P3 Functions and Graph Transformations

Question 1

Remainder Theorem: F(X) =

Answer 1

Q(x) x divisor + remainder
where Q(x) is the quotient

Question 2

Solving Partial Fractions

Answer 2

1) substitution - eliminate by substituting a x value

2) equating coefficients

Question 3

Repeated Denominator of Partial Fraction etc. (x+1) squared

Answer 3

Write one denominator as (x+1) and another as (x+1) squared

Question 4

Function

Answer 4

A mapping such that every element of the domain is mapped to exactly one element of the range notation

Question 5

One base function

Answer 5

One input for one output

Question 6

Many to one function

Answer 6

Multiple different inputs give the same output

Question 7

One of many function

Answer 7

NOT A FUNCTION

multiple outputs for a single input

Question 8

Domain

Answer 8

set of possible inputs = x values

Question 9

Range

Answer 9

set of possible outputs = y values

Question 10

Vertical Line Test

Answer 10

• To test whether a graph is a function
• Draw a vertical line on the graph
• If the vertical line crosses more than one point on the graph it is not a function

Question 11

Inverse Functions

Answer 11

• always a reflection in x=y
• switch domain and range
• maps that output values to the input values

Question 12

Exponential functions

Answer 12

e.g. 2 to the power of x

output gets multiplied by some constant each time the input increases by a unit

Question 13

Euler’s constant

Answer 13

e = 2.718282….
y = e to the power of x
the gradient is identical to the function

Question 14

ln =

Answer 14

loge

Question 15

inverse of e to power x

Answer 15

ln(x)

Question 16

sketching f(x) = e to the power x

Answer 16

asymptote x = 0
y intercept y = 1
graph keeps rising upwards

Question 17

sketching f(x) = ln(x)

Answer 17

graph y = e power x reflected in y = x
asymptote y = 0
crosses x axis at x=1
graph keeps moving right

Question 18

Modulus of a number (absolute value)

Answer 18

its positive numerical value

eg. |-5| = 5 and |5| = 5

Question 19

Modulus Function of graph y = |f(x)|

Answer 19

“reflection of graph in the y axis and deletion of parts below”

• sketch graph of y = f(x)
• reflect in the x-axis any parts where f(x) < 0 (below the x-axis)
• delete parts below x-axis

Question 20

Modulus Function of graph y = f(|x|)

Answer 20

“reflection in y axis of the positive x values”

• sketch graph of y = f(x) for x bigger than 0
• reflect this in the y axis

Question 21

Solve equation of the type |f(x)| = g(x) or |f(x)| = |g(x)|

Answer 21

• use a sketch to locate the roots

- solve algebraically, using the -f(x) for the reflected parts of y = f(x) and -g(x) for reflected parts of y = g(x)

Question 22

f(x + a)

Answer 22

a horizontal translation of -a (LEFT)

Question 23

f(x) + a

Answer 23

a vertical translation of +a (UPWARDS)

Question 24

f(ax)

Answer 24

a horizontal stretch of scale factor 1/a

Question 25

af(x)

Answer 25

a vertical stretch of scale factor a