Pure Chapters 1-7 Flashcards
What is the domain?
The set of possible inputs for a function
What is the range?
The set of possible outputs for a function
Where are the asymptotes of the graph y=k/x or y=k/x^2?
x = 0 y = 0
Transformations: y = f(x) + a
Translation by the vector (0 a)
Transformations: y = f(x + a)
Translation of (-a 0)
Transformations: y = af(x)
Stretch by scale factor a in the vertical direction
Transformations: y = f(ax)
Stretch by scale factor 1/a in the horizontal direction
Transformations: y = -f(x)
Reflection in the x-axis
Transformations: y = f(-x)
Reflection of the graph in the y-axis
What happens to asymptotes in translations?
They are translated in the same way as the graph
What forms can the equation of a line take? (2)
y = mx + c
0 = ax + by + c
What is similar about parallel lines?
They have the same gradient
What are the gradients of perpendicular lines? (2)
A perpendicular to a line of gradient m has a gradient of -1/m
The product of the gradients of two perpendicular lines is -1
What does a graph of direct proportion look like? (2)
The two quantities increase at the same rate meaning the graph is a straight line
The graph passes through the origin
How is an algebraic fraction simplified?
Factorise the numerator and denominator wheee possible and cancel
What is the factor theorem?
If f(x) is a polynomial then: If f(p)=0, then (x-p) is a factor of f(x) If (x-p) is a factor of f(x), then f(p)=0
What is proof by deduction?
Start from known facts or definitions and use logical steps to reach the correct conclusion
What must a mathematical proof contain? (4)
Any information or assumptions being used
Every step shown clearly
All possible cases covered
A statement of proof at the end of the working
How do you prove an identity? (3)
Start with the expression on one side of the identity
Manipulate that expression until it matches the other side
Leave the other expression unchanged
What is proof by exhaustion?
Break the statement into each case and prove them all separately
How do you find a remainder in polynomial division? (2)
Use the grid method and then work out what is needed to reach the desired constant
Substitute the result of the dividing term into the initial equation to find the remainder
What is a counter-example? (2)
One example that does not work for the statement
Only one example is needed to disprove a statement completely
What is a theorem?
A statement that has already been proven
What is a conjecture?
A statement that has yet to be proven
What is the quotient?
The result of polynomial division
What is a polynomial?
A finite expression with positive whole number indices
