Questions and Answers
What are the x-values in a relationship?
What is the definition of Domain?
All the possible x-values in a relationship.
What are Y-values?
Dependent variable in a relationship.
What defines the Range in a relationship?
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What does f(x) represent?
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What is an ordered pair?
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What does the vertical line test determine?
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The rate of change is represented as ΔY / ΔX.
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What does the slope formula calculate?
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What does the slope-intercept form represent?
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What does the equation Ax + By = C represent?
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What does a horizontal line's slope represent?
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What indicates the graph of a vertical line?
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What is the standard form of a quadratic function?
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What does the value of a indicate in a quadratic function?
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What does the zero-product property state?
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What does f(x) = a(x - r)(x - s) represent?
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What is the formula for finding the y-value of the vertex in vertex form?
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What does the formula x = -b ± √(b^2 - 4ac) / 2a represent?
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What is the meaning of correlation coefficient?
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What is the five number summary?
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What is an outlier?
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Study Notes
X-values and Y-values
- X-values represent the independent variable (input) in a relationship, essential for defining positions on the x-axis in graphs.
- Y-values represent the dependent variable (output) and are used to denote positions on the y-axis.
Domain and Range
- Domain includes all possible x-values in a relationship, which can be specified (e.g., all real numbers, positive integers).
- Range encompasses all possible y-values and resonates with the same types as domain (e.g., real numbers, positive & negative integers).
Function Notation
- The equation f(x) = y is known as function notation, where f(x) can be interpreted as y, facilitating comparisons among functions.
Ordered Pairs
- An ordered pair (X, Y) or coordinate indicates a specific point on a graph, with X always being the first value (e.g., (-2, 3)).
- In function notation, (x, f(x)) illustrates specific values evaluated within a function (e.g., (10, -5)).
Function Characteristics
- A relationship is confirmed as a function if x-values do not repeat; each input corresponds to a single output.
- The Vertical Line Test visually determines whether a curve represents a function, confirming one output per unique input.
Slope and Rate of Change
- The formula ΔY/ΔX is used to calculate the rate of change—essential for determining slope in linear relationships.
- Slope can also be calculated using the formula (Y2 - Y1) / (X2 - X1), enabling analysis of two given points.
Linear Functions
- The slope-intercept form is expressed as y = mx + b (m being slope and b the y-intercept), facilitating the graphing of linear functions.
- Standard form, Ax + By = C, describes linear functions where slope can be identified as -A/B.
Line Characteristics
- Horizontal lines have a slope of zero, expressed as y = b, and vertical lines do not function as a relation, with an undefined slope, expressed as x = a.
Quadratic Functions
- Standard form of a quadratic function is f(x) = ax² + bx + c, representing a parabola shaped graph.
- The parameter 'a' determines the orientation and width of the parabola (a > 0: opens up; a < 0: opens down).
Vertex and Axis of Symmetry
- The vertex of the parabola can be identified using h = -b / (2a) for the x-value, and k = c - (b² / 4a) for the y-value.
- Vertex form of a quadratic function is f(x) = a(x - h)² + k, which directly reveals the vertex coordinates (h, k).
Factored Form and Zeros
- The factored form f(x) = a(x - r)(x - s) facilitates the identification of x-intercepts (zeros, roots).
- The Zero Product Property indicates that if the product equals zero, at least one factor must be zero.
Exponential Functions
- The standard form for exponential functions is f(x) = a * b^x, with 'a' as the initial value and 'b' as the growth/decay factor.
Growth and Decay Models
- Growth is expressed as y = a(1 + r)², while decay is denoted as y = a(1 - r)², where 'r' is the rate of growth or decay.
Statistical Measures
- The correlation coefficient quantifies the strength of a relationship but does not imply causation.
- Mean signifies average, while standard deviation provides insight into data dispersal.
Median and Interquartile Range (IQR)
- Median is the middle value of ordered data; IQR is calculated using the five-number summary (min, max, median, Q1, Q3).
Outliers and Five Number Summary
- Outliers are extreme values that can influence the mean significantly; they are identified through specific tests.
- A five-number summary includes minimum, maximum, median, lower quartile, and upper quartile, useful for data comparison and box plot construction.
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Description
This quiz covers the essential concepts of functions, including x-values and y-values, domain and range, and function notation. Test your understanding of ordered pairs and the characteristics that define a function. Perfect for students looking to enhance their knowledge in algebra.
