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Questions and Answers

What is the domain in relation to a function?

Any set of ordered pairs

Set of all inputs or independent variable values, x's (correct)

Ordered pairs where every x is paired with only one y

Set of all outputs or dependent variable values, y's

What is the range in relation to a function?

Set of all inputs or independent variable values, x's

Any set of ordered pairs

Set of all outputs or dependent variable values, y's (correct)

Ordered pairs where every x is paired with only one y

What is a relation?

Any set of ordered pairs

What is a function?

Ordered pairs where every x is paired with only one y Signup and view all the answers

What is the vertical line test?

Test to determine if a graph is a function Signup and view all the answers

How do you find the x-intercept of an equation when no points are given?

Find the value of x when y=0 Signup and view all the answers

What is the point-slope form of a linear equation?

y - y1 = m(x - x1) Signup and view all the answers

What is the standard form of a linear equation?

ax + by = c Signup and view all the answers

What is the best use for slope-intercept form?

Graphing Signup and view all the answers

What is the best use for point-slope form?

Writing equations for lines Signup and view all the answers

What is the best use for standard form?

Finding x and y intercepts Signup and view all the answers

What is a linear sequence?

Sequence with constant difference between terms Signup and view all the answers

What is the graph of an arithmetic sequence?

Scatterplot with linear points and domain of counting numbers Signup and view all the answers

How do you find the first term if given the 10th and 20th term?

Find the ratio of term to term number Signup and view all the answers

What are the steps to solve using linear combination?

Put equations into standard form, find opposites, combine equations together, solve Signup and view all the answers

How do you write matrix equations?

[coefficients]*[variables in correct order] = [answers given without using calculator] Signup and view all the answers

What does the discriminant represent?

b^2 - 4ac Signup and view all the answers

What constitutes a polynomial?

Exponents are whole numbers and not x Signup and view all the answers

Describe the shape of the graph for x^2.

Bowl shape Signup and view all the answers

Describe the shape of the graph for x.

Linear Signup and view all the answers

Describe the shape of the graph for x^3.

Disco left to right Signup and view all the answers

Describe the shape of the graph for |x|.

V shape Signup and view all the answers

Describe the shape of the graph for the square root of x.

Stretch (arrow in upper right quadrant pointing to right) Signup and view all the answers

Describe the shape of the graph for b^x when b > 1.

Hill going down right to left (upper 2 quadrants) Signup and view all the answers

Study Notes

Key Concepts in Advanced Algebra

Domain: Represents the complete set of inputs or values of the independent variable, typically denoted as x.
Range: Encompasses all possible outputs or values of the dependent variable, typically denoted as y.
Relation: Defined as any collection of ordered pairs.
Function: A special type of relation where each x-value is associated with exactly one y-value.
Vertical Line Test: A method to verify if a graph represents a function; a vertical line should intersect the graph at only one point.
Finding x-intercept: Determine the x-value where y equals zero.

Forms of Linear Equations

Point-Slope Form: Expressed as (y - y_1 = m(x - x_1)), useful for writing equations based on one known point and the slope.
Standard Form: Written as (ax + by = c), typically used for identifying x and y-intercepts.
Slope-Intercept Form: Used primarily for graphing; represented as (y = mx + b), where m is the slope and b is the y-intercept.

Sequences and Their Graphs

Linear Sequence: This involves a sequence with a constant difference between consecutive terms.
Graph of Arithmetic Sequence: Depicted as a scatterplot that forms linear points with a domain consisting of counting numbers.
Finding the First Term: To retrieve the first term from specified later terms (like the 10th and 20th terms), calculate the ratio of the changes between terms.

Solving and Writing Equations

Linear Combination Steps: Start by placing equations in standard form, find opposite coefficients, combine the equations effectively, and then solve for the variable.
Writing Matrix Equations: Represented as ([coefficients] * [variables] = [answers]) while avoiding calculator use.

Polynomial Characteristics

Discriminant: Given by the formula (b^2 - 4ac), key for determining the nature of roots in a quadratic equation.
Polynomial Degree: Identified as the highest exponent present in the polynomial.
Polynomial Criteria: An expression qualifies as a polynomial if all exponents are whole numbers.

Graph Shapes

Quadratic Function (x^2): Typically forms a bowl shape.
Linear Function (x): Results in a straight line.
Cubic Function (x^3): Features a distinct "disco" pattern across the graph.
Absolute Value (|x|): Characterized by a V shape, displaying symmetry about the y-axis.
Square Root of x: Shows a stretched pattern, with an arrow pointing in the upper right quadrant.
Exponential Function (b^x) (b > 1): Descends in a "hill" shape sloping down from left to right in the upper two quadrants.