Maths Flashcards
How do you test for functions?
The vertical line test.
Run ruler across graph, if x-value has more than one corresponding y-value then it is NOT a function
Define a function
A function is a relation such that each element of the domain is associated with exactly one element of the range
What is and how do you write the equation of an asymptote?
Asymptote: a straight line that the graph tends to but never touches
E.g x=0 and y=0
What do the different brackets (..) and [..] mean when writing the domain and range of a function
Curved brackets (..) mean value is excluded from domain/range
Squares brackets [..] mean value is included in domain/range
Composite functions
f[g(x)] = (fog)(x) -> a function of a function
If f(x)=x+3 and g(x)=2x +1 then:
(fog) (x)=(2x+1)+3
(fog) (x)=2x+4
If: (fog)(x)=x
and (gof)(x)=x
What is the relationship between these functions?
They are inverse functions of one another
How do you calculate inverse functions?
let f(x)=y then swap x->y and solve for y
e.g: f(x)=2x+5 y=2x+5 x=2y+5 x-5=2y (x-5)/2=y
So inverse function of f(x)=2x+5 is f^-1(x)=(x-5)/2
How can you test whether a function will have an inverse?
The horizontal line test.
Run ruler down graph - if it hits two points on the graph at same time then it has no inverse function
f(x)+a
What does this do to the graph?
It vertically shifts the graph
+a -> upwards (towards positive)
-a -> downwards (towards negative)
af(x)
What does this do to the graph?
Stretches the graph vertically
Critical points of the x-axis do not change
+a -> steeper - divide by a
-a -> flatter - multiply by a
f(x+a)
What does this do to the graph?
Horizontally shifts the graph
+a -> moves left (towards negative)
-a -> moves right (towards positive)
Opposite to sign
f(ax)
What does this do to the graph?
Stretches the graph horizontally - x values
Reciprocal of a number is…
1/number
A number multiplied by its reciprocal is..
1
In reciprocal functions, what does increasing the value of the numerator do to the graph?
It stretches the graph - the curve gets further away from the origin
For reciprocal functions, what does having a negative value on the numerator do to the graph?
It inverses the graph
What is a polynomial?
An equation in which x is raised to a power in series
e.g: g(x)=3x^3 + 4x^2 +6x
How do you find the vertical asymptote of a reciprocal graph?
By taking the opposite value of ‘+c’ in the denominator
e.g: y=x+7 -> y=1/(x+7) -> x= -7
How do you find the horizontal asymptote of a reciprocal graph?
Taking the coefficient of x in the numerator
e.g: y=x/3-x -> y=1
How to complete the square
x^2 + bx + c = 0
=> (x + b/2)^2 - (b/2)^2 +c=0
Solve for x
Or take c away from both sides
x^2+bx=-c -> take 1/2 coefficient of b, square it, then add to both sides
Solve for x
Depends if you want all on one side
-f(x)
What does this do to the graph?
It reflects the graph f(x) in the x-axis
f(-x)
What does this do to the graph?
It reflects the graph f(x) in the y-axis
One way to find the axis of symmetry is to use a formula derived from the quadratic formula. What is this?
-b/2a
How do you find the equation if a tangent?
- Differentiate equation of graph
- Sub x point into differentiation to find gradient
- Sub x point into original graph equation and solve to find y point
- Sub all values into y-y1=m(x-x1)
What is the product rule?
If f(x)= u(x) • v(x)
Then f’(x)= u(x)•v’(x) + v(x)•u’(x)
What is the quotient rule?
If f(x)= u(x)/v(x)
Then f’(x)= (v(x)•u’(x) - u(x)•v’(x))/[v(x)]^2
What is the chain rule?
If f(x)= u(v(x))
Then f’(x)= u’(v(x)) • v’(x)
How to write a horizontal asymptote/line for a line for a value (k)
y = k
How to write a vertical asymptote/line for a value (k)
x = k
Asin(Bx + C) + D
A = amplitude (* y-values stretched by factor A)
B = period -> 2/ |B| for sin&cos
/ |B| for tan
C = Horizontal shift by C/B opposite to sign: - (right)/+ (left)
D = Vertical shift: +(up)/ - (down)
