Math Revision

point-slope form

(y -y) = m(x - x)

completed square form

y = a(x - h)² + k, where vertex = (h,k)

factorized form

a(x - p)(x - q) = y, where p and q are x-intercepts

the axis of symmetry is x = (p + q)/2

completing the square

when y = bx + c

completed square: y = (x + b/2)² - (b/2)² + c

cutting

2 points of intersection, if the line touches the curve at least once, it is a tangent to the curve

touching

1 point of intersection, if the line touches the curve at least once, it is a tangent to the curve

missing

0 points of intersection

natural domain of f(x) = x²

x ∈ R

natural domain of f(x) = √x

x ≥ 0

natural domain of f(x) = 1/x

x ≠ 0

natural domain of f(x) = 1/√x

x > 0

many to one function

there are multiple inputs to the same output (e.g. parabola), does not pass horizontal line test

inverse is not a function

one to one function

every x goes to a unique y (e.g. straight line), passes horizontal line test.

inverse is a function

graph of f(x) = x³

graph of f(x) = -x³

graph of f(x) = x³ + anything

graph of f(x) = -x³ + anything

reciprocal function

a function of the form y = k/x, where k ≠ 0

graph is a rectangular hyperbola which has…

• 2 branches

• a horizontal and vertical asymptote

• the coordinates of points on the asymptotes are (1,k), (k,1), and (-k,-1), (-1,-k)

• if you increase k, the curve becomes flatter as the graph moves further from the origin

asymptotes for function in form y = b/(cx + d) + a, where b & c ≠ 0

vertical asymptote: x = -d/c

horizontal asymptote: y = a

asymptotes for function in form y = (ax + b)/(cx + d), where c ≠ 0

vertical asymptote: -d/c

horizontal asymptote: a/c

if y = undefined for the domain in f^-1(x)…

y = ∞