L.J.E GRADE 9 GRAPH SUMMARY p1 1. THE STRAIGHT LINE GRAPH: Std form: y = mx + c gradient y-intercept

Know standard form : y = mx + c 1.1 Be able to draw any straight line

a) Using Intercept-Intercept Method (when there is an x-term, a y-term and a number) b) Gradient-Intercept Method (Used when the graph passes through (0;0) i.e when x = 0, y = 0). c) Special Lines: y = c, x = k, y = x and y = -x

NOTE: The gradient of a line is 12

12

xx

yym

−

−= =

𝑟𝑖𝑠𝑒

𝑟𝑢𝑛

Parallel lines have ……………… gradient So if m1 = 3 , m2 = Perpendicular lines: m1 × m2 = ……………….. So if m1 = ½ , m2 = ……………. 1.2 a) Finding the Equation of a Straight Line off a sketch

b) Finding the Equation of a Line, Given m or c and a point: c) Finding the gradient of a line between two points

1.3 Finding the Point of Intersection of 2 Straight Lines (Simultaneous Equations): 1.4 Determining whether or not a point lies on a line. NB LHS = RHS = 1.1 Drawing Straight Lines:

a) Intercept-Intercept Method: Use this when graph cuts both axes in different places – will have an x-, y-term and a number.

Step 1: to find y-int, Let ………………………. Step 2: to find x-int, Let ………………………. Step 3: Join the 2 intercepts and label the points on the graph. NOTE: Equation does not have to be in std form. Always show the x and y intercepts. Eg: On the same axes, draw rough graphs of: 2x + 4y = 8 and y = 2x - 3 2x + 4y = 8: y-int: if …………, ……………………………

x-int: if …………, ……………………………. y = 2x - 3 y-int: if …………, ……………………………

x-int: if …………, ……………………………. Do Exercise 1 on p5 for homework.

L.JE GRADE 9 GRAPHS p2

b) Gradient-Intercept Method: Use this when the graph passes through (0;0) – will have an x-term, y-term and no other number other than 0.

NOTE: - The equation MUST be in standard form - You will know to use this method – when x = 0, you get y = 0.

- Always show the gradient

Step 1: Get equation into std form Step 2: mark c = .. Step 3: Apply gradient m = … and move from c Eg: On the same set of axes, draw rough graphs of x – 2y = 0 and y = -2x x – 2y = 0 y = -2x std form: ………………. is in std form c = c = m = m = Homework: Do p6 Exercise 2 NB: Show gradient

c) Horizontal and Vertical Graphs: y = c

and x = k .

y = c cuts ……-axis at c. NB: m = ……

x = k cuts ……-axis at k. NB: m is ………………………………

Eg: On the same set of axes, draw rough graphs of y = 2 and x = -3.

Graphs of y = x and y = -x.

y = x y = -x

If x = 0 , y = If x = 0 , y =

... grad-int ... grad-int

c = 0 c = 0 (0;0)

m = m =

SHOW GRADIENT & LABEL GRAPH Homework: Do Exercise 3 on p7 and Exercise 4 on p8

LJE GRADE 9 GRAPHS P3 1.2 a) Finding the Equation of a Straight Line Given m or c and a point. HW: Ex5 on P9

2 UNKNOWNS NEED 2 BITS OF INFO a) m and c / b) m & point / c) c & point / d) 2 points a) Find the equations of the following lines a, b, c, d: a: b: For c and d, do not use the Method. Just write the equation

straight away for horizontal and vertical lines.

c:

d: b) & c) Egs : Given c and a point: Find the equation of the line passing through: EXTENSION WORK (a) (0;2) and (3;4). (b) (-1; 3) with gradient -3 I write y = mx + c y = mx + c

Ii c = parallel ... gradients ,,,,,,,,,,,,: m =

Iii Subst into y = mx + c for x, y & c Subst into y = mx+c for x, y & m

(3;4) ………………… (-1; 3) ……………………….

... c = ……… ... c =

... y = ………… ... y = ……………. c) (-1;3) and parallel to y = -3x+1

y = mx + c || y = -3x+1 parallel, m = ….. Substitute for m, x and y into y = mx + c (-1 ; 3) x y d) Finding the gradient of a line between 2 points: Find the gradient of the line that passes through a) (-4; -1) and (-2; 2). b) (-4; -1) and (2; 1).

12

12

xx

yym

−

−= =

Homework: Do Exercise 6 on P10

1.3 Finding the Point of Intersection of 2 Straight Lines (Simultaneous Equations): LJE p4

Worked example: Find the point of intersection of the lines y = x + 2 and y = -x + 4 NB: i Both equations must be in standard form !! ii Cut where equal iii Substitute for x in y = x+2 (or other 1)

…………………….. y = ………………..

x =

Homework: Do Exercise 7 on p10

1.4 Determining whether or not a point lies on a line.

Worked example: Determine whether or not the point (2;4) lies on the line y = 2x + 2.

Solution:

LHS = y =

RHS = 2x + 2

=

= =/= LHS Therefore (2;4) ………………………………… on the line y = 2x+2. 1.5 EXAMPLES OF USING GRAPHS IN LIFE SITUATIONS Homework: p11 Exercise 8 The following graph represents the distance of a traveller from PE over a period of 20 hours . The vertical axis represents the distance in kilometres. The horizontal axis represents the time in hours.

a) What was the distance travelled between D and E ? b) How long did it take for the traveller to go from E to F ? c) Calculate the gradient of the line between C and D. d) Calculate the speed of the traveller between A and B

LJE p5 Exercise 1 on Drawing Straight line graphs using the Intercept-Intercept Method 1. On the same set of axes, draw rough graphs of : i) a) y = 2x – 4 and b) y + x = 4 ii) a) y = - 2x + 4 and b) y - x = 4 y-int: y-int: x-int: x-int:

p6

.

iii) a) y = 2x – 3 and b) y + x = 3 iv) a) y = - 2x + 3 and b) y - x = 3 y-int: y-int: x-int: x-int:

Exercise 2 on Drawing Straight line graphs using Gradient-Intercept Method & Intercept-Intercept LJE p6 1. On the same set of axes, draw rough graphs of : i) a) y = 2x + 4 and b) y = - x ii) a) y = - 2x - 4 and b) y = x y-int: y-int: x-int: x-int:

p6

.

iii) a) y = 2x + 3 and b) y + x = 0 iv) a) y = - 2x - 3 and b) y - 2x = 0 y-int: y-int: x-int: x-int:

Exercise 3 on Drawing ANY Straight line graphs LJE p7 1. On the same set of axes, draw rough graphs of : i) a) y = 2x - 4 b) y = - 2x c) x = 2 ii) a) y = 2x - 1 b) y = - x c) y = 2 y-int: y-int: x-int: x-int:

p6

.

iii) a) y = 2x + 1 b) y + x = 0 c) y = -1 iv) a) y = - 2x - 3 b) y - 2x = 0 c) x = 1,5 y-int: y-int: x-int: x-int:

Exercise 4 on Drawing Straight line graphs using Gradient-Intercept Method & Intercept-Intercept LJE p8 1. On the same set of axes, draw rough graphs of : i) a) y = x + 4 and b) y = - 2x c) y = 4 ii) a) y = - 2x - 1 b) y = x c) y = -1 y-int: y-int: x-int: x-int:

.

iii) a) y + x = 4 and b) y - 2x = 0 c) x = 4 iv) a) y + 2x = 1 b) y - x = 0 c) x = 0,5 y-int: y-int: x-int: x-int:

After Exercise 4 Revise daily using exercises from Textbook.

LJE GRADE 9 GRAPHS After Ex5 do Exercises from Textbook on Finding eqn P9 Exercise 5: Find the Equation of the following straight lines 1) Find the equations of the following lines a, b, c, d: a: b: For c, do not use the Method. Just write the equation

for horizontal and vertical lines. c:

2) Find the equations of the following lines a, b, c, d: a: b:

For c do not use the Method. Just write the equation straight away for horizontal and vertical lines. c:

3) Find the equations of the following lines a, b, c, d: a: b: For c do not use the Method. Just write the equation

straight away for horizontal and vertical lines. c:

Exercise 6. Find the gradient of the line that passes through LJE GR 9 GRAPHS P10

a) (-4; -1) and (-2; 2). b) (-4; -1) and (2; 1).

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yym

−

−= =

c) (-4; -1) and (-2; 2). d) (-4; -1) and (2; 1). e) (-4; -1) and (-2; 2). f) (-4; -1) and (2; 1).

g) (-4; -1) and (-2; 2). h) (-4; -1) and (2; 1).

Exercise 7. Find the point of intersection of the lines:

1. y1 = 2x and y2 = - x + 3

NB: i Both equations must be in standard form !! ii Cut where equal iii Substitute for x in y = x+2 (or other 1)

…………………….. y = ………………..

x =

2. y = - 2x and y = - x + 3

3. y1 = x + 5 and y2 = - x + 3

Exercise 8. Answer the questions on the following real-life graphs LJE GR 9 GRAPHS P11

1. The following graph represents the distance of a traveller from PE over a period of 20 hours . The vertical axis represents the distance in kilometres. The horizontal axis represents the time in hours.

a) What was the distance travelled between B and C ? b) How long did it take for the traveller to go from C to D ? c) Calculate the gradient of the line between E and F. d) Calculate the speed of the traveller between C and D. 2. Mr x travels from PE. The graph shows the distance he travelled over 8 hours. Describe from the graph: a) Give the co-ordinates at A, B, C and D. b) For how many hours and how far he travelled c) Between which two points he was travelling fastest. d) His speed (gradient) between A and B. e) His speed between C and D. f) What was happening between B and C.

2 8 10 6 4

LJE GR 9 GRAPHS P11

3 Mr. Driver who runs a taxi service had to drop a passenger at the airport. The graph below shows the

distance from the taxi depot during the drive. Refer to the graph and answer the questions that follow:

a) How far is the airport from the taxi depot ? …………………………………….. b) How many minutes did it take him to get to the airport from the taxi depot ? c) What was the average speed of the taxi for the first 2 minutes in km/min ? d) Calculate his average speed during the last four minutes before he arrived at the airport.

Time in mins Extension Exercise: Find the equation of the line passing through: (a) (0; -2) and (3;4). (b) (1; -3) with gradient 3 I write y = mx + c y = mx + c

Ii c = parallel ... gradients ,,,,,,,,,,,,: m =

Iii Subst into y = mx + c for x, y & c Subst into y = mx+c for x, y & m

(3;4) ………………… (1; -3) ……………………….

... m = ……… ... c =

... y = ………… ... y = ……………. c) (1;3) and parallel to y = 3x+1

y = mx + c || y = -3x+1 parallel, m = ….. Substitute for m, x and y into y = mx + c (1 ; 3) x y

6

3

2

1

Extra Exercises to add: p13

LJE p7