Completing the square and turning points: practice

3 marksHigherGrade 6Non-calculator

y = x^2 + 6x + 2

(a)

Write

x^2 + 6x + 2

in the form

(x + a)^2 + b

.

(2 marks)

(b)

Write down the coordinates of the turning point of

y = x^2 + 6x + 2

.

(1 mark)

2 marksHigherGrade 6CalculatorProblem solving

Find the coordinates of the turning point of

y = x^{2} + 10x + 17

.

(2 marks)

4 marksHigherGrade 7Non-calculatorProblem solving

The curve

y = x^2 + px + q

has a turning point at

(5, -3)

.

(a)

Find the value of

p

.

(2 marks)

(b)

Find the value of

q

.

(2 marks)

3 marksHigherGrade 7Non-calculatorProblem solving

A rectangular enclosure is built against a straight wall. Exactly 60 m of fencing is used for the other three sides. Find the greatest possible area of the enclosure.

(3 marks)

4 marksHigherGrade 7Non-calculator

y=x^2-6x+10

.

(a)

Write y in completed-square form.

(2 marks)

(b)

Find the minimum point.

(2 marks)

5 marksHigherGrade 7Non-calculatorProblem solving

Exactly 24 m of fencing forms three sides of a rectangle against a straight wall. Each side perpendicular to the wall has length x.

(a)

Write the area in completed-square form.

(3 marks)

(b)

Find the greatest possible area.

(2 marks)

5 marksHigherGrade 7Non-calculator

x^2+10x+18=0

.

(a)

Solve by completing the square. Give exact answers.

(3 marks)

(b)

Find the sum of the two roots.

(2 marks)

4 marksHigherGrade 7Non-calculator

Consider

y = -x^2 + 8x - 11

.

(a)

Write y in the form

a - (x - b)^2

.

(2 marks)

(b)

Write down the coordinates of the maximum point of the curve.

(1 mark)

(c)

Explain why

-x^2 + 8x - 11 \le 5

for every value of x.

(1 mark)

5 marksHigherGrade 8Non-calculator

2x^2 - 12x + 5

(a)

Write

2x^2 - 12x + 5

in the form

a(x + b)^2 + c

.

(3 marks)

(b)

Hence solve

2x^2 - 12x + 5 = 0

. Give your answers in exact form.

(2 marks)

4 marksHigherGrade 8Non-calculatorProblem solving

E = x^2 - 8x + 20

(a)

By completing the square, show that

E > 0

for all values of

x

.

(2 marks)

(b)

Find the greatest value of

\dfrac{1}{x^2 - 8x + 20}

.

(2 marks)