Plotting a Graph

x y y coordinate

0

1

2

3

4

Remember coordinates

are plotted ( , )x y

We can work out points to plot a line

y=2x+3The value of y depends on x

For example:If x=1 then y=2(1)+3 =5

35

7

9

11

2(0)+3

2(1)+3

2(4)+3

2(3)+3

2(2)+3

(0,3)

(1,5)(2,7)

(3,9)

(4,11)

y

x

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2 3y x

x y

0 3

1 5

2 7

3 9

4 11

Gradients

Gradient=slope =rate of change =how steep the line is =rise run

y

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m= 5 2

To read a gradient: we start on the line and go up or down first and then back to the line

y

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y

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y

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m= 2/3m= 3/1

m= 3

How’s this graph different from the other 2 graphs?

m= 3/4 ??

m= - 3/4

Draw lines with the following gradients

2/3 3 -1/2 -2

1) Decide which way the line goes by the 1) Decide which way the line goes by the + , – sign+ , – sign

2) Count rise and run2) Count rise and run

Jonathan Climbs a hill. The gradient is 3/2 when he goes uphill, the gradient is -2/5 when he goes downhill. Draw the hill Jonathan climbs. (Assume that it’s a very pointy hill)

y

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y= 3x-2y= -x+5y=1/2x+2y= 4-x

Gradient Intercept Method

gradient y intercept

y

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y

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3 2y x 2

13

y x

2 points determine a line!!!!!2 points determine a line!!!!!(you need 2 points to draw a line)(you need 2 points to draw a line)

The equation is in the form of y=mx+c

y

x

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y=3x-2

y=-x+5

y=3 x + 6 5

y=4 - 1x 2

Application of linear graphs

Applications of GraphsGradient

The gradient is the rate of change of the graph

It is the change in y over the change in x

We can describe it in y units per x unit

Cost$

Timehours

Cost$

Weighttonnes

WaterLevellitres

Time miniutescost

gradienttime

$

hr

charge per hour

costgradient

weight

$

tonne

charge per tonne

water levelgradient

time

litres

minute

water lost per minute

=flow rate

meter

days

Water level of a reservoir

Cooking time

hours

kg

Temperature of an item in the freezertemp

minutes

Interpreting Gradient

Gradient=m/day How many meters the water level drops per day

Gradient=hour/kgHow many hours does it take to cook 1 kg of food.

Gradient=temp/min How much the temp dropsper minute

3

4

5

4

5

2

2

5

7

6.5

5

If the equation of the line is y= -x+5How long would it take before the reservoir is empty?

y

x

The pizzeria Italiano specializes in selling large size PizzasThe relationship between x ( the number of pizzas they sell) andy (their daily costs) is given by the equation y=10x+50.

a) Draw a graph showing the number of pizzas on the x-axis, and the daily costs on the y x-axis.

b) What are their costs if they sell 8 pizzas?

c) If their costs are $100, how many pizzas did they sell.

d) What’s the y-intercept, What does it represent?

e) Give the gradient. What does it represent?

.The time it takes for a block of ice to melt can be represented by a straight-line graph.

x=time (hours)y=volume of the block of ice (litres)

There are two blocks of ice, A and B. The equation for block A is y=6-1/2x. The equation of block B is y=8-xa) Draw the lines for the two blocks of iceb) How large is block A to start with?c) Explain what the y-intercept for block B graph represents?d) When are the two blocks the same volume?e) What do the x-intercepts for the 2 lines tell you?

Linear graph revision

y

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y

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y

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Parabola factorised form

Draw the following parabola

y=3x2

y=(x+2)2 – 4

y=-x2 + 5

y= -(x+3)2 + 2

y

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y=(x-2)(x-3)y=(x-2)(x-3)Is it a parabola???Is it a parabola???

y

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y=(x-1)(x-5) Step a Find the two x interceptsPut y=0 (x-1)(x-5)=0 x=1 and x=5

Step bFind the y interceptPut x=0y=(0-1)(0-5)y= 5

Step cFind the line of symmetryMid-way between 1 and 5x=(1+5)÷2=3

Step dFind the turning pointSubstitute x=3 into y=(x-1)(x-5) y=(3-1)(3-5) y=2×-2 y=-4Turning point (3,-4)

Step (a)Find the x- and y-intercepts by putting y = 0 and x = 0.Step (b) Find the axis of symmetrymidway between the two x-intercepts.Step (c) Find the coordinates of turning point.Substitute the mid-point of two X-intercepts in to the equation to get y

This gives the x coordinate, sub it back into

the original equation to find the y coordinate

( 3 1)( 3 5)

( 2)(2)

4 Vertex ( 3, 4)

y

y

x

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2) Cuts axis 0

(0 1)(0 5)

1 5

5

y x

y

y

y

Parabolas in factorised form

1) Cuts axis 0

( 1)( 5) 0

1 0, 5 0

1, 5

x y

x x

x x

x x

y

x

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3) To find the vertex.

This must be half way between the x intercepts

-5 -1 63

2 2x x x

( 1)( 5)y x x

To find the symmetry

y

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y=(x-4)(x+2) Step (a)Find the x-intercepts by putting y = 0 Step Step (b) Find the y-intercept by putting x=0Step (c)Find the axis of symmetrymidway between the two x-intercepts.Step (d) Find the coordinates of turning point.( , )Substitute the mid-point of two X-intercepts in to the equation to get y

y

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y=(x-4)(x+2) Step a Find the two x interceptsPut y=0 (x-4)(x+2)=0 x=4 and x=-2

Step bFind the y interceptPut x=0y=(0-4)(0+2)y= -8

Step cFind the line of symmetryMid-way between 2 and -3x=(4+-2)÷2=1

Step dFind the turning pointSubstitute x=1 into y=(x-4)(x+2) y=(1-4)(1+2) y=-3×3 y=-9Turning point (1,-9)

y

x

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– 1

2

1) Cuts axis 0

( 3) 0

3 0

3

x y

x

x

x

2( 3)y x

2

2) Cuts axis 0

(0 3)

3

9

y x

y

y

y

y

x

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4) Turning point is at (-3,0)

3) Line of symmetry is at y=-3

P259

14.3

Q1 and Q5

The sketch shows the functiony = x(x - 2)

(a) What are the coordinates of A?(b) What are the coordinates of B?(c) What is the equation of m?(d) What are the coordinates of the turning point of the curve?(e) What is the minimum value of the function? 

The sketch shows the functiony = (x-2)(x+3)

(a) What are the coordinates of A?(b) What are the coordinates of B?(c) What is the equation of D?(d) What are the coordinates of the turning point of the curve?(e) What is the minimum value of the function? 

Match up each of the graphs with the following functions

(a) y = x2 - 2

(b) y - 2 = (x + 1)2

(c) y = (x + 2)(x - 1)

(i)                         (ii)                        

(iii)

                        (iv)                        

(v)                         (vi)                        

(d) y = (x - 2)(x + 1)

(e) y = - x2 + 2

(f) y = x2 + 2

y

x

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– 1

2

1) Cuts axis 0

( 3) 0

3 0

3

x y

x

x

x

2( 3)y x

2

2) Cuts axis 0

(0 3)

3

9

y x

y

y

y

y

x

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1

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– 1

4) Turning point is at (-3,0)

3) Line of symmetry is at y=-3

Parabola applications

x-intercept:(0,0)Y-intercept:NoneVertex:(0,0)

x-intercept:(-4,0) (0,0)y-intercept:(0,0)Vertex:(-2, -4)

y

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X-intercpet:(-3,0) (3,0)Y-intercpet:(0,5)Vertex:(5,0)

Alternatively Draw parabola:y= x2 + 2x - 8

Find

1) X intercepts2) Y intercept3) Vertex (turning point)

For the helicopter to fly above the rainbow parabola, how high must the helicopter fly? (In other words what is the maximum value of the parabola)

Problem 2) Nick threw a ball out of a window that is 4 units high. The position of the ball is determined by the parabola y = -x² + 4.At how many feet from the building does the ball hit the ground?

You need to draw a parabola

There are two solutions. 2 and − 2.

This picture assumes that Nick threw the ball to the rightso that the balls lands at 2 feet away from the building.

Problem 4) A ball is dropped from a height of 36 feet. The quadratic equation d = -t² + 36 provides the distance, d, of the ball, after t seconds. After how many seconds, does the ball hit the ground?

d

t

d = -t² + 36

When the ball hits the groundd=0So we are looking for X intercpt

0= -t² + 36t= ±6

t=-6 is not a sensible answerSo t=6After 6 seconds

?

The stream of water from a fountain can be modelled by a parabola with equation:

H=(2 -d)( 1+d ) H is the height of the water

stream above the ground and d is the distance from the wall (both in meters)

a) Calculate the height of the fountain’s spout.

b) How far from the wall does the water stream hit the ground?

c) Blake stands 1.3m from the wall. He is 1.7m tall. Will he get wet ?

H

d

Wall

ground1.3

Equation for parabola

In an instantWhat do you know about the

following three parabola?

y=(x-3)(x-4) y=(x-2)2 + 3y=x2

Write an equation for a parabola There are 2 forms of parabola

y=(x+a)(x+b) y=(x+1)(x-3)

When you know x-intercepts

y=(x-a)2 + by=(x-4)2 + 1

When you know the turning point

3-1

4

1(4,1)

a cb

g

3-3

-3 1

e f

h i

d

-3

1

-3

3-13

3

1

-3 -1

j

-3 -1

k

3

1

l

3

m

3

n

-3

End of the year revision

0 1 2 3 4 5 6 7 8 9 10-9 -8 -7 -6 -5 -4 -3 -2 -1-10 x

y

1

2

3

4

5

6

7

8

9

10

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

y = x

y = 2x

Drawing Straight Line Graphs

y = -2/3x+4

y = 7

x=-7

0 1 2 3 4 5 6 7 8 9 10-9 -8 -7 -6 -5 -4 -3 -2 -1-10 x

y

1

2

3

4

5

6

7

8

9

10

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

3y-2x=12Intercept Method

2y+4x=8

741 2 3 5 6 8 100

1

2

3

4

5

6

7

8

9

10

9 x

All straight lines have the equation of the form

y = ax + b

GradientWhere linemeets y-axis

y

ww

w.m

ath

srevis

ion

.com

Straight Line Equation

Find the equations of the following lines

y = x y = x+4

Line are parallel same

gradient

y = 4x+2 y = -2x+2

10 20 30 400

10

20

30

40

50

50 E

CQ1. Find the connection between

cost (C) and electricity used (E)

Since line passes through 15 on the y-axis b =15

Gradient a =35 15 20 1

40 0 40 2

y

x

Equation is

ww

w.m

ath

srevis

ion

.com

GeneralStraight Line Equation

115

2C E

10 20 30 400

10

20

30

40

50

50 T

WQ2. The graph shows the

connection between water flowing out of a tank (W)and time (T)

Since line passes through 40 on the y-axis b = 40

Gradient a =40

140

y

x

Equation is

ww

w.m

ath

srevis

ion

.com

GeneralStraight Line Equation

40W T

Downward slope

If we add a number in the brackets graph is shifted horizontallyIf we add a number at the end, the graph is shifted vertically

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2( 3) 4y x

2( 2) 5y x

2( 1) 2y x

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We can also translate the reflected graphs

2 4y x

2( 2) 3y x

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y=3x2

y=-1/2x2

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y=(x-4)(x+2) Step (a)Find the x-intercepts by putting y = 0 Step Step (b) Find the y-intercept by putting x=0Step (c)Find the axis of symmetrymidway between the two x-intercepts.Step (d) Find the coordinates of turning point.( , )Substitute the mid-point of two X-intercepts in to the equation to get y

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y=(x-4)(x+2) Step a Find the two x interceptsPut y=0 (x-4)(x+2)=0 x=4 and x=-2

Step bFind the y interceptPut x=0y=(0-4)(0+2)y= -8

Step cFind the line of symmetryMid-way between 2 and -3x=(4+-2)÷2=1

Step dFind the turning pointSubstitute x=1 into y=(x-4)(x+2) y=(1-4)(1+2) y=-3×3 y=-9Turning point (1,-9)

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1) Cuts axis 0

( 3) 0

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2) Cuts axis 0

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4) Turning point is at (-3,0)