Graph Sketching and expressions

Question 1

Factorising cubics

Answer 1

Factorize by doing first half of cubic then second half to find common bracket

try 1, 2, -1, -2, 1/2, -1/2

Question 2

Sketching y =sin^2((root x))

Answer 2

look at +ve, -ve, max, min, zeros

Question 3

If constant at end, when sketching…

Answer 3

sketch without constant then shift it

Question 4

Cubics (odd powers) will always have at least one solution because

Answer 4

the y-value goes from -ve infinity to +ve infinity

Question 5

Look at when x tends to +/- infinity

Answer 5

discriminant, roots, asymptote, zeroes, axis

Question 6

Single/odd factor (x-a) means

Answer 6

the curve crosses at x=a

Question 7

Repeated/even factor (x-a)^2 means

Answer 7

the curve touches at x=a

Question 8

Find the asymptopes of the curve, find intercepts, turning points

Answer 8

pay attention to domains to have correct amount of required solutions

Question 9

Is the graph a series of transformations of a simpler graph

Answer 9

Symmetry (is the function odd or even)

Question 10

To find max/min of quadratic

Answer 10

complete the square
max if -ve
min if +ve

Question 11

When two lines of touch

Answer 11

both the y-value and the gradient are the same

Question 12

When they intersect,

Answer 12

only the y-values are the same

Question 13

Ensure you look if question says intersect or touch

Answer 13

as they mean different things

Question 14

Look for dominant parts of graph when combining functions

Answer 14

graphs of x^2 + root(x)*x
sin^2(x)
sin(root x)
…

Question 15

Odd powers =

Answer 15

infinite range, min roots = 1, max roots = max power

Question 16

Maximum turning points =

Answer 16

maximum power - 1

Question 17

Even =

Answer 17

finite range, min roots = 0, max roots = max power

Question 18

(x-a) means

Answer 18

line crosses at x=a

Question 19

(x-a)^2 means

Answer 19

line touches at x=a

Question 20

(x-a)^3 means

Answer 20

points of inflection at x=a

Question 21

Sin(x^2) –>

Answer 21

x^2 causes oscillation period (width of hill) to decrease

Question 22

2^-x peaks

Answer 22

gradually become shallower

Question 23

Limit(1/x) =

Answer 23

0

Question 24

Limit f(x)/g(x) =

Answer 24

limite f’(x)/g’(x)

Question 25

When finding the number of distinct roots for a changing constant -

Answer 25

sketch without constant then shift up or down

Question 26

a^x grows quicker than x^b

Answer 26

x^x grows quicker than x!

Question 27

(x–>0)limit(x^x) =

Answer 27

1

Question 28

Odd function:

Answer 28

f(-x) = -f(x) sine

Question 29

Even function:

Answer 29

f(-x) = f(x) cosine

Question 30

sin(2pi - x) =

Answer 30

sin(-x) = -sin(x)

Question 31

(x-1)(x-2)….(x-n) = k

Answer 31

n=3 then cubic which always has a solution from -ve infinity to +infinity

n is even then may have no solution depending if minimum is below the x axis i.e. only for certain values of k

k>=0 then if n is odd then would have a solution

The equation could have a repeated solution depending on n and k

Question 32

x^3:

Answer 32

max –> min

Question 33

x^4:

Answer 33

min –> max –> min

Question 34

x^5:

Answer 34

max –> min –> max –> min

Question 35

combined functions

Answer 35

sub in values
look at ranges of individual parts
which bit is the dominant part of the graph

Question 36

Cubics will always have

Answer 36

at least one solution because the y-value goes from -ve infinity to +ve infinity (in general this is true when the greatest power is an odd number)

Question 37

When you have f(x) + g(x)

Answer 37

its often useful to consider the graphs separately, and what happens when you put in values, then add them

Question 38

Cubic stationary point:

Answer 38

max –> min

Question 39

Quartic stationary point:

Answer 39

min –> max –> min

Question 40

Quintic stationary point:

Answer 40

max –> min –> max –> min

Question 41

When two lines touch:

Answer 41

both y value and gradient area equal

Question 42

When two lines interest

Answer 42

only the y value is equal

Question 43

when they touch

Answer 43

equate y values and gradient (use discriminant for gradient)

Question 44

sketch x^4 - y^2 = 2y +1

Answer 44

sketch x^6 + y^6 = 1

Question 45

sketch |x+y| = 1

Answer 45

Compare coefficients

Question 46

y=2^-x * sin^2(x^2) –>

Answer 46

(0,0) y>0, max+min, period, values, dominant part

Question 47

I(c) >=0 as

Answer 47

no solutions so above the x axis

Question 48

sketch y=sin^2(root x)

Answer 48

sketch y=sin(x)/x

Question 49

Sketch y= root (2-x^2),

Answer 49

x+(root2 - 1)y = root 2

Question 50

sketch y=log(f(x)) –> solutions, y axis, quadrants

Answer 50

Dominant part of graph

Question 51

y= 2^(x^2 - 4x + 3):

Let a=2^x^2 = 2^f(x) therefore f(x-2)^2 +1

Answer 51

Think of graph to justify max/min

Question 52

Repeated root touches (x+a)^2n

Answer 52

root crosses (x+a)^2n+1

Question 53

Transforming functions

Answer 53

f(x+-a) -f(x) f(-x) f(1/x)

Question 54

sub in values for graphs, think of direction, turning points

Answer 54

when inequalities, think of discriminant

Question 55

Simplify complex transformations

Answer 55

Try and simplify transformations e.g. reflection in both axes and a shift (2, 0) = 180 degree rotation about (1,0)

Question 56

f(x) = k

Answer 56

therefore k is a negative shift if k is positive

Question 57

sketch quartics using calculus

Answer 57

sin(x) = sin(y) (0,0, (pi, 0)

Question 58

(x^3 - 1)^2, sketch (x^(2n-1))^2

Answer 58

f(x) = k therefore k is a shift of y co-ordinates

Question 59

Sketch a^2 - a^1.5 - 8 = 0

Answer 59

Asymptopes, axis’, x –> infinity

Question 60

To find where f(x)-g(x) crosses the x axis,

Answer 60

sketch both functions and look for intersction

Question 61

Sketch cos(pi*x)

Answer 61

Tan values = 1/root3, 1, root3,

Question 62

Sketching x^2, x^3, x^4 and x^5 together -

Answer 62

all cross (0, 0) and intersect at (1, 1)

Between 0 < x < 1 results in smaller value

-1 < x < 0 greater power

x^2 has greatest values so is above the rest between -1 and 1, then after 1 and before -1 it is lower

x^5 is below the rest up until 1 then goes above it

in negative part, x^3 is inside of x^5 up to -1 and then it is outside