Unit 4 - Elementary Functions Flashcards
Equation to find the vertex of a quadratic equation
x = -(b/2a)
The clause with “∧” in the set-builder notation in means that
in each element (x, y) ∈ (f ∩ g), the values of x and y must satisfy the following system of equations:
y = f(x)
y = g(x)
(Systems of linear equations)
We say that a function f : R → R is quadratic if and only if:
there are a, b, c in R such that for all x ∈ R, f(x) = ax2 + bx + c
We say that a function f : R → R is linear if and only if:
there exist m, c in R such that for all x ∈ R, f(x) = mx + c
The vertex of f
is the turning point where the parabola changes from increasing to decreasing (or vice versa).
if a is positive, then the parabola is
convex: its vertex lies below all other points;
if a is negative, then the parabola is
concave: its vertex lies above all other points;
Equation to calculate the discriminant
D = b^2 − 4ac
The roots of a quadratic equation can be calculated using
x = (−b ± √D)/2a
where D is the discriminant (b^2 − 4ac)
If the discriminant > 0 then f has
2 distinct roots
Correspondingly, its parabola has 2 intersections with the x-axis
If the discriminant is 0
then the 2 roots of f coincide
Correspondingly, the parabola has one intersection with
the x-axis
If the discriminant < 0
then √D is undefined
hence, there are no values of x that satisfy equation, so f has no real roots, and its parabola has no intersections with the x-axis
What does the symbol R+ refer to in set notation
R+ = {x ∈ R | x > 0}
positive real numbers