Exam-Style Questions.

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

Worked Solution

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

(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

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

Given that \(x^2 – 8x + 3 = (x – a)^2 – b\) for all values of x,

(a) find the value of a and the value of b.

(b) Hence write down the coordinates of the turning point on the graph of \(y = x^2 – 8x + 3\)

Worked Solution

(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

(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

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

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