collect the following data about the river Afon, where x is the distance in metres from the source and y is the depth in centimetres.

This data is represented in the following scatter diagram.

Kiran knows that the depth of the river is 0 cm at the source.
Kiran calculates $\bar{x}$ and $\bar{y}$ for the seven points given in the table on page 2 and draws a line on the scatter diagram through the mean point $(\bar{x}, \bar{y})$ and the point (0,0).

(a)

Find

[ 3 ]

(i)

the equation of Kiran's line.

For the seven points given in the table Logan finds the regression line of y on x with equation y=a x+b, where $a, b \in \mathbb{R}$ .

[ 3 ]

[Maximum number: 5]

The following table shows values of f(x) and g(x) for different values of x.
Both f and g are one-to-one functions.

(a)

Find g(0).

[ 1 ]

(b)

Find $(f \circ g)(0)$ .

[ 2 ]

(c)

Find the value of x such that f(x)=0.

[ 2 ]

[Maximum number: 5]

The graph of y=f(x) for $-4 \leq x \leq 6$ is shown in the following diagram.

(a)

Write down the value of

[ 2 ]

(i)

f(2);

(ii)

$(f \circ f)(2)$ .

[ 2 ]

(b)

Let $g(x)=\frac{1}{2} f(x)+1$ for $-4 \leq x \leq 6$ . On the axes above, sketch the graph of g.

[ 3 ]

[Maximum number: 4]

Consider the function $f(x)=x^{2}+x+\frac{50}{x}, x \neq 0$ .

(a)

Find f(1).

[ 2 ]

(b)

Solve f(x)=0.

The graph of f has a local minimum at point A .

[ 2 ]

[Maximum number: 7]

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

(a)

For the graph of f

[ 5 ]

(i)

find the x-coordinates of the x-intercepts;

(ii)

find the coordinates of the vertex.

The function f can be written in the form $f(x)=-2(x-h)^{2}+k$ .

[ 5 ]

(b)

Write down the value of h and the value of k.

[ 2 ]

[Maximum number: 10]

Let $f(x)=8 x-2 x^{2}$ . Part of the graph of f is shown below.

(a)

Find the x-intercepts of the graph.

[ 4 ]

(b)

(i)

Write down the equation of the axis of symmetry.

[ 3 ]

(ii)

Find the y-coordinate of the vertex.

[ 3 ]

[Maximum number: 6]

Let f(x)=p(x-q)(x-r). Part of the graph of f is shown below.

The graph passes through the points (-2,0),(0,-4) and (4,0).

(a)

Write down the value of q and of r.

[ 2 ]

(b)

Write down the equation of the axis of symmetry.

[ 1 ]

(c)

Find the value of p.

[ 3 ]

[Maximum number: 6]

Let f be a quadratic function. Part of the graph of f is shown below.

The vertex is at P(4,2) and the y-intercept is at Q(0,6).

(a)

Write down the equation of the axis of symmetry.

The function f can be written in the form $f(x)=a(x-h)^{2}+k$ .

[ 1 ]

(b)

Write down the value of h and of k.

[ 2 ]

(c)

Find a.

[ 3 ]

[Maximum number: 6]

Let $f(x)=\sqrt{x+2}$ for $x \geq-2$ and g(x)=3 x-7 for $x \in \mathbb{R}$ .

(a)

Write down f(14).

[ 1 ]

(b)

Find $(g \circ f)(14)$ .

[ 2 ]

(c)

Find $g^{-1}(x)$ .

[ 3 ]

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