Sketching Graphs

Question 1

Sketch y = x^2 - 9

Answer 1

• ( First identify what type of graph it is ( Quadratic ) )
• ( The find the x intercepts by making y = 0 )
• 0 = x^2 - 9
• x^2 = 9
• x = +- 3
• ( Then we find the y intercept by making x = 0 )
• y = ( 0 )^2 - 9
• y = - 9
• ( Then draw out the quadratic )

Question 2

The graph of function y = f ( x ) has intersections with the axes only at the points ( - 2, 0 ), ( 0, 8 ) and ( 2, 0 ).
Constants p, q and r are all positive integers.
Which of the following functions could be f ( x )?
For those functions, find the values of p, q and r.

• px + q
• p( q - x^2 )
• p( x^2 - q )
• x^3 + px + qx + r
• x^3 - px^2 - qx + r
• p + qx - rx^2 - x^3

Answer 2

px + q:

• Not possible because it is a linear equation

p ( q - x^2 ):

• ( If we expand it )
• pq - px^2, ( since p and q are integers, it appears to be a quadratic graph equation )
• y = -px^2 + pq
• ( We can plot in any of the two points to make a simultaneous equation )
• 8 = - p( 0 )^2 + pq ( 0, 8 )
• 0 = - p( 2 )^2 + pq ( 2, 0 )
• 8 = pq
• 0 = - 4p + pq
• ( Minus them )
• 8 = - 4p
• p = - 2
• ( However p is positive so p = 2 )
• ( input p = 2 into one of the equations )
• 8 = ( 2 )q
• 2q = 8
• q = 4
• ( No r values for this equation )

p ( x^2 - q ):

• Not possible

x^3 + px^2 + qx + r:

• Not possible
• x^3 - px^2 - qx + r:
• ( This is a cubic graph )
• ( Points: ( 2, 0 ), ( - 2, 0 ) and ( 0, 8 ) )
• ( That means we have: )
• ( x + 2 ) and ( x - 2 )
• ( One of these coordinates must be squared as it’s a three point graph, since the graph is positive, it goes in the positive direction, therefore goes up from - 2 on the x axis and through ( 0, 8 ) and to 2 on the x axis, meaning this last coordinate must be the squared one )
• y = ( x - 2 )^2 ( x + 2 )
• ( Now we can expand it to identify p, q and r )
• x^ 3 - 2x^2 - 4x + 8
• p = 2 ( p, q and r are all positive integers )
• q = 4
• r = 8

p + qx - rx^2 - x^3:

• ( Graph is going the opposite direction to our previous answer )
• ( So equation is the opposite as well )
• y = ( x - 2 ) ( x + 2 )^2
• ( Expand to find p, q and r )
• 8 + 4x - 2x^2 - x^3
• p = 8
• q = 4
• r = 2