Unit 4 - Elementary Functions

Question 1

Equation to find the vertex of a quadratic equation

Answer 1

x = -(b/2a)

Question 2

The clause with “∧” in the set-builder notation in means that

Answer 2

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)

Question 3

We say that a function f : R → R is quadratic if and only if:

Answer 3

there are a, b, c in R such that for all x ∈ R, f(x) = ax2 + bx + c

Question 4

We say that a function f : R → R is linear if and only if:

Answer 4

there exist m, c in R such that for all x ∈ R, f(x) = mx + c

Question 5

The vertex of f

Answer 5

is the turning point where the parabola changes from increasing to decreasing (or vice versa).

Question 6

if a is positive, then the parabola is

Answer 6

convex: its vertex lies below all other points;

Question 7

if a is negative, then the parabola is

Answer 7

concave: its vertex lies above all other points;

Question 8

Equation to calculate the discriminant

Answer 8

D = b^2 − 4ac

Question 9

The roots of a quadratic equation can be calculated using

Answer 9

x = (−b ± √D)/2a

where D is the discriminant (b^2 − 4ac)

Question 10

If the discriminant > 0 then f has

Answer 10

2 distinct roots

Correspondingly, its parabola has 2 intersections with the x-axis

Question 11

If the discriminant is 0

Answer 11

then the 2 roots of f coincide

Correspondingly, the parabola has one intersection with
the x-axis

Question 12

If the discriminant < 0

Answer 12

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

Question 13

What does the symbol R+ refer to in set notation

Answer 13

R+ = {x ∈ R | x > 0}

positive real numbers