Graphical Functions

Worksheets for GCSEMathematics

Mr Black'sMaths Resources for TeachersGCSE 1-9

Algebra

Coordintes and Graphical Functions Worksheets

Contents

Differentiated Independent Learning Worksheets

Solutions

• Coordinates in one quadrant• Coordinates in all four quadrants• Mapping diagrams using function machines• Plotting straight line graphs in the form y = mx + c• Plotting linear functions in the form ax + by = c• Gradient of straight line graphs• Deriving the equation of straight line graphs• Parallel Gradients• Perpendicular Gradients• Plotting quadratic graphs• Sketching quadratic graphs• Plotting cubic graphs• Plotting reciprocal graphs• Plotting exponential graphs• Equation of a circle• Equation of a tangent

• Coordinates in one quadrant• Coordinates in all four quadrants• Mapping diagrams using function machines• Plotting straight line graphs in the form y = mx + c• Plotting linear functions in the form ax + by = c• Gradient of straight line graphs• Deriving the equation of straight line graphs• Parallel Gradients• Perpendicular Gradients• Plotting quadratic graphs• Sketching quadratic graphs• Plotting cubic graphs• Plotting reciprocal graphs• Plotting exponential graphs• Equation of a circle• Equation of a tangent

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2

Coordinates in One Quadrant Q1. Write down the coordinates of the following points:

A = E =

B = F =

C = G =

D = H =

Q2.

a) ABCR is a rectangle. Plot thecoordinate R

b) GCBS is a parallelogram. Plot thecoordinate S

c) T is the midpoint of FA. What arethe coordinates of T?

d) U is the midpoint of FB. What arethe coordinates of U?

e) AFGV is a square.i) Plot the coordinate V.ii) State the coordinates for the centreof the square

Q3. a) Join (1,1), (4,2), (1,3), (1,1)

Shape = ______________

b) Join (4,0), (6,0), (8,5), (6,5), (4,0)

Shape = _______________

c) Join ( 10,2), (9,4), (7,2), (9,0),(10,2)

Shape = _______________

3

Coordinates in Four Quadrants Q1. Write down the coordinates of the following points:

A = F =

B = G =

C = H =

D = I =

E =

Q2. a) ABEQ is a rectangle. Plot the coordinate Q.. b) GFDR is a square. Plot the coordinate R c) S is the midpoint of GF. What are the coordinates of S? d) T is the midpoint of CI. What are the coordinates of T? e) Which two coordinates make the line GH into two different right-angled triangles? f) Point U centre of the rectangle ABEQ. State its coordinates.

Q3. a) Join (0,0), (3,0), (0,2), (0,0) Shape = ______________ b) Join (0,0), (-2,0), (-5,2), (-5,0), (0,0) Shape = _______________ c) Join ( -4,-3), (2,-3), (3,-1), (-3,-1), (-4,-3) Shape =

4

Linear Functions

𝑦 = 2𝑥 − 5

𝑦 = 20 − 5𝑥 𝑦 = 0.5𝑥 + 1.5 𝑦 = 3𝑥 + 14

Q1. Generate the outputs for the given functions and inputs. a) 𝑦 = 6𝑥 − 1

b) 𝑦 = 2𝑥 + 8

c) 𝑦 = 3𝑥 + 5

d) 𝑦 = 10𝑥 − 7

c) 𝑦 = 4 + 5𝑥

d) 𝑦 = 5(𝑥 + 2)

Q2. Match the function with the table of results.

x y

x y

x y

x y

0 14

0 -5

0 20

0 1.5

1 17

1 -3

1 15

1 2

2 20

2 -1

2 10

2 2.5

3 23

3 1

3 5

3 3

4 26

4 3

4 0

4 3.5

Q3. Complete the mapping diagram for these functions. a) 𝑦 = 3𝑥 + 2

b) 𝑦 = 2𝑥 + 5

y x

0 1 2 3 4

y

x

0 1 2 3 4

y

x

0 1 2 3 4

y x

0 1 2 3 4

y

x

0 1 2 3 4

y x

0 1 2 3 4

5

Plotting Linear Graphs

Q1. On separate copies of the grid to the left draw the graphs: a) y = x + 5 b) y = 2x + 1 c) y = 3x + 2 d) y = 2(x + 2)

Q2. a) Draw the graph of 𝑦 = 3𝑥 − 2 for 𝑥 values

from -3 to 3.

b) Draw the graph of 𝑦 = 4𝑥 − 1 for 𝑥 values from -3 to 3.

Q3. Draw the graph of 𝑦 = 2𝑥 − 5 for −3 ≤ 𝑥 ≤ 3.

Q4. Draw the graph of 𝑦 =𝑥

2+ 3 for −2 ≤ 𝑥 ≤ 4.

Q5. Draw the graph of 𝑦 =𝑥

3− 5 for −3 ≤ 𝑥 ≤ 2.

Q6. Match the letter of the following graphs with their linear function.

i) 𝑦 = 2𝑥 + 5 ii) 𝑦 = 3𝑥 + 4 iii) 𝑦 = 𝑥 − 2

iv) 𝑦 =𝑥

2+ 2

6

Linear Functions from Two Points Q1. Plot these linear functions using the axes provided. a) 3𝑥𝑥 + 5𝑦𝑦 = 15

b) 4𝑥𝑥 + 2𝑦𝑦 = 8

c) 6𝑥𝑥 − 3𝑦𝑦 = 18

d) 2𝑦𝑦 − 3𝑥𝑥 = 6

e) 3𝑥𝑥 + 4𝑦𝑦 = −12

f) 2𝑥𝑥 + 𝑦𝑦 = −9

Q2. a)

2𝑥𝑥 + 3𝑦𝑦 = 18

4𝑥𝑥 + 5𝑦𝑦 = 32

b)

𝑥𝑥 + 2𝑦𝑦 = 7

2𝑥𝑥 − 𝑦𝑦 = −1

c)

3𝑥𝑥 − 2𝑦𝑦 = −14

𝑥𝑥 + 4𝑦𝑦 = 7

d)

3𝑥𝑥 − 2𝑦𝑦 = −14

𝑥𝑥 + 4𝑦𝑦 = 7

7

Gradient of Linear Graphs Q1. Derive the gradient for each of these linear functions.

a)

b)

c)

Q2. Derive the equation in the form 𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑐𝑐 for each of these line segments.

Q3. Calculate the gradients of the following line segments.

a) ( 1, 7 ) & ( -6, 3 ) b) ( -7, 1 ) & ( -5, -2 ) c) ( -9, -3 ) & ( 0, 5 )

d) ( 1, -5 ) & ( -7, 7 ) e) ( 7, -1 ) & ( 10, -5 ) f) ( 1, -2 ) & ( 10, -6 )

g) ( 5, 3 ) & ( -4, -8 ) h) ( 3, -7 ) & ( 7, -10 ) i) ( 3, 2 ) & ( -4, -10 )

8

Equation of Straight Line Graphs Q1. Derive the equation in the form 𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑐𝑐 for each of these graphs.

a)

b)

c)

Q2. Derive the equation in the form 𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑐𝑐 for each of these line segments.

𝑖𝑖) 𝑎𝑎 =

𝑖𝑖𝑖𝑖) 𝑏𝑏 =

𝑖𝑖𝑖𝑖𝑖𝑖) 𝑐𝑐 =

𝑖𝑖𝑖𝑖) 𝑑𝑑 =

Q3. Derive the equation of the straight line graph that passes through these coordinates:

a) (0,1) and (2,11)

b) (0,4) and (3,0)

c) (4,7) and (10,10)

9

Parallel Gradients

Q1. Write down the equations of the lines that run parallel to 𝑦 = 2𝑥 + 1.

Q2. Write down the equation of a line parallel to the line with the equation

a) 𝑦 = 6𝑥 + 14, b) 𝑦 = 9𝑥 − 8 a) 𝑦 =𝑥

4+ 4

c) 𝑦 =𝑥

6+ 1 d) 𝑦 =

3𝑥

2+ 46 e) 𝑦 = 3 −

2𝑥

5

Q3. Write down the equations of the lines that run parallel to

a) 𝑦 = 3𝑥 + 4, passing through (0 , 6) b) 𝑦 = 2𝑥 − 5, passing through (1 , -5)

c) 𝑦 = 6𝑥 + 5, passing through (2 ,9) d) 𝑦 =𝑥

2+ 6, passing through (10 , 15)

e) 𝑦 = 4 − 5𝑥, passing through (2 , -2) f) 𝑦 = 2(3𝑥 + 7) , passing through (-2 , -16)

Q4. Match the equations that would run parallel to each other.

Q5 Write down the equation of the line that runs parallel to the equation

and passes through the point (-10, -4)

3𝑥 − 5𝑦 = 8

10

Perpendicular Gradients

Q1. Write down the gradient of a straight line that is perpendicular to a line of gradient:

a) 2 b) 5 c) -3

d) 1

4 e) 0.8 f)

2

3

g) 5

2 h)1

1

2 i) −

2

7

Q2. Draw the line 𝑦 = 6 − 5𝑥.

a) State its gradient.

b) Draw a line perpendicular to this and state its gradient.

Q3. Draw the line 𝑥 + 2𝑦 = 6. a) State its gradient. b) Draw a line perpendicular to this and state its gradient.

Q4. Match the equations which perpendicular intersections.

Q5.

a) Find the equation of the line perpendicular to 𝑦 =𝑥−2

2 which passes through (0 , -1)

b) If point A has coordinates (4 , 1) and point B has coordinates (2 , 7) find:

i) the coordinates of the mid-point of AB.

ii) the gradient of AB.

iii) the equation of the perpendicular bisector of AB.

𝑦 = 3𝑥 + 2 𝑦 = 2 − 𝑥

3

𝑦 = 2(𝑥 − 3) 𝑦 =2

3𝑥 + 5

3𝑥 + 2𝑦 − 3 = 0 𝑦 = 8 − 1.5𝑥

𝑦 =2𝑥 + 3

3 𝑦 = 5 − 0.5𝑥

11

Plotting Quadratics Graphs Q1. Complete the tables and draw the graphs on the axes below. a)

x -2 -1 0 1 2

𝑦𝑦 = 𝑥𝑥2 − 3 -2

b)

x -2 -1 0 1 2

𝑦𝑦 = 𝑥𝑥2 + 1 1

c)

x -2 -1 0 1 2

𝑦𝑦 = 𝑥𝑥2 − 𝑥𝑥 0

Q2. Complete the tables and draw the graphs on the axes below. a)

x -3 -2 -1 0 1

𝑦𝑦 = 𝑥𝑥2 + 2𝑥𝑥 0

b)

x -2 -1 0 1 2

𝑦𝑦 = 2𝑥𝑥2 − 3 5

c)

x -1 0 1 2 3

𝑦𝑦 = 2𝑥𝑥2 − 3𝑥𝑥 − 1 4

Q3. Draw the graph of 𝑦𝑦 = 𝑥𝑥2 − 𝑥𝑥 − 2.

𝑥𝑥 𝑦𝑦 -3 -2 -1 0 1 2 3

Use the graph to solve the equations: a) 0 = 𝑥𝑥2 − 𝑥𝑥 − 2 b) 8 = 𝑥𝑥2 − 𝑥𝑥 − 2 c) −1 = 𝑥𝑥2 − 𝑥𝑥 − 2

12

Sketching Quadratics

Q1. Match the equation to each graph.

a)

b)

c)

d)

i) 𝑦 = 𝑥2 − 4

ii) 𝑦 = −(𝑥 + 2)2

iii) 𝑦 = 4 − 𝑥2

iv) 𝑦 = (𝑥 − 2)2

Q2. Sketch the graphs of the following functions.

State the roots, y-intercept and turning point for each graph.

a) 𝑦 = 𝑥2 − 9

b) 𝑦 = 𝑥2 − 3𝑥 + 10

c) 𝑦 = 𝑥2 + 2𝑥 − 15

d) 𝑦 = 𝑥2 − 3𝑥 + 2

e) 𝑦 = 12 − 𝑥 − 𝑥2

f) 𝑦 = 2𝑥2 − 3𝑥 − 4

g) 𝑦 = 7 + 4𝑥 − 3𝑥2

13

Plotting Cubic Graphs Q1. Complete the tables and draw the graphs on the axes below. a)

x -2 -1 0 1 2

𝑦𝑦 = 𝑥𝑥3 − 2 -3

b)

x -2 -1 0 1 2

𝑦𝑦 = 𝑥𝑥3 + 3 3

c)

x -2 -1 0 1 2

𝑦𝑦 = 𝑥𝑥3 − 2𝑥𝑥 4

d)

x -2 -1 0 1 2

𝑦𝑦 = 𝑥𝑥3 − 4𝑥𝑥 + 5 5

Q2. Use graphical methods to solve the following equations within the range −3 ≤ 𝑥𝑥 ≤ 2 a) 0 = 𝑥𝑥3 + 2𝑥𝑥2 − 𝑥𝑥 + 1

b) −1 = 𝑥𝑥3 + 2𝑥𝑥2 − 𝑥𝑥 + 1

c) 4 = 𝑥𝑥3 + 2𝑥𝑥2 − 𝑥𝑥 + 1

d) 2.5 = 𝑥𝑥3 + 2𝑥𝑥2 − 𝑥𝑥 + 1

14

Plotting Reciprocal Graphs Q1. Complete the tables and draw the graphs on the axes below. a)

x -4 -3 -2 -1 0 1 2 3

𝑦𝑦 = 2 𝑥𝑥�

b)

x -4 -3 -2 -1 0 1 2 3

𝑦𝑦 = 5 𝑥𝑥�

c)

x -4 -3 -2 -1 0 1 2 3

𝑦𝑦 = −1 𝑥𝑥�

Q2. An insect colony decreases after the spread of a virus. Its population, P, after m months is given by the equation

𝑃𝑃 =2500𝑚𝑚

valid for 1 ≤ 𝑚𝑚 ≤ 6 a) Complete this table to examine the change in population. m (months) 1 2 3 4 5 6 P (Population) 1250 b) Model the change in population over the six months as a graph.

c) Use the graph to estimate how long it takes for the population to reach 1000.

d) How long does it take for the population to decrease from 3500 to 1500?

15

Exponential Graphs

Q1. Draw the graph of 𝑦 = 5𝑥 for the values of x between -2 and 3.

𝑥 -2 -1 0 1 2 3 𝑦 5 125

Use this graph to find approximate solutions for the equations: a) 5𝑥 = 40 b) 5𝑥 = 100

Q2. Draw the graph of 𝑦 = 2−𝑥 for the values of x between -

3 and 2.

𝑥 -3 -2 -1 0 1 2 𝑦 8 0.5

Use this graph to find approximate solutions for the equations: a) 2−𝑥 =3 b) 2−𝑥 = 5.5

Q3. The formula 𝐶 = 3600 × 0.85𝑦 gives the value of a car

after y years. a) What is the value of the car when new? b) What is the value of the car after 7 years? c) Draw the graph of C against y to model for cost over 10 years. d) Use the graph to estimate when the car loses 60% of its initial value.

16

Equation of a Circle

Q1. State the equation of each of these circles.

a)

b)

Q2. Use the grid to sketch the following graphs.

a) 𝑥2 + 𝑦2 = 36 b) 𝑥2 + 𝑦2 = 81 c) 𝑥2 + 𝑦2 = 64

Q3. Use graphical methods to solve the following pairs of simultaneous equations

a) 𝑥2 + 𝑦2 = 36 & 𝑥 + 𝑦 = 5 b) 𝑥2 + 𝑦2 = 25 & 4𝑥 + 3𝑦 = 12 c) 𝑥2 + 𝑦2 = 49 & 𝑦 = 3𝑥 − 4 d) 𝑦2 = 36 − 𝑥2 & 𝑦 = 𝑥2 − 1

17

Equation of a Tangent

Q1. For questions a - d find:

i) the gradient of the radius, r

ii) the gradient of the line L1 and

iii) the equation of the line L1 in the form ax + by + c = 0.

a)

b)

c)

d)

Q2. Use algebra to show that 4𝑥 − 3𝑦 + 13 = 0 is tangential to the circle 𝑥2 + 𝑦2 = 25 and find the point of intersection.

Q3. Use algebra to show that 5𝑥 + 12𝑦 = 169 is tangential to the circle 𝑥2 + 𝑦2 = 169 and find the point of intersection.

Q4. Use algebra to show that 3𝑥 + 4𝑦 − 50 = 0 is tangential to the circle 𝑥2 + 𝑦2 = 100 and find the point of intersection.

Q5. Use algebra to prove that 𝑦 = 𝑥 + 1 is not tangential to the circle 𝑥2 + 𝑦2 = 16.

18

Coordinates in One Quadrant Solutions

Q1.

A = (1,1) E = (0,0)

B = (4,2) F = (0,2)

C = (0,5) G = (3,5)

D = (3,0) H = (5,4)

Q2. a) Coordinate R = (0,4) b) Coordinate S = (2,0) c) Coordinate T = (0,2) d) Coordinate U = (2,2) e) i) V = (3,3) ii) Centre = (1.5,1.5)

Q3. a) Shape = Isosceles Triangle b) Shape = Parallelogram c) Shape = Kite

19

Coordinates in Four Quadrants Q1.

A = (-3,5) F = (4,0)

B = (3,4) G = (0,0)

C = (-3,-2) H = (0,-4)

D = (3,-3) I = (-5,-4)

E = (-3,1)

Q2. a) Coordinate Q = (1,2) b) Coordinate R = (0,-4) c) Coordinate S = (4,-2) d) Coordinate T = (-3,-1) e) GHF & GHI f) Coordinate U = (-1.5, 3.5)

Q3. a) Shape = Right-Angled Triangle b) Shape = Trapezium c) Shape = Parallelogram

20

Linear Functions

𝑦 = 2𝑥 − 5

𝑦 = 20 − 5𝑥 𝑦 = 0.5𝑥 + 1.5 𝑦 = 3𝑥 + 14

Q1. a) 𝑦 = 6𝑥 − 1

b) 𝑦 = 2𝑥 + 8

c) 𝑦 = 3𝑥 + 5

d) 𝑦 = 10𝑥 − 7

c) 𝑦 = 4 + 5𝑥

d) 𝑦 = 5(𝑥 + 2)

Q2.

x y

x y

x y

x y

0 14

0 -5

0 20

0 1.5

1 17

1 -3

1 15

1 2

2 20

2 -1

2 10

2 2.5

3 23

3 1

3 5

3 3

4 26

4 3

4 0

4 3.5

Q3. a) 𝑦 = 3𝑥 + 2

b) 𝑦 = 2𝑥 + 5

x 6 -1

y -1 5 11 17 23

x

0 1 2 3 4

x 2 + 8

y 8 10 12 14 16

x

0 1 2 3 4

x 3 + 5

y 5 8 11 14 17

x

0 1 2 3 4

x10 -7

y -7 3 13 23 33

x

0 1 2 3 4

+4 x 5

y 4 9 14 24 29

x

0 1 2 3 4

+ 2 x 5

y 10 15 20 25 30

x

0 1 2 3 4

21

Plotting Linear Graphs

Solutions

Q1.

Q2. a) b)

Q3. Q4. Q5.

Q6.

a = i b = ii c = iv d = iii

22

Linear Functions from Two Points

Solutions Q1.

a) 3𝑥𝑥 + 5𝑦𝑦 = 15

b) 4𝑥𝑥 + 2𝑦𝑦 = 8

c) 6𝑥𝑥 − 3𝑦𝑦 = 18

d) 2𝑦𝑦 − 3𝑥𝑥 = 6

e) 3𝑥𝑥 + 4𝑦𝑦 = −12

f) 2𝑥𝑥 + 𝑦𝑦 = −9

Q2.

a) 𝑥𝑥 = 3,𝑦𝑦 = 4

b) 𝑥𝑥 = 1,𝑦𝑦 = 3

c) 𝑥𝑥 = −3.𝑦𝑦 = 2.5

d) 𝑥𝑥 = −4,𝑦𝑦 = −2

23

Gradient of Linear Graphs Solutions

Q1.

a) 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 = 2 b) 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 = 0.5 c) 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 = 3

Q2.

a) 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 = 4

b) 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 = 2

c) 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 = 12

d) 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 = −1

e) 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 = −2

Q3.

a) Gradient = 0.57 b) Gradient = -1.5 c) Gradient = 0.89

d) Gradient = -1.5 e) Gradient = -1.33 f) Gradient = -0.44

g) Gradient = 1.22 h) Gradient = -0.75 i) Gradient = 1.71

24

Equation of Straight Line Graphs Solutions

Q1

a) 𝑦𝑦 = 4𝑚𝑚 − 1 b) 𝑦𝑦 = 3𝑚𝑚 + 2 c) 𝑦𝑦 = 8 − 3𝑚𝑚

Q2.

𝑖𝑖) 𝑦𝑦 = 𝑚𝑚 − 1 (𝑎𝑎)

𝑖𝑖𝑖𝑖) 𝑦𝑦 = 2𝑚𝑚 + 6 (𝑏𝑏)

𝑖𝑖𝑖𝑖𝑖𝑖) 𝑦𝑦 = 8 − 3𝑚𝑚 (𝑐𝑐)

𝑖𝑖𝑖𝑖) 𝑦𝑦 = −𝑚𝑚 − 2 (𝑑𝑑)

Q3.

a) 𝑦𝑦 = 5𝑚𝑚 + 1 b) 𝑦𝑦 = 6 − 2𝑚𝑚 c) 𝑦𝑦 = 𝑥𝑥2

+ 5

25

Parallel Gradients

Solutions

Q1. 𝑦 = 2𝑥 + 6, 𝑦 = 2𝑥 − 5

Q2. a) 𝑚 = 6 b) 𝑚 = 9 a) 𝑚 =1

4

c) 𝑚 =1

6 d) 𝑚 =

3

2 e) 𝑚 = −

2

5

Q3. a) 𝑦 = 3𝑥 + 6 b) 𝑦 = 2𝑥 − 7

c) 𝑦 = 6𝑥 − 3 d) 𝑦 =𝑥

2+ 10

e) 𝑦 = 8 − 5𝑥 f) 𝑦 = 6𝑥 − 4

Q4. 𝑦 =

𝑥 + 2

2→ 𝑦 = 0.5𝑥 − 4

𝑦 = 2𝑥 + 1 → 2𝑦 − 4𝑥 = 3

𝑦 = 4𝑥 − 9 → 𝑦 = 2(2𝑥 + 1)

5𝑥 − 4𝑦 − 2 = 0 → 𝑦 = 1.25𝑥 + 8

Q5. 𝑦 =

3𝑥

5− 2

26

Perpendicular Gradients

Solutions

Q1. a) -0.5 b) -0.2 c) 1

3

d) −4 e) -1.25 f) −1.5

g) −0.4 h)- 2

3 i)

7

2

Q2.

b) 𝑦 = 0.2𝑥

Q3.

b) 𝑦 = 2𝑥

Q4.

Q5.

a) 𝑦 = −2𝑥 − 1

b) If point A has coordinates (4 , 1) and point B has coordinates (2 , 7) find:

i) (3 , 4)

ii) Gradient = -3

iii) 𝑦 = 𝑥

3+ 5

𝑦 = 3𝑥 + 2 𝑦 = 2 − 𝑥

3

𝑦 = 2(𝑥 − 3) 𝑦 =2

3𝑥 + 5

3𝑥 + 2𝑦 − 3 = 0 𝑦 = 8 − 1.5𝑥

𝑦 =2𝑥 + 3

3 𝑦 = 5 − 0.5𝑥

27

Plotting Quadratics Graphs Solutions Q1. a)

x -2 -1 0 1 2

𝑦𝑦 = 𝑥𝑥2 − 3 1 -2 -3 -2 1

b) x -2 -1 0 1 2

𝑦𝑦 = 𝑥𝑥2 + 1 5 2 1 2 5

c) x -2 -1 0 1 2

𝑦𝑦 = 𝑥𝑥2 − 𝑥𝑥 6 2 0 0 2

Q2. a)

x -3 -2 -1 0 1

𝑦𝑦 = 𝑥𝑥2 + 2𝑥𝑥 3 0 -1 0 3

b) x -2 -1 0 1 2

𝑦𝑦 = 2𝑥𝑥2 − 3 -5 -1 -3 -1 5

c) x -1 0 1 2 3

𝑦𝑦 = 2𝑥𝑥2 − 3𝑥𝑥 − 1 4 -1 -2 1 8

Q3. Draw the graph of 𝑦𝑦 = 𝑥𝑥2 − 𝑥𝑥 − 2. 𝑥𝑥 𝑦𝑦 -3 10 -2 4 -1 0 0 -2 1 -2 2 0 3 4

Use the graph to solve the equations: a) 𝑥𝑥 = −1, 2b) 𝑥𝑥 = −2.7, 3.65c) 𝑥𝑥 = −0.7, 1.6

28

Sketching Quadratics

Solutions Q1.

a = i b = iv c = iii d = ii Q2

a) 𝑦 = 𝑥2 − 9

b) 𝑦 = 𝑥2 − 3𝑥 + 10

c) 𝑦 = 𝑥2 + 2𝑥 − 15

d) 𝑦 = 𝑥2 − 3𝑥 + 2

e) 𝑦 = 12 − 𝑥 − 𝑥2

f) 𝑦 = 2𝑥2 − 3𝑥 − 4

g) 𝑦 = 7 + 4𝑥 − 3𝑥2

29

Sketching Quadratics

30

Plotting Cubic Graphs Solutions Q1. a)

x -2 -1 0 1 2

𝑦𝑦 = 𝑥𝑥3 − 2 -10 -3 -2 -1 6

b)

x -2 -1 0 1 2

𝑦𝑦 = 𝑥𝑥3 + 3 -5 2 3 4 11

c)

x -2 -1 0 1 2

𝑦𝑦 = 𝑥𝑥3 − 2𝑥𝑥 -4 1 0 -1 4

d)

x -2 -1 0 1 2

𝑦𝑦 = 𝑥𝑥3 − 4𝑥𝑥 + 5 5 8 5 2 5

a) 𝑥𝑥 = −2.55 b) 𝑥𝑥 = −2.65 c) 𝑥𝑥 = 1.15 d) 𝑥𝑥 = −2.14,−0.78

31

Plotting Reciprocal Graphs

Q1. a)

-4 -3 -2 -1 0 1 2 3

-0.5 -0.7 -1 -2 ER 2 1 0.67

b) -4 -3 -2 -1 0 1 2 3

-1.25 -1.67 -2.5 -5 ER 5 2.5 1.67

c) -4 -3 -2 -1 0 1 2 3

0.25 0.33 0.5 1 ER -1 -0.5 -0.33

Q2. a)

m (months) 1 2 3 4 5 6 P (Population) 2500 1250 833 625 500 417

b)

c) 2.5 months

d) Approximately 3.25 months

32

Exponential Graphs

Solutions

Q1. 𝑥 -2 -1 0 1 2 3 𝑦 0.04 0.2 1 5 25 125

a) 𝑥 ≈ 2.4 b) 𝑥 ≈ 2.8

Q2.

𝑥 -3 -2 -1 0 1 2 𝑦 8 4 2 1 0.5 0.25

a) 𝑥 ≈ −1.5 b) 𝑥 ≈ −3.4

Q3. The formula 𝐶 = 3600 × 0.85𝑦 gives the value of a

car after y years. a) £3600 b) £1154.08 c) d) Approx. 5.5 years

33

Equation of a Circle

Solutions

Q1. a) 𝑥2 + 𝑦2 = 49

b) 𝑥2 + 𝑦2 = 100

Q2. a)

b)

c)

Q3. a) (−0.93 , 5.93) & (5.93 , −0.93)

b) (−0.71 , 4.95) & (4.56 , −2.01) c) (−0.98 , −6.93) & (3.38 , 6.13) d) (−2.54 , 5.43) & (2.54 , 5.44)

34

Equation of a Tangent

Solutions

Q1. a)

i) 𝑀 = −3

4

ii) L1(M) =4

3

iii) 4𝑥 − 3𝑦 + 25 = 0

b)

i) 𝑀 =3

4

ii) L1(M) = −4

3

iii) 4𝑥 + 3𝑦 − 37 = 0

c)

i) 𝑀 =4

3

ii) L1(M) = −3

4

iii) 3𝑥 + 4𝑦 + 24 = 0

d)

i) 𝑀 =12

5

ii) L1(M) = −5

12

iii) 5𝑥 + 12𝑦 − 169 = 0

Q2. Point of intersection= (-4 , 3)

Q3. Point of intersection= (5 , 12)

Q4. Point of intersection= (6 , 8)

Q5. Solutions at (-3.28 , -2.28) & (2.28 , 3.28)

35