Linear equations are the building blocks of algebra, describing straight lines on a graph. They're essential for understanding relationships between variables and making predictions.
Graphing linear equations using is a powerful technique. By finding where a line crosses the x and y axes, you can quickly plot its path. This method simplifies graphing and helps visualize equations in real-world contexts.
Graphing Linear Equations Using Intercepts
Locating graph intercepts
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The represents the point where a line intersects the (horizontal axis)
At this point, the y-coordinate equals 0
Expressed as an (a,0), where a is the x-coordinate value
The indicates the point where a line crosses the (vertical axis)
At this point, the x-coordinate equals 0
Expressed as an ordered pair (0,b), where b is the y-coordinate value
Intercepts provide crucial information about a line's position on the
Calculating linear equation intercepts
The of a is y=mx+b
m denotes the line's slope (steepness and direction)
b represents the y-intercept (where the line crosses the y-axis)
To determine the y-intercept, set x=0 in the equation and solve for y
The obtained y value corresponds to the y-coordinate of the y-intercept (0,y)
Example: For the equation y=2x+3, setting x=0 yields y=3, so the y-intercept is (0,3)
To find the x-intercept, set y=0 in the equation and solve for x
The obtained x value corresponds to the x-coordinate of the x-intercept (x,0)
Example: For the equation y=2x+3, setting y=0 gives 0=2x+3, solving for x yields x=−23, so the x-intercept is (−23,0)
Graphing lines using intercepts
Identify the x and y intercepts from the given linear equation
Calculate the y-intercept by setting x=0 and solving for y
Calculate the x-intercept by setting y=0 and solving for x
Plot the y-intercept on the y-axis and the x-intercept on the x-axis
Locate the y-intercept point (0,b) and mark it on the y-axis
Locate the x-intercept point (a,0) and mark it on the x-axis
Connect the plotted intercepts with a straight line using a straightedge
Extend the line beyond the intercepts in both directions to represent the line's infinite nature
Ensure the line passes through both intercept points
Label the graphed line with its equation, if provided
Write the equation near the line for clear identification
Example: To graph the line y=−21x+2, find the intercepts:
y-intercept: (0,2)
x-intercept: (4,0)
Plot the intercepts and connect them with a straight line, labeling it with the equation
Coordinate System and Graph Components
The coordinate plane is divided into four quadrants by the x and y axes
The is the point where the x and y axes intersect, represented as (0, 0)
are used to specify the location of points on the plane
A is a relationship between variables where each input has exactly one output
Key Terms to Review (13)
Cartesian Coordinates: Cartesian coordinates are a system used to locate points on a two-dimensional plane by specifying their horizontal and vertical positions. This coordinate system is essential for graphing functions and understanding the relationship between variables in the context of algebra.
Coordinate Plane: The coordinate plane, also known as the Cartesian coordinate system, is a two-dimensional plane used to represent and analyze the relationship between two variables. It consists of a horizontal x-axis and a vertical y-axis, which intersect at a point called the origin, forming a grid-like structure that allows for the precise location and graphing of points, lines, and other mathematical objects.
Function: A function is a special relationship between two or more variables, where the value of one variable (the dependent variable) depends on the value of the other variable(s) (the independent variable(s)). Functions are essential in mathematics, physics, and many other fields, as they allow us to model and analyze the behavior of various phenomena.
Intercepts: Intercepts refer to the points where a graph intersects the coordinate axes, providing important information about the behavior and characteristics of a function. They are crucial in understanding and interpreting graphical representations of mathematical relationships.
Linear Equation: A linear equation is a mathematical equation that represents a straight line on a coordinate plane. It is characterized by a constant rate of change, or slope, and a starting point, or y-intercept, that together define the line's position and orientation.
Ordered Pair: An ordered pair is a set of two numbers that represent a specific location on a coordinate plane. It consists of an x-coordinate and a y-coordinate, which together define a unique point in the rectangular coordinate system.
Origin: The origin is a fundamental point of reference in the Cartesian coordinate system, where the x-axis and y-axis intersect at the point (0, 0). This point serves as the starting point for measuring and graphing coordinates in the rectangular coordinate plane.
Quadrant: A quadrant is one of the four equal parts into which a plane or a sphere is divided by two intersecting lines or planes that are perpendicular to each other. It is a fundamental concept in the Cartesian coordinate system and is essential for understanding the graphing of linear and quadratic equations in two variables.
Slope-Intercept Form: Slope-intercept form is a way to represent the equation of a linear line in the form $y = mx + b$, where $m$ represents the slope of the line and $b$ represents the $y$-intercept, or the point where the line crosses the $y$-axis.
X-Axis: The x-axis is the horizontal line on a coordinate plane that represents the independent variable. It is the horizontal reference line that intersects the origin (0,0) and extends infinitely in both the positive and negative directions.
X-intercept: The x-intercept of a line is the point where the line crosses the x-axis, indicating the value of x when the value of y is zero. This point provides important information about the behavior and characteristics of the line.
Y-axis: The y-axis is one of the two primary axes in the rectangular coordinate system. It is the vertical line that runs from the bottom to the top of the coordinate plane, perpendicular to the x-axis. The y-axis is used to represent the vertical or up-and-down position of a point on the coordinate plane.
Y-intercept: The y-intercept is the point where a line or curve intersects the y-axis on a coordinate plane. It represents the value of the function at the point where the input (x-value) is zero, providing important information about the behavior and characteristics of the function.