Graphical Methods Questions and Answers

Graphical Methods Questions

1. The equation of a circle is given as

2x² + 2y² – 8x + 5y + 10 = 0. Find the radius of the circle and the coordinates of its centre. (3 mks)

2. The equation of a circle is given by x² + 4x + y² – 5 = 0. Find the centre of the circle and its

radius.

3. The equation of a circle is x² + y² + 6x – 10y – 2 = 0. Determine the co-ordinates of the

centre of the circle and state its radius

4. In the diagram below ABE is a tangent to a circle at B and DCE is a straight line.

If ABD = 60^o, BOC = 80^o and O is the centre of the circle, find with reasons BEC

5. Obtain the centre and the radius of the circle represented by the equation:

x² + y² – 10y + 16 = 0

6. Complete the table below, for the function y = x³ + 6x² + 8x

 (a) Draw a graph of the function y = x³ + 6x² + 8x for – 5 x 1 and use the graph to estimate

the roots of the equation x³ + 6x² + 8x = 0

 (b) Find which values of x satisfy the inequality x³ + 6x² + 8x -1 > 0

7. Sketch the curve of the function y = x³– 3x + 2 showing clearly minimum and maximum points

and the y – intercept.

8. Show that 4y² + 4x² = 12x – 12y + 7 is the equation of a circle, hence find the co-ordinates

of the centre and the radius

9. Two variables R and P are connected by a function R = KPⁿ where K and n are constants.

The table below shows data involving the two variables

(a) Express R = KPⁿ
in a linear form

(b) Draw a line graph to represent the information above

(c) Find the values of constants K and n

(d) Write down the law connecting R and P

(e) Find the value of P when R = 900

10. A circle of radius 3cm has the centre at (-2, 3) . Find the equation of the circle in the

form of x² + y² + Px + qy + c = 0

11. In an experiment, the values of two quantities V and T were observed and the results recorded as

shown below.

It is known that T and V are related by a law of the form a

b + V

where a and b are constants.

a) Draw the graph of I against V

T

 b) Use your graph to find;

i) The values of a and b.

ii) V when T = 0.38

iii) T when V = 4.5

12. Find the equation of the tangent to the curve y = 2x³ + x² + 3x – 1 at the point (1, -5)

expressing you answer in the form y = mx + c

13. Given that :- 243 = (81)^-1 x ( ¹/₂₇) ^x determine the value of x

14. Show that 3x² + 3y
² + 6x – 12y – 12 = 0 is an equation of a circle hence state the radius and

 centre of the circle

15. (a) Fill in the table below for the function y = -6 + x + 4x² + x³ for -4 
x 
2

(b) Using the grid provided draw the graph for y = -6 + x + 4x² + x³ for -4 x  2

 (c) (i) Use the graph to solve the equations:-

 (i) x³ + 4x² + x – 4 = 0

 (ii) -6 + x + 4x² + x³ = 0

 (iii) -2 + 4x² + x³ = 0

16. The table below shows the results obtained from an experiment to determine the relationship

between the length of a given side of a plane figure and its perimeter

(a) On the grid provided, draw a graph of perimeter P, against 

 (b) Using your graph determine;

(i) the perimeter of a similar figure of side 2.5cm

(ii) the length of a similar figure whose perimeter is 9.43cm

(iii) the law connecting perimeter p and the length 

 (c) If the law is of the form P = 2k + c where k and c are constants, find the value of k

17. In an experiment with tungsten filament lamp, the reading below of voltage (V) current (I),

power (P) and resistance (R)were obtained. It was established that P was related to R by

a law P = a Rⁿ – 0.6. Where a and n are constants.

Plot a suitable line graph and hence use it to determine the value of a and n

18. Find the gradient of a line joining the centre of a circle whose equation is x² + y² – 6x = 3 – 4y

and a point P(6,7) outside the circle..

19. a) Complete the table below for the function y = -x³ + 2x² – 4x + 2.

 b) On the grid provided below draw the graph of -x³ + 2x² – 4x + 2 for – 3 ≤ x ≤ 4.

 c) Use the graph to solve the equation -x³ + 2x² – 4x + 2
= 0.

 d) By drawing a suitable line on the graph solve the equation. –x³ + 2x² – 5x + 3 = 0.

20. Determine the turning point of the curve y = 4x³ – 12x + 1. State whether the turning

point is a maximum or a minimum point.

21. (a) Complete the table below for the equation of the curve given by y = 2x³ – 3x² + 1

 (b) Use the table to draw the graph of the function y = 2x³ – 3x² + 1

c) Use your graph to find the values of x for :-

 (i) y > 0

 (ii) The roots of the equation 2x³ – 3x² + 1 = 0

 (iii) 2x³ – 3x² = 9

22. Find the radius and the centre of a circle whose equation is :

2x² + 2y² – 6x + 10y + 9 = 0

Graphical methods Answers

1. x² + 4x + y² = 5

x² + 4x + ( ½ x 4)² + y² = 5 + (½ x 4)²

(x + 2)² + (y + 0)² = 5 + 4

(x + 2)² + (y + 0)² = 9

Centre (-2,0)

Radius 9

r = 3 units

2. x² + 6x + (3)² + y² – 10y + (^–5) = 2 + 9 + 25

(x + 3)² + (y – 5)² = 36

(x – ^–3)² + (y – +5)² = 6²

∴ centre (-3, 5)

Radius 6 units

3. CBE = 40⁰ ( alt.segiment theoren)

BCE = 120⁰ (Suppl. To BCD = 60⁰alt. seg.)

(40 + 120 + E) = 180⁰ (Angle sum of  )

 BEC = 20⁰

4. X² +Y² – 10Y + 25 = 25 – 16

(X -0)² + (Y – 5)² = 9

(X – 0)² + (Y – 5)² = 3²

Centre (0, 5)

Radius = 3

5.

x³ + 6x² + 8x >1

Between

(i) x = -3.85 0.1 and x -2.15 0.1

(ii) x > 0.5 0.1

6. y = x³ – 3x + 2

x = 0, y = 2

(0, 2) ⇒ y – intercept.

dy = 3x² – 3 = 0

 dx x² = 1

x = ∓ 1

 x = 1 y = 0

Point (1, 0) min point

x = -1, y= 4

Point (-1, 4) max point.

7. 4x₂ – 12x + 4y² + 12y = 7

x² – 3x + y² + 3y = ⁷/₄

x² – 3x + (³/₂)² + y² + 3y + (³/₂)² = ⁷/₄ + ⁹/₄ + ⁹/₄ = ²⁵/₄

(x – ³/₂)² + (y + 3/2)2 = 25/4

Centre (1,5, -1.5) Radius 2.5units

8. Log R =nlog p + log K

Gradient = 2 – 0.6

0.7

= 1.4 = 2

0.7

Log R intercepts = 0.6 = logk

K= 4

The law connecting R and P is R=4P²

900 = 4P²

P² = 900

4

225 = P²

9. (x +2)² (y-3)² = 3²

X² + 4x + 4 + y² – 6y + 9 = 3²

X² + y² + 4x – 6y + 4 = 0

10.

 T = a

b + V

I = b + V

 T a

I = 1V + b

T a a

 y = mx + C

 b) (i) 1 = Grad ⇒ ∆y = 7.3 – 5 = 2.3 = 0.575

a ∆x 10 – 6 4

a = 1.739

 b = y – Intercept ⇒2.04

 a

b = 2.04 b = 2.04 x 1.739

1.739 = 3.547556

b ≃ 3.548

(ii) T = 0.38

I = 2.63 shown on graph

T

V = 1

-1

(iii) I = 4.45

 T

 T = (4.45)

= 0.2247

≃0.22

11. y = 2x³ + x² + 3x -1

dy = 6x² + 2x + 3

dx

gradeindent at (1, -5)

= 6 + 2 + 3= 11

y-(^–5) = 11

x -1

y + 5 =11x -11

y = 11x -16

12. 3⁵ = 3^-4 x 3^-x

3⁵ = 3^-4-x

-4 –x = 5

-x = 9

x =-9

13. x² + 2x + 1 + y² – 4y + 4 = 4 + 1 + 1

(x+1)² + (y-2)² = 9

Centre (-1, 2)

Radius 3units

14. c)

y = x³ + 4x² + x -6

0 = x³ + 4x² + x -4

y = -2

(iii) y = x³ + 4x² + x – 6

0 = x³ + 4x² + 0 – 2

y = x – 4

x 1 0 -2

y -3 -4 -8

c (i) solution 0.8

-1.5

And -3.2

(c) 1, -2, -3

15.

(i) P = 15.75cm

(ii) l=1.5cm

(iii) m = 35- 25 = 10 = 6.667

5.5 – 4.0 1.5

(c) choose P(5,31.4)

p – 31.4 = 10

l -5 1.5

p-31.4 = 100

 l-5 1.5

15p – 471 = 100k – 500

15p = 100l – 29

15 15

2k = 100

15

k= 100 = 3.33

2 x 15

c =1.93

P + 0.6 = ar^h

Log (P + 0.6) = log a + n log R

= n log R + log 9

Log 0.3 = ¼ = 0.25

Log a = 0.3

17. x² + y² – 6x = 3 – 4y

x² – 6x + (^-6/₂)² + y² + 4y + (⁴/₂)² = 3 + (^-6/₂)² + (⁴/₂)²

 (x – 3)² (y + 2)² = 3 + 9 = 4

(x – 3)² (y + 2)² = 16

 C (3, -2)

 Gradient ∆y = 7 – -2 = 3

∆x 6 – 3

 b) Check on the graph paper.

 c) x = 0.5 + 0.1

d) –x³ + 2x² – 5x + 3 = 0

Line to allow: y = x – 1

x 0 1

y -1 0

x = 0.65

19. ^Dy/_dx = 12x² – 12

12x² – 12 = 0

12(x² – 1) =0

x = 1

 x = -1

At x = 1 At x = -1

– 0 + + 0 –

(1,7) (-1, 9)

Minimum maximum

20. (a) table

(b) plotting

scale

smooth curve

(c) (i) -0.5 < x < 1 and x>1

(iii) x = 2.5  0.1

21. 2x² + 2y² – 6x + 10y + 9 = 0

x² + y² – 3x + 5y + 9/2 = 0

x² + y² – 3x + 5y = -9/2

x² – 3x + 9 + y² + 5y + 25 = 8.5 – 4.5

4 4

(x – 3)² + (y + 5)² = 4

2 2

Radius = 2 units

Centre = (1.5, -2.5)