3)some important graphs

Question 1

3.2.1 In which quadrant is xy greater than 0

Answer 1

1st and 3rd, because positivepositive= positive and negativenegative=negative

Question 2

3. 2.2 a)picture the COMPLETE graph of y=f(x) , where f(x)=the sqrt of 4-x^2
   b) why does it have this shape?

Answer 2

a) A graph in which the line is a semicircle between the points x=-2 and x=2 where there are no points when y= a negative integer ,
b) x coordinate were worked out as square root cannot be negative so 4-x^2= positve ans , so x^2

Question 3

3. 4/5, a)Where is the vertex of the parabola if a= positive?
   b) Where is the vertex of the parabola if a= negative?

Answer 3

a) u, at the lowest part

b) upside down u, at highest part

Question 4

3.4/5 How do you find where a parabola (so will be of the form y=x^2+bx+c) intercepts the y axis?

Answer 4

substitute x= 0 into the equation, so will =c

Question 5

3.4/5 How do you find where a parabola intercepts the x axis?

Answer 5

first check it does usng b^2-4ac= 0 or more ,substitute y=0 into the equation, then factorise or quadratic equation( but you don’t have a calculator for this exam)

Question 6

3.4/5 How do you find where a parabolas line of symmetry (x value for vertex) is?

Answer 6

Its x= the midpoint of the two values of the x intercept or x= the value of the only x intercept or if there are no points of the x intercept x=the midpoint of two values where y = same thing for each coordinate

Question 7

3.4/5 How do you find the equation of the line of symmetry?

Answer 7

Its x= the midpoint of the two values of the x intercept or x= the value of the only x intercept or if there are no points of the x intercept its x=the midpoint of two values where y = same thing for each coordinate,REMEMBER TO SAY X=BLAHBLAH….NOT Y=BLAHBLAH

Question 8

3.4/5 How do you find the coordinate of the vertex of a parabola?

Answer 8

find the line of symmetry , then sub in x into the equation to find y , then right out the cordinate

Question 9

3.4/5 What is a very quick method to find the the coordinate of the vertex of a parabola?

Answer 9

complete the square, whatever is added or subtracted from x within the bracket multiply by -1 that is the x coordinate( even if squared bracket is then taken away e.g. y=11-(x+3)^2), whatever is outside the bracket is the y coordinate

Question 10

3. 7.1 Draw a sketch of the graph f(x)= 3x^2-2x-1:

hint: use completing sqaure,only need to find vertex ,y intercept and line of symmetry

Answer 10

1st complete the square)f(x)=3x^2-2x-1
 f(x)={3(x-1/3)^2-1/6}-1
f(x)=3(x-1/3)^2-7/6
2nd find vertex)
 -1/3 * -1=1/3 x=1/3
and y= -7/6 vertex= (1/3, -7/6)
3rd use equation to find y intercept) sub in x= 0 , easiest to do in the form f(x)= ax^2+bx+c as then y=c
y=-1 y intercept= (0,-1)
4th) draw sktech using this info, remeber since in completed square form brakcet is multiplied parabola is a little stretched but doesn't matter when plotting if you include y intercept

Question 11

3.3 What coordinates do all graphs of y=x^n have?

Answer 11

(0,0) and (1,1)

Question 12

3. 3 If n is greater in the the equation y=x^n, what happen to the positioning of the graph a)between x=0 and x=1 and b)x>1
   c) explain why this happens

Answer 12

a) graphs stays closer as c) x^n+1 =x*x^n so when 0x^n when x>1

Question 13

3.3 Define an even function

Answer 13

Function with the property of f(-a)=f(a) for all values of a e.g. f(x)=x^2

Question 14

3.3 What properties do even function graphs have?

Answer 14

the graph is symmetrical about the y-axis, graph can be any shape as long as it has the line of symmetry y=0

Question 15

3.3 Define an odd function

Answer 15

Functions with the property f(x)= -f(-x) or -f(x)=f(-x) e.g. y=x or y=x^3

Question 16

3.3 What properties do odd function graphs have?

Answer 16

They appear to be inverted about the x axis then reflect about the y or you could say symmetrical bout the origin

Question 17

3.6.1 Find the point if intersection of y=2 and y=x^2-3x+4

Answer 17

1. solve them simultaneously) x^2-3x+4=2
   x^2-3x+2=0
   (x-1)(x-2)=0
   so x=1 and x=2
2. sub in x to find the value of y) use the easiest equation which is in the case y=2 therefore y=2 in both instances
3. write out coordinates) (1,2) and (2,2)

Question 18

3.6.2 a) Find the point if intersection of y=2x-1 with the graph y=x^2

Answer 18

1. solve them simultaneously) x^2= 2x-1
x^2-2x+1=0
(x-1)(x-1)=0
so x=1
2. sub in x to find the value of y ,use the easiest equation which is in the case y=2x-1 )
2x-1= (2*1)-1
=2-1= 1   so y=1
3. write out coordinate) (1,1)

Question 19

3.6.2 b)You have found the point if intersection of y=2x-1 with the graph y=x^2 , YOu have found there is only one point of intersection, what does this tell you?

Answer 19

the line is a tangent to the graph, the point they touch is the POINT OF CONTACT of the tangent and the curve

Question 20

3.6 What are the general steps to find the point of intersection of two graphs?

Answer 20

1) solve them simultaneously
2) substitute in x to find y
3) write out coordinates

Question 21

3.7 Use functions to explain your sketchings of graphs

Answer 21

1) explain how you found y intercept
e. g.f(0)=4 (0,4)= y intercept
2) explain how you find the vertex of a graph:
a) you completed the square
b) then you inputted each step into the function
e. g. y=(x-1)+3 that is the same as f(x+1)-3=0
c) you translated the graph so the vertex is in position
e. g. i this case vertex coordinate = (-1,3)

Question 22

3. 7. Use factors to sketch graphs

hint: basicaly means factorize to sketch graph

Answer 22

1) factorize to find x intercept
e.g. y= x^2-4x-5
y=(x+1)(x-5) so x=-1 and x=5 so x intercept is (-1,0) (5,0)
2) use midpoint to find line of symmetry
e.g. midpoint of -1 and 5 is 2 so ans= x=2
3) find vertex : by sub in x=2
4)find y intercept by sub in x=0
5) plot and draw line of best fit

Question 23

3.8 Find the equation of the graph of the type y=ax^2+bx+c which crosses the x-axis at the points (1,0) and (4,0), and also passes through the point (3,-4)

Answer 23

a) draw out graph
b) write out the form of the equation using the x intercept values
e. g.since the curve cuts the x-axis at (1,0) and (4,0) the equation has the form: y=a(x-1)(x-4)
c) sub in values of x=3 and y=-4 to find the value of a
d) sub in a value into the form of the equation
e) put equation into desired form e.g. for this you’d need to expand the brackets

Question 24

3.8 What does a ,r, s,t tell you in the equation

f(x)= a(x-r)(x-s)(x-t)?

Answer 24

that the graph passes through the points (r,0), (s,0) and (t,0)
a positive sign of the constant a tells you the graph increases from left to right
a negative= it decreases ,

Question 25

3.8 sketch the graph y=-(x+2)(x-1)^2

Answer 25

a) find the x intercepts (remember r,s,t):
in this example 2 and -1 are in the brackets so the coordinates for the x intercept are: (-2,0) and (1,0)
b) find the y intercept ( multiply numbers and don’t forget those pesky signs):
y=-12-1^2 y=-2
c) plot coordinates and draw line of best fit don’t forget its curve

Question 26

Excersice 3D que.9 ) ensure you get 100 percent on these and that you understand the theory behind these questions needed

Answer 26

the questions ask for you to suggest a possible equation for each graph always check your answer.