Graphs and Sketching

• Co-ordinates of key points must be marked in a sketch
• Always check the co-efficient of X^2
• Always identify intersections with the y and x axid
• b^2 - 4ac is known as the discriminant
• If the discriminant is equal to 0 than f(x) has 1 repeated root
• If the discriminant is below 0 then f(x) has no real roots
• The number of turning points = The coefficient of x^2 - 1
• If the order of the power on ax^n is odd, the shape goes uphill
• If the order of the power on ax^n is even, the tails go upwards
• What graph is this

  A) 2

  
• An x^4 graph

  
• An x^5 graph

  
• When completing the square, always factor out the coefficient of x^2 first
• The gradient is delta y over x
• y = mx + c -> equation of a straight lin
• The points of intersection of two lines are found by setting them equal to one another
• m = (y2 - y1)/(x2 - x1) -> gradient formula
• distance -> d = √(x2 - x1)^2 + (y2 - y1)^2
• A change inside f(x) is an opposite move
• A change outside f(x) is a literal change
• Inside the bracket changes the x co-ordinate
• Outside the bracket changes the y co-ordinate
• The gradient is a constant in a straight line
• Midpoint, M = (((x1 + x2)/2) + ((y1 + y2)/2))
• The different types of proof

  A) counterexample

  B) deduction

  C) exhaustion
• Perpendicular lines give negative gradients
• Parallel lines give the same gradient
• Euler's number = 2.71828
• ln (e) = 1