3)some important graphs Flashcards
3.2.1 In which quadrant is xy greater than 0
1st and 3rd, because positivepositive= positive and negativenegative=negative
- 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?
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
- 4/5, a)Where is the vertex of the parabola if a= positive?
b) Where is the vertex of the parabola if a= negative?
a) u, at the lowest part
b) upside down u, at highest part
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?
substitute x= 0 into the equation, so will =c
3.4/5 How do you find where a parabola intercepts the x axis?
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)
3.4/5 How do you find where a parabolas line of symmetry (x value for vertex) is?
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
3.4/5 How do you find the equation of the line of symmetry?
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
3.4/5 How do you find the coordinate of the vertex of a parabola?
find the line of symmetry , then sub in x into the equation to find y , then right out the cordinate
3.4/5 What is a very quick method to find the the coordinate of the vertex of a parabola?
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
- 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
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 intercept3.3 What coordinates do all graphs of y=x^n have?
(0,0) and (1,1)
- 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
a) graphs stays closer as c) x^n+1 =x*x^n so when 0x^n when x>1
3.3 Define an even function
Function with the property of f(-a)=f(a) for all values of a e.g. f(x)=x^2
3.3 What properties do even function graphs have?
the graph is symmetrical about the y-axis, graph can be any shape as long as it has the line of symmetry y=0
3.3 Define an odd function
Functions with the property f(x)= -f(-x) or -f(x)=f(-x) e.g. y=x or y=x^3
3.3 What properties do odd function graphs have?
They appear to be inverted about the x axis then reflect about the y or you could say symmetrical bout the origin
3.6.1 Find the point if intersection of y=2 and y=x^2-3x+4
- solve them simultaneously) x^2-3x+4=2
x^2-3x+2=0
(x-1)(x-2)=0
so x=1 and x=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
- write out coordinates) (1,2) and (2,2)
3.6.2 a) Find the point if intersection of y=2x-1 with the graph y=x^2
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)
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?
the line is a tangent to the graph, the point they touch is the POINT OF CONTACT of the tangent and the curve
3.6 What are the general steps to find the point of intersection of two graphs?
1) solve them simultaneously
2) substitute in x to find y
3) write out coordinates
3.7 Use functions to explain your sketchings of graphs
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)
- Use factors to sketch graphs
hint: basicaly means factorize to sketch graph
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
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)
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
3.8 What does a ,r, s,t tell you in the equation
f(x)= a(x-r)(x-s)(x-t)?
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 ,
3.8 sketch the graph y=-(x+2)(x-1)^2
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
Excersice 3D que.9 ) ensure you get 100 percent on these and that you understand the theory behind these questions needed
the questions ask for you to suggest a possible equation for each graph always check your answer.
