Algebra Class: Parabolas and Linear Functions

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Questions and Answers

What type of vertex does a parabola have when the coefficient 'a' is greater than 0?

Minimum (correct)

Maximum

Neither

Undefined

How is the axis of symmetry of a parabola defined?

x = 2a

x = -4a

x = -b/2a (correct)

x = b/4a

What equation represents the vertex of a parabola in standard form?

(b/2a, -4a)

(−2a, b/(4a)) (correct)

(−b/(2a), 4a)

(2a, -b/(4a))

In the optimization problem presented, what were the two numbers whose sum is 30, which minimizes the sum of their squares?

15 and 15 Signup and view all the answers

What is the first step in proving a function is one-to-one?

Demonstrate that if f(x1) = f(x2), then x1 = x2. Signup and view all the answers

Which of the following accurately describes a one-to-one correspondence?

Each element in the codomain corresponds to exactly one element in the domain. Signup and view all the answers

For the function f(x) = 2x - 3, what is the value of x if f(x) equals 1?

3 Signup and view all the answers

When is a function classified as an even function?

If f(-x) = f(x) for all x. Signup and view all the answers

What is the general form of a quadratic function?

f(x) = ax^2 + bx + c Signup and view all the answers

Which statement is true about the solutions of the equation y > 0 in terms of x for a quadratic function?

It has two solutions. Signup and view all the answers

What does the 'm' represent in the linear function f(x) = mx + c?

The slope of the line. Signup and view all the answers

What is the shape of the graph of a quadratic function?

Parabola Signup and view all the answers

In relation to quadratic functions, what is the significance of the term 'discriminant'?

It determines if the function has real solutions. Signup and view all the answers

Which of the following describes a one-to-one correspondence in relation to functions?

Every element in the domain corresponds to a unique element in the range. Signup and view all the answers

What does the notation -b ± √(b² - 4ac) represent in the context of quadratic functions?

The general solution for x in a quadratic equation. Signup and view all the answers

In terms of optimization within quadratic functions, what would represent the maximum or minimum value?

The vertex of the parabola. Signup and view all the answers

Which of the following statements about subsets is true?

Every set is a subset of itself. Signup and view all the answers

What is the power set of the set {1, 2}?

{ {}, {1}, {2}, {1, 2} } Signup and view all the answers

If A = {2, 4, 6} and B = {4, 5, 6}, what is A ∩ B?

{4, 6} Signup and view all the answers

What does A - B represent?

All elements that are in A but not in B. Signup and view all the answers

Which of the following describes the De Morgan law correctly?

(A ∪ B)' = A' ∩ B' Signup and view all the answers

If the universal set U contains 5 elements and A contains 3 elements, how many elements are in the complement of A?

2 Signup and view all the answers

Given two sets A and B, which statement correctly describes the union of these sets?

Includes elements from both A and B, with no duplicates. Signup and view all the answers

For the sets A = {1, 2, 3} and B = {2, 3, 4}, what is A − B?

{1} Signup and view all the answers

If A = {x | x is a counting number less than 4}, what is 𝑃(A)?

{ {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} } Signup and view all the answers

Which of the following is NOT a property of the intersection of sets?

A ∩ B includes elements that are only in A. Signup and view all the answers

Study Notes

Parabola Properties

A parabola with a positive coefficient a will have a minimum point
A parabola with a negative coefficient a will have a maximum point
The axis of symmetry is found with the formula x = -b / 2a
The vertex of a parabola is located at (- b / 2a, (4ac - b^2) / 4a)

One-to-One Correspondence

A one-to-one correspondence is a function with both one-to-one and onto properties
In a one-to-one correspondence, for every element in the codomain, there is exactly one corresponding element in the domain
Example:
- The function f(x) = 2x - 3 is a one-to-one correspondence because for every x in the domain, there exists exactly one y (in the codomain) and for every y, there exists exactly one x.

Straight Line Graphs

The graph of a linear function f(x) = mx + c is a straight line
The constant c represents the y-intercept (where the line crosses the y-axis)
The slope of the line is represented by m, which is calculated as the change in y divided by the change in x between any two points on the line

Quadratic Functions

A quadratic function is defined by the general form f(x) = ax^2 + bx + c where a, b, and c are real numbers with a not equal to 0
The graph of any quadratic function is called a parabola.
The solution to any quadratic function can be found using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a

Sets

A set is a collection of objects
A subset of a set A includes all elements of set A, denoted by A ⊂ B
The power set of a set A is the collection of all subsets of A, denoted by P(A) = 2^n, where n is the number of elements in A.
- Every set is a subset of itself
- The empty set is a subset of every set

Operations on Sets

The union of sets A and B combines all elements from both sets, denoted by A ∪ B
The intersection of sets A and B includes only elements present in both sets, denoted by A ∩ B
The difference of sets A and B includes only elements present in A but not in B, denoted by A - B
The complement of set A with respect to the universal set U (denoted by A' or U - A) includes all elements in U not in A

Laws of Algebra of Sets

Distributive Laws:
- A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
- A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
De-Morgan's Law:
- (A ∪ B)' = A' ∩ B'
- (A ∩ B)' = A' ∪ B'

Ordered Pairs

The order of elements in an ordered pair is significant, unlike sets
For example: (1, 2, 3) is different from (2, 3, 1)

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Description

This quiz covers key properties of parabolas, one-to-one correspondences, and straight line graphs. You'll learn how to find the vertex of a parabola, understand the concept of one-to-one correspondence in functions, and analyze linear functions and their graphs. Test your knowledge on these essential algebra topics!