Maths

Question 0

How do you test for functions?

Answer 0

The vertical line test.

Run ruler across graph, if x-value has more than one corresponding y-value then it is NOT a function

Question 1

Define a function

Answer 1

A function is a relation such that each element of the domain is associated with exactly one element of the range

Question 2

What is and how do you write the equation of an asymptote?

Answer 2

Asymptote: a straight line that the graph tends to but never touches

E.g x=0 and y=0

Question 3

What do the different brackets (..) and [..] mean when writing the domain and range of a function

Answer 3

Curved brackets (..) mean value is excluded from domain/range

Squares brackets [..] mean value is included in domain/range

Question 4

Composite functions

Answer 4

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

Question 5

If: (fog)(x)=x
and (gof)(x)=x
What is the relationship between these functions?

Answer 5

They are inverse functions of one another

Question 6

How do you calculate inverse functions?

Answer 6

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

Question 7

How can you test whether a function will have an inverse?

Answer 7

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

Question 8

f(x)+a

What does this do to the graph?

Answer 8

It vertically shifts the graph
+a -> upwards (towards positive)
-a -> downwards (towards negative)

Question 9

af(x)

What does this do to the graph?

Answer 9

Stretches the graph vertically
Critical points of the x-axis do not change
+a -> steeper - divide by a
-a -> flatter - multiply by a

Question 10

f(x+a)

What does this do to the graph?

Answer 10

Horizontally shifts the graph
+a -> moves left (towards negative)
-a -> moves right (towards positive)
Opposite to sign

Question 11

f(ax)

What does this do to the graph?

Answer 11

Stretches the graph horizontally - x values

Question 12

Reciprocal of a number is…

Answer 12

1/number

Question 13

A number multiplied by its reciprocal is..

Answer 13

1

Question 14

In reciprocal functions, what does increasing the value of the numerator do to the graph?

Answer 14

It stretches the graph - the curve gets further away from the origin

Question 15

For reciprocal functions, what does having a negative value on the numerator do to the graph?

Answer 15

It inverses the graph

Question 16

What is a polynomial?

Answer 16

An equation in which x is raised to a power in series

e.g: g(x)=3x^3 + 4x^2 +6x

Question 17

How do you find the vertical asymptote of a reciprocal graph?

Answer 17

By taking the opposite value of ‘+c’ in the denominator

e.g: y=x+7 -> y=1/(x+7) -> x= -7

Question 18

How do you find the horizontal asymptote of a reciprocal graph?

Answer 18

Taking the coefficient of x in the numerator

e.g: y=x/3-x -> y=1

Question 19

How to complete the square

Answer 19

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

Question 20

-f(x)

What does this do to the graph?

Answer 20

It reflects the graph f(x) in the x-axis

Question 21

f(-x)

What does this do to the graph?

Answer 21

It reflects the graph f(x) in the y-axis

Question 22

One way to find the axis of symmetry is to use a formula derived from the quadratic formula. What is this?

Answer 22

-b/2a

Question 23

How do you find the equation if a tangent?

Answer 23

1. Differentiate equation of graph
2. Sub x point into differentiation to find gradient
3. Sub x point into original graph equation and solve to find y point
4. Sub all values into y-y1=m(x-x1)

Question 24

What is the product rule?

Answer 24

If f(x)= u(x) • v(x)

Then f’(x)= u(x)•v’(x) + v(x)•u’(x)

Question 25

What is the quotient rule?

Answer 25

If f(x)= u(x)/v(x)

Then f’(x)= (v(x)•u’(x) - u(x)•v’(x))/[v(x)]^2

Question 26

What is the chain rule?

Answer 26

If f(x)= u(v(x))

Then f’(x)= u’(v(x)) • v’(x)

Question 27

How to write a horizontal asymptote/line for a line for a value (k)

Answer 27

y = k

Question 28

How to write a vertical asymptote/line for a value (k)

Answer 28

x = k

Question 29

Asin(Bx + C) + D

Answer 29

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)