2.1 The Rectangular Coordinate Systems and Graphs
3 min read•june 24, 2024
The is the foundation for graphing in algebra. It lets us points and equations on a , using x and y coordinates to pinpoint locations. This system is crucial for visualizing mathematical relationships and solving problems graphically.
Interpreting graphs involves analyzing key features like , distances, and midpoints. By examining these elements, we can extract valuable information about equations and functions, helping us understand their behavior and make predictions about real-world situations they represent.
The Cartesian Coordinate System
Points in Cartesian coordinates
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- Two-dimensional plane formed by intersecting horizontal () and vertical () number lines at the (0, 0)
- represents horizontal distance from
- Positive values to the right, negative values to the left
- represents vertical distance from origin
- Positive values above, negative values below
- To plot a point, start at origin, move horizontally by , then vertically by y-coordinate
- Coordinates written as
- First value is x-coordinate, second value is y-coordinate
- Examples: (3, 4) is 3 units right and 4 units up, (-2, -5) is 2 units left and 5 units down
Graphing equations with technology
- To by plotting points:
- Create table of x and y values satisfying equation
- Plot points from table on ()
- Connect points with smooth curve or straight line
- Graphing calculators and software graph equations efficiently
- Enter equation into technology
- Adjust to view desired portion of graph
- Examples: (linear), ()
Interpreting Graphs
Intercepts of graphs
- : point where graph crosses x-axis
- y-coordinate always 0
- To find, set y = 0 and solve for x
- : point where graph crosses y-axis
- x-coordinate always 0
- To find, set x = 0 and solve for y
- Examples: of is (3, 0), y- of is (0, 4)
Distance formula for lengths
- Calculates length of line segment between points and
- To use:
- Identify coordinates of two points
- Substitute coordinates into formula
- Simplify and calculate result
- Example: Distance between (1, 2) and (4, 6) is
Midpoint formula for segments
- Finds coordinates of point dividing line segment into two equal parts
- For segment with endpoints and , midpoint is
- To use:
- Identify coordinates of endpoints
- Substitute coordinates into formula
- Simplify and calculate result
- Example: Midpoint of segment with endpoints (-3, 1) and (5, 7) is
Information from graph analysis
- Shape and behavior determine equation or relationship type
- Linear equations: straight lines
- Quadratic equations: parabolas
- Exponential equations: curves increasing or decreasing rapidly
- Identify key features
- Intercepts, symmetry, increasing or decreasing behavior, turning points (maximums or minimums)
- Use information to make predictions or draw conclusions about relationships
- Examples: opens upward (positive leading ), line with positive slope (increasing )
Functions and their properties
- A function (function) is a rule that assigns each input value to exactly one output value
- The graph of a function is a visual representation of its behavior
- : set of all possible input values for a function
- : set of all possible output values for a function
- Understanding these properties helps analyze and interpret functions in various contexts
Key Terms to Review (60)
(x, y): The notation (x, y) represents an ordered pair of numbers used to define a point in a two-dimensional rectangular coordinate system. Each point is uniquely identified by its position along the horizontal axis (x-axis) and the vertical axis (y-axis), where 'x' indicates the horizontal displacement from the origin and 'y' indicates the vertical displacement.
Absolute value function: An absolute value function is a type of piecewise function that returns the non-negative value of its input. It is denoted as $f(x) = |x|$ and has a V-shaped graph.
Area of a circle: The area of a circle is the amount of space enclosed within its circumference. It is calculated using the formula $A = \pi r^2$, where $r$ is the radius of the circle.
Binomial coefficient: A binomial coefficient is a coefficient of any of the terms in the expansion of a binomial raised to a power, typically written as $\binom{n}{k}$ or $C(n,k)$. It represents the number of ways to choose $k$ elements from a set of $n$ elements without regard to order.
Cartesian coordinate system: A Cartesian coordinate system is a two-dimensional plane defined by an x-axis (horizontal) and a y-axis (vertical), used to graphically represent algebraic equations. Each point in the plane is identified by an ordered pair of coordinates $(x, y)$.
Cartesian Coordinate System: The Cartesian coordinate system is a two-dimensional coordinate system that uses perpendicular x and y axes to define the location of a point in a plane. It provides a systematic way to represent and analyze spatial relationships and graphical information.
Center of an ellipse: The center of an ellipse is the midpoint of both the major and minor axes, serving as the point of symmetry for the ellipse. It is typically denoted by a coordinate pair $(h, k)$ in the Cartesian plane.
Circle: A circle is a closed, two-dimensional shape formed by a set of points that are all equidistant from a fixed point called the center. Circles are fundamental geometric shapes with numerous applications in mathematics, science, and everyday life.
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression. It represents the magnitude or strength of the relationship between the variable and the overall expression. Coefficients are essential in various mathematical contexts, including polynomial factorization, linear equations, quadratic equations, and the graphing of polynomial functions.
Constant: A constant is a value that does not change. In algebra, it is often a number without any variables attached to it.
Constant: A constant is a fixed value that does not change within a given context or problem. It is a fundamental quantity that remains the same regardless of the circumstances or variables involved.
Coordinate plane: A coordinate plane is a two-dimensional surface formed by the intersection of a vertical line (y-axis) and a horizontal line (x-axis). These axes divide the plane into four quadrants used for graphing equations and geometric shapes.
Coordinate Plane: The coordinate plane, also known as the Cartesian coordinate system, is a two-dimensional plane that uses a horizontal (x-axis) and a vertical (y-axis) line to define the position of a point. It provides a way to represent and analyze relationships between variables in a visual and mathematical manner.
Dependent variable: The dependent variable is the output of a function, whose value depends on the input or independent variable. It is usually represented as $y$ in the equation $y = f(x)$.
Direct variation: Direct variation describes a linear relationship between two variables where one variable is a constant multiple of the other. Mathematically, it is expressed as $y = kx$, where $k$ is the constant of variation.
Direct Variation: Direct variation is a mathematical relationship between two variables where one variable is directly proportional to the other. This means that as one variable increases, the other variable increases at the same rate, and vice versa. Direct variation is a fundamental concept in understanding the behavior of linear functions and modeling real-world situations involving proportional relationships.
Distance formula: The distance formula calculates the distance between two points in a plane. It is derived from the Pythagorean theorem.
Distance Formula: The distance formula is a mathematical equation used to calculate the straight-line distance between two points on a coordinate plane. It is a fundamental concept in the study of coordinate geometry and is essential for understanding the properties and behaviors of various functions and graphs.
Domain: The domain of a function is the complete set of possible input values (x-values) that allow the function to work within its constraints. It specifies the range of x-values for which the function is defined.
Domain: The domain of a function refers to the set of input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is the set of all possible values that can be plugged into the function to produce a meaningful output.
Ellipse: An ellipse is a closed, two-dimensional shape that resembles an elongated circle. It is one of the fundamental conic sections, which are the shapes formed by the intersection of a plane and a cone.
Equation in quadratic form: An equation is in quadratic form if it can be written as $a(x^2 + bx + c) = 0$ where $a, b,$ and $c$ are constants. It often involves substituting a variable to simplify into a standard quadratic equation.
Equation in two variables: An equation in two variables is a mathematical statement that relates two different quantities, typically represented as $x$ and $y$. These equations can be graphed on a coordinate plane showing the relationship between the variables.
F(x): f(x) is a function notation that represents a relationship between an independent variable, x, and a dependent variable, f. It is a fundamental concept in mathematics that underpins the study of functions, their properties, and their applications across various mathematical topics.
Function: A function is a mathematical relationship between two or more variables, where one variable (the dependent variable) depends on the value of the other variable(s) (the independent variable(s)). Functions are a fundamental concept in mathematics and are essential in understanding various topics in college algebra, including coordinate systems, quadratic equations, polynomial functions, and modeling using variation.
Graph: A graph is a visual representation of the relationship between variables, typically displayed on a coordinate plane. It allows for the depiction of patterns, trends, and data points in a clear and concise manner.
Graph in two variables: A graph in two variables is a visual representation of all the possible solutions to an equation involving two variables, typically plotted on a rectangular coordinate system. The horizontal axis represents one variable, while the vertical axis represents the other.
Graphing Calculator: A graphing calculator is a type of handheld electronic device that can be used to perform a variety of mathematical functions, including graphing equations, analyzing data, and solving complex problems. These calculators are widely used in mathematics and science education, as well as in various professional settings.
Intercept: The intercept is the point at which a line or curve intersects one of the coordinate axes in a rectangular coordinate system. It represents the value of the dependent variable when the independent variable is zero.
Intercepts: Intercepts are points where a graph crosses the x-axis or y-axis. The x-intercept occurs when $y = 0$, and the y-intercept occurs when $x = 0$.
Inverse variation: Inverse variation occurs when one variable increases while the other decreases, following the form $y = \frac{k}{x}$ where $k$ is a constant. This relationship creates a hyperbolic graph.
Inverse Variation: Inverse variation is a relationship between two variables where as one variable increases, the other variable decreases proportionally. This concept is fundamental to understanding the behavior of functions and how variables interact in various real-world applications.
Linear equation: A linear equation in one variable is an algebraic equation that can be written in the form $ax + b = 0$, where $a$ and $b$ are constants, and $x$ is the variable. The solution to the equation is the value of $x$ that makes the equation true.
Linear Equation: A linear equation is a mathematical expression that represents a straight line on a coordinate plane. It is an equation where the variables are raised to the power of one and the variables are connected by addition or subtraction operations.
Midpoint formula: The midpoint formula calculates the exact center point between two defined points on a coordinate plane. The formula is $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$, where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points.
Ordered pair: An ordered pair is a set of two elements written in a specific order, typically as (x, y), where x represents the horizontal coordinate and y represents the vertical coordinate.
Ordered Pair: An ordered pair is a set of two numbers, typically represented as (x, y), that uniquely identifies a point on a coordinate plane. The first number, x, represents the horizontal position, while the second number, y, represents the vertical position of the point.
Origin: In the rectangular coordinate system, the origin is the point where the x-axis and y-axis intersect. It is denoted by the coordinates (0, 0).
Origin: The origin is a specific point in a coordinate system that serves as the reference point for all other points. It is the intersection of the x-axis and y-axis, and is typically denoted as the point (0, 0). The origin is a fundamental concept in various mathematical and scientific contexts, as it provides a common starting point for measurement and analysis.
Parabola: A parabola is a symmetric curve that represents the graph of a quadratic function. It can open upward or downward depending on the sign of the quadratic coefficient.
Parabola: A parabola is a curved, U-shaped line that is the graph of a quadratic function. It is one of the fundamental conic sections, along with the circle, ellipse, and hyperbola, and has many important applications in mathematics, science, and engineering.
Plot: In the context of the rectangular coordinate system and graphs, the term 'plot' refers to the visual representation of data or mathematical functions on a coordinate plane. Plotting involves mapping points or curves based on their x and y coordinates to create a graphical depiction of the relationship between variables.
Pythagorean Theorem: The Pythagorean Theorem is a fundamental relationship in geometry that describes the connection between the lengths of the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
Quadrant: A quadrant is one of the four sections of the coordinate plane that are divided by the x-axis and y-axis. Each section is defined by its position relative to these axes.
Quadrants: Quadrants are the four equal divisions of a coordinate plane, created by the intersection of the x-axis and y-axis. They are a fundamental concept in understanding the rectangular coordinate system and are also crucial in the study of right triangle trigonometry.
Quadratic: A quadratic is a polynomial function of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$. Quadratics are characterized by a parabolic shape and have many important applications in mathematics and the sciences.
Range: In mathematics, the range refers to the set of all possible output values (dependent variable values) that a function can produce based on its input values (independent variable values). Understanding the range helps in analyzing how a function behaves and what values it can take, connecting it to various concepts like transformations, compositions, and types of functions.
Slope-intercept form: Slope-intercept form is a way to express the equation of a straight line using the formula $y = mx + b$. In this formula, $m$ represents the slope and $b$ represents the y-intercept.
Slope-Intercept Form: Slope-intercept form is a way of representing a linear equation in two variables, typically written as $y = mx + b$, where $m$ represents the slope of the line and $b$ represents the $y$-intercept. This form allows for easy interpretation of the line's characteristics and is widely used in the study of linear functions and their applications.
Two-Dimensional Plane: A two-dimensional plane is a flat surface that extends infinitely in two perpendicular directions, representing the x and y coordinates. It is the fundamental coordinate system used in various mathematical and scientific disciplines, including the study of rectangular coordinate systems and graphs.
Variable: A variable is a symbol or letter that represents an unknown or changeable value in a mathematical expression, equation, or function. It serves as a placeholder for a value that can vary or be assigned different values within a given context.
Window Settings: Window settings refer to the parameters that define the visible portion of the coordinate plane displayed on a graph or plot. These settings determine the scale, range, and perspective of the graphical representation, allowing users to focus on specific regions of interest within the overall coordinate system.
X-Axis: The x-axis is the horizontal axis on a coordinate plane, typically running left to right. It is used to represent the independent variable in a graph and helps visualize the relationship between two or more variables.
X-coordinate: The x-coordinate is the first value in an ordered pair $(x, y)$ representing a point's horizontal position on the Cartesian plane. It indicates how far left or right the point is from the origin (0, 0).
X-Coordinate: The x-coordinate is the horizontal position of a point on a coordinate plane. It represents the distance from the origin (0,0) to the point along the horizontal x-axis. The x-coordinate is a crucial component in understanding and working with various mathematical concepts, including coordinate systems, graphs, unit circles, and systems of linear equations.
X-intercept: The x-intercept is the point where a graph crosses the x-axis, where the y-coordinate is zero. It represents the solution(s) to an equation when $y = 0$.
X-Intercept: The x-intercept of a graph is the point where the graph of a function or equation intersects the x-axis, indicating the value of x when the function's output or the equation's value is zero. The x-intercept is a crucial concept in understanding the behavior and properties of various mathematical functions and equations.
Y-axis: The y-axis is the vertical axis in a rectangular coordinate system, which represents the dependent variable and is typically used to plot the values or outcomes of a function. It is perpendicular to the x-axis and provides a visual reference for the range of values a function can take on.
Y-coordinate: The y-coordinate is the vertical position of a point on a coordinate plane, measured as the distance from the x-axis. It represents the up-down position of a point and is used to describe the location of objects or data points within a two-dimensional coordinate system.
Y-intercept: The y-intercept is the point at which a line or curve intersects the y-axis, representing the value of the dependent variable (y) when the independent variable (x) is zero. It is a crucial concept in understanding the behavior and properties of various mathematical functions and their graphical representations.