plotting a graph
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Plotting a Graph. y=2x+3. The value of y depends on x For example: If x=1 then y=2(1)+3 =5. (0,3). 2(0)+3. 3. 5. 2(1)+3. (1,5). 2(2)+3. (2,7). 7. 2(3)+3. (3,9). 9. 2(4)+3. (4,11). 11. Gradients. Gradient=slope =rate of change  PowerPoint PPT PresentationTRANSCRIPT

Plotting a Graphy=2x+3The value of y depends on x
For example:If x=1 then y=2(1)+3 =53579112(0)+32(1)+32(4)+32(3)+32(2)+3(0,3)(1,5)(2,7)(3,9)(4,11)
xyycoordinate01234

xy03152739411

GradientsGradient=slope =rate of change =how steep the line is =rise runm= 5 2

To read a gradient: we start on the line and go up or down first and then back to the line
m= 2/3m= 3/1
m= 3Hows 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 + , sign2) 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 its a very pointy hill)

y= 3x2y= x+5y=1/2x+2y= 4x

Gradient Intercept Methodgradienty intercept2 points determine a line!!!!!(you need 2 points to draw a line)The equation is in the form of y=mx+c

y=3x2
y=x+5
y=3 x + 6 5
y=4  1x 2

Application of linear graphs

Applications of GraphsGradientWaterLevellitres
Time miniutes

meterdaysWater level of a reservoirCooking timehourskgTemperature of an item in the freezertempminutesInterpreting GradientGradient=m/day How many meters the water level drops per dayGradient=hour/kgHow many hours does it take to cook 1 kg of food.
Gradient=temp/min How much the temp dropsper minute
3454522576.55If the equation of the line is y= x+5How long would it take before the reservoir is empty?yx

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.
Draw a graph showing the number of pizzas on the xaxis, and the daily costs on the y xaxis.
b) What are their costs if they sell 8 pizzas?
c) If their costs are $100, how many pizzas did they sell.
d) Whats the yintercept, 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 straightline 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=61/2x. The equation of block B is y=8xDraw the lines for the two blocks of iceHow large is block A to start with?Explain what the yintercept for block B graph represents?When are the two blocks the same volume?What do the xintercepts for the 2 lines tell you?

Linear graph revision

Parabola factorised form

Draw the following parabola y=3x2y=(x+2)2 4y=x2 + 5y= (x+3)2 + 2

y=(x2)(x3)Is it a parabola???

y=(x1)(x5)Step a Find the two x interceptsPut y=0 (x1)(x5)=0 x=1 and x=5
Step bFind the y interceptPut x=0y=(01)(05)y= 5
Step cFind the line of symmetryMidway between 1 and 5x=(1+5)2=3
Step dFind the turning pointSubstitute x=3 into y=(x1)(x5) y=(31)(35) y=22 y=4Turning point (3,4)

Step (a)Find the x and yintercepts by putting y = 0 and x = 0.Step (b) Find the axis of symmetrymidway between the two xintercepts.Step (c) Find the coordinates of turning point.Substitute the midpoint of two Xintercepts in to the equation to get y

Parabolas in factorised formTo find the symmetry

y=(x4)(x+2) Step (a)Find the xintercepts by putting y = 0 Step Step (b) Find the yintercept by putting x=0Step (c)Find the axis of symmetrymidway between the two xintercepts.Step (d) Find the coordinates of turning point.( , )Substitute the midpoint of two Xintercepts in to the equation to get y

y=(x4)(x+2)Step a Find the two x interceptsPut y=0 (x4)(x+2)=0 x=4 and x=2
Step bFind the y interceptPut x=0y=(04)(0+2)y= 8
Step cFind the line of symmetryMidway between 2 and 3x=(4+2)2=1
Step dFind the turning pointSubstitute x=1 into y=(x4)(x+2) y=(14)(1+2) y=33 y=9Turning point (1,9)

4) Turning point is at (3,0)
3) Line of symmetry is at y=3

P25914.3Q1 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 = (x2)(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

4) Turning point is at (3,0)
3) Line of symmetry is at y=3

Parabola applications

xintercept:(0,0)Yintercept:NoneVertex:(0,0)
xintercept:(4,0) (0,0)yintercept:(0,0)Vertex:(2, 4)
Xintercpet:(3,0) (3,0)Yintercpet:(0,5)Vertex:(5,0)
Alternatively Draw parabola:y= x2 + 2x  8Find
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 = x + 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 = t + 36 provides the distance, d, of the ball, after t seconds. After how many seconds, does the ball hit the ground? dtd = t + 36When the ball hits the groundd=0So we are looking for X intercpt
0= t + 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)
Calculate the height of the fountains spout.How far from the wall does the water stream hit the ground?Blake stands 1.3m from the wall. He is 1.7m tall. Will he get wet ?
HdWallground1.3

Equation for parabola

In an instantWhat do you know about the following three parabola?
y=(x3)(x4)
y=(x2)2 + 3
y=x2

Write an equation for a parabola There are 2 forms of parabola y=(x+a)(x+b) y=(x+1)(x3)
When you know xintercepts
y=(xa)2 + by=(x4)2 + 1
When you know the turning point3141(4,1)

acbg3331efhid3133133131

j31k31l3m3n3

End of the year revision

y = xy = 2x Drawing Straight Line Graphsy = 2/3x+4y = 7x=7

3y2x=12Intercept Method2y+4x=8

74123568100123456789109xAll straight lines have the equation of the form y = ax + bGradientWhere linemeets yaxisywww.mathsrevision.comStraight Line EquationFind the equations of the following linesy = xy = x+4Line are parallel same gradienty = 4x+2y = 2x+2

102030400102030405050ECQ1.Find the connection betweencost (C) and electricity used (E)Since line passes through 15 on the yaxis b =15 Gradient a =Equation is www.mathsrevision.comGeneralStraight Line Equation

102030400102030405050TWQ2.The graph shows theconnection between water flowing out of a tank (W)and time (T)Since line passes through 40 on the yaxis b = 40 Gradient a =Equation is www.mathsrevision.comGeneralStraight Line EquationDownward 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

We can also translate the reflected graphs

y=3x2
y=1/2x2

y=(x4)(x+2) Step (a)Find the xintercepts by putting y = 0 Step Step (b) Find the yintercept by putting x=0Step (c)Find the axis of symmetrymidway between the two xintercepts.Step (d) Find the coordinates of turning point.( , )Substitute the midpoint of two Xintercepts in to the equation to get y

y=(x4)(x+2)Step a Find the two x interceptsPut y=0 (x4)(x+2)=0 x=4 and x=2
Step bFind the y interceptPut x=0y=(04)(0+2)y= 8
Step cFind the line of symmetryMidway between 2 and 3x=(4+2)2=1
Step dFind the turning pointSubstitute x=1 into y=(x4)(x+2) y=(14)(1+2) y=33 y=9Turning point (1,9)

4) Turning point is at (3,0)