Unit 4 - Elementary Functions Flashcards

1
Q

Equation to find the vertex of a quadratic equation

A

x = -(b/2a)

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2
Q

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

A

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)

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3
Q

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

A

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

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4
Q

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

A

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

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5
Q

The vertex of f

A

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

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6
Q

if a is positive, then the parabola is

A

convex: its vertex lies below all other points;

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7
Q

if a is negative, then the parabola is

A

concave: its vertex lies above all other points;

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8
Q

Equation to calculate the discriminant

A

D = b^2 − 4ac

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9
Q

The roots of a quadratic equation can be calculated using

A

x = (−b ± √D)/2a

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

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10
Q

If the discriminant > 0 then f has

A

2 distinct roots

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

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11
Q

If the discriminant is 0

A

then the 2 roots of f coincide

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

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12
Q

If the discriminant < 0

A

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

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13
Q

What does the symbol R+ refer to in set notation

A

R+ = {x ∈ R | x > 0}

positive real numbers

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