Graphs curve sketching + factor theorem BARNES Flashcards
y = x^2
happy/sad face curve (negative or positive)
1 turning point
0, 1 or 2 real roots
y = x^3
waterfall (if positive going up, if negative going down)
0 or 2 turning points
1, 2 or 3 real roots
y = x^4
y = ax^4
flat happy face
or W shape (if negative M, upside down)
1 or 3 turning points
0, 1, 2, 3, or 4 real roots
A factor (x - a) ⇒
crosses x-axis at a
Eg: y = (x - 2)(x + 2)
A factor (x - a)^2 ⇒
touches x-axis at a
Eg: y = -(x + 3)^2
A factor (x - a)3 ⇒
crosses x-axis & flattens at a
Eg: y = (x - 1)^3(x + 2)
graph 1/x
Both have asymptotes at x = 0 and y = 0
like curly x on the side
graph 1/x^2
Both have asymptotes at x = 0 and y = 0
1/x2 is always positive, and symmetrical
y = x^1/2
increasing and getting less steep, starts at origin
x^1/3
increasing and getting less steep
an S shape
How do we find an inverse function geometrically?
y = x^2 and y = x^3, mirrored in y = x
x^1/2 stops at origin because with inverse functions like square roots, we restrict to one output
Proportional means
two quantities are related through a constant, k
y is directly proportional to x means
y = kx
y is inversely proportional to x means
y = k/x
factor theorem
If (ax - b) is a factor of the polynomial f(x) then f(b/a) = 0
Using the factor theorem, show that (x + 1) is a factor of x^3 + 3x^2 - x - 3
if x+1 is a factor of fx, then f(-1) = 0
Write x3 - x2 - 14x + 24 as a product of linear factors
Use trial and error to find the first factor, and algebraic division thereafter.
one factor is usually x, (x ± 1) or (x ± 2)
ans- (x-2) (x+4) (x-3)
Two factors of q(x) = x3 + 4x2 + ax + b are (x - 1) and (x + 1).
Find the values of the constants a and b.
factor (x-1) goes to q(1)=0
factor (x-1) goes to q(-1)=0
solve simultaneously
b=-4
a=-1
