Graph Sketching and expressions Flashcards
Factorising cubics
Factorize by doing first half of cubic then second half to find common bracket
try 1, 2, -1, -2, 1/2, -1/2
Sketching y =sin^2((root x))
look at +ve, -ve, max, min, zeros
If constant at end, when sketching…
sketch without constant then shift it
Cubics (odd powers) will always have at least one solution because
the y-value goes from -ve infinity to +ve infinity
Look at when x tends to +/- infinity
discriminant, roots, asymptote, zeroes, axis
Single/odd factor (x-a) means
the curve crosses at x=a
Repeated/even factor (x-a)^2 means
the curve touches at x=a
Find the asymptopes of the curve, find intercepts, turning points
pay attention to domains to have correct amount of required solutions
Is the graph a series of transformations of a simpler graph
Symmetry (is the function odd or even)
To find max/min of quadratic
complete the square
max if -ve
min if +ve
When two lines of touch
both the y-value and the gradient are the same
When they intersect,
only the y-values are the same
Ensure you look if question says intersect or touch
as they mean different things
Look for dominant parts of graph when combining functions
graphs of x^2 + root(x)*x
sin^2(x)
sin(root x)
…
Odd powers =
infinite range, min roots = 1, max roots = max power
Maximum turning points =
maximum power - 1
Even =
finite range, min roots = 0, max roots = max power
(x-a) means
line crosses at x=a
(x-a)^2 means
line touches at x=a
(x-a)^3 means
points of inflection at x=a
Sin(x^2) –>
x^2 causes oscillation period (width of hill) to decrease
2^-x peaks
gradually become shallower
Limit(1/x) =
0
Limit f(x)/g(x) =
limite f’(x)/g’(x)
When finding the number of distinct roots for a changing constant -
sketch without constant then shift up or down
a^x grows quicker than x^b
x^x grows quicker than x!
(x–>0)limit(x^x) =
1
Odd function:
f(-x) = -f(x) sine
Even function:
f(-x) = f(x) cosine
sin(2pi - x) =
sin(-x) = -sin(x)
(x-1)(x-2)….(x-n) = k
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
x^3:
max –> min
x^4:
min –> max –> min
x^5:
max –> min –> max –> min
combined functions
sub in values
look at ranges of individual parts
which bit is the dominant part of the graph
Cubics will always have
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)
When you have f(x) + g(x)
its often useful to consider the graphs separately, and what happens when you put in values, then add them
Cubic stationary point:
max –> min
Quartic stationary point:
min –> max –> min
Quintic stationary point:
max –> min –> max –> min
When two lines touch:
both y value and gradient area equal
When two lines interest
only the y value is equal
when they touch
equate y values and gradient (use discriminant for gradient)
sketch x^4 - y^2 = 2y +1
sketch x^6 + y^6 = 1
sketch |x+y| = 1
Compare coefficients
y=2^-x * sin^2(x^2) –>
(0,0) y>0, max+min, period, values, dominant part
I(c) >=0 as
no solutions so above the x axis
sketch y=sin^2(root x)
sketch y=sin(x)/x
Sketch y= root (2-x^2),
x+(root2 - 1)y = root 2
sketch y=log(f(x)) –> solutions, y axis, quadrants
Dominant part of graph
y= 2^(x^2 - 4x + 3):
Let a=2^x^2 = 2^f(x) therefore f(x-2)^2 +1
Think of graph to justify max/min
Repeated root touches (x+a)^2n
root crosses (x+a)^2n+1
Transforming functions
f(x+-a) -f(x) f(-x) f(1/x)
sub in values for graphs, think of direction, turning points
when inequalities, think of discriminant
Simplify complex transformations
Try and simplify transformations e.g. reflection in both axes and a shift (2, 0) = 180 degree rotation about (1,0)
f(x) = k
therefore k is a negative shift if k is positive
sketch quartics using calculus
sin(x) = sin(y) (0,0, (pi, 0)
(x^3 - 1)^2, sketch (x^(2n-1))^2
f(x) = k therefore k is a shift of y co-ordinates
Sketch a^2 - a^1.5 - 8 = 0
Asymptopes, axis’, x –> infinity
To find where f(x)-g(x) crosses the x axis,
sketch both functions and look for intersction
Sketch cos(pi*x)
Tan values = 1/root3, 1, root3,
Sketching x^2, x^3, x^4 and x^5 together -
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
