Completing the square

Write 28 24x 6x² in the form a b(x c)² where a, b and c are integers.

On the axes below, sketch the graph of y 28 24x 6x²

Show clearly the coordinates of the turning point and the coordinates of the point of intersection of the graph with the y-axis.

Express 2x² 11x 9 in the form a(x b)² c where a, b and c are numbers to be found.

The curve C has equation y 2(x 3)² 11(x 3) 9

The point P is the minimum point on C

Find the coordinates of P

25f (x) = 17 − 3x² + 12x

Write f (x) in the form a – b(x − c)² where a, b and c are constants.

f (x) =_........

Given that

2ⁿ = 2^x2 16^x × 8

and

x 0

find an expression for x in terms of n State any restrictions on n

19Express 3x² – 6x + 5 in the form a(x – b)² + c

The function f is such that

Find f^–1(x)

The function g is such that

Express g(x) in the form a(x – b)² + c

find the value of p

The line with equation y = k is a tangent to the curve with equation y = x² –

Find the value of k

Express 2x² – 12x + 3 in the form a(x + b)² + cwhere a, b and c are integers.

The curve C has equation y = 2(x + 4)² – 12(x + 4) + 3

The point M is the minimum point on C

Find the coordinates of M

Express each of a, b and c in terms of q so that

can be written as a − b (x − c)²

Express 7 + 12x – 3x² in the form a + b(x + c)² where a, b and c are integers.

C is the curve with equation y = 7 + 12x – 3x² The point A is the maximum point on C

Use your answer to part (a) to write down the coordinates of A

Factorise 6y² – y – 5

Express 4x² – 8x + 7 in the form a(x + b)² + c where a, b and c are integers.

ABCED is a five-sided shape.

ABCD is a rectangle. CED is an equilateral triangle.

AB = xcm BC = ycm

The perimeter of ABCED is 100 cm. The area of ABCED is Rcm²

Find the value of x for which R has its maximum value.

Explain why the maximum value of R is given by this value of x.

Expand and simplify (x + 4)(x – 2)(x + 1)

Express x² – 10x + 40 in the form (x + a)² + b, where a and b are integers.

21The function f is such that f(x) = 5 + 6x – x²for x ≤ 3

Express 5 + 6x – x² in the form p – (x – q)² where p and q are constants.

Using your answer to part (a), find the range of values of x for which f^–1(x) is positive.

the coordinates of any points of intersection of C with the coordinate axes,

the coordinates of the turning point.

Find the value of a and the value of b.

Another curve C has equation y= f (x) The coordinates of the minimum point of C are (4, 5)

Write down the coordinates of the minimum point of the curve with equation

y= f (2x)

y= f (x) − 7

The graph of y = sinx° for 0 x ≤ 360 is drawn on the grid.

On the grid, draw the graph of y = 2sin(x + 30)° for 0 x ≤ 360

Write x²x + 10 in the form (xa)² + b where a and b are integers.

Hence, describe fully the single transformation that maps the curve with equation y = x² onto the curve with equation y = x²x + 10

The diagram shows a solid cuboid.

The total surface area of the cuboid is A cm²

Find the maximum value of A.

Write 5 + 12xx² in the form a + b(x + c)² where a, b and c are integers.

Show your working clearly.

where p and q are integers. Show your working clearly.

Show your working clearly.

Express 7 – 4x – x² in the form p – (x + q)² where p and q are constants.

Use your answer to part (a) to solve the equation 7 – 4(y + 3) – (y + 3)² = 0

The curve C has equation y = 3 – 5(x + 1)² The point A is the maximum point on C.

Write down the coordinates of A.