Fiveable
Fiveable
Fiveable
Fiveable

Pre-Algebra

11.3 Graphing with Intercepts

3 min readLast Updated on June 25, 2024

Linear equations are the building blocks of graphing. They show how two variables relate on a coordinate plane. Understanding x and y-intercepts is key to plotting these equations accurately and efficiently.

Graphing linear equations helps visualize mathematical relationships. By calculating intercepts and using different forms of equations, you can plot lines quickly. This skill is crucial for more complex math and real-world problem-solving.

Graphing Linear Equations

X and y-intercepts on coordinate planes

Top images from around the web for X and y-intercepts on coordinate planes
Top images from around the web for X and y-intercepts on coordinate planes
  • The x-intercept represents the point where a graph intersects the x-axis (horizontal axis)
    • At the x-intercept, the y-coordinate equals 0
    • The coordinate point for the x-intercept is written as (x,0)(x, 0) (e.g., (3,0)(3, 0))
  • The y-intercept signifies the point where a graph intersects the y-axis (vertical axis)
    • At the y-intercept, the x-coordinate equals 0
    • The coordinate point for the y-intercept is written as (0,y)(0, y) (e.g., (0,2)(0, -2))

Calculation of linear equation intercepts

  • For a linear equation in slope-intercept form, y=mx+by = mx + b
    • The y-intercept is the value of bb, which represents the constant term (e.g., in y=2x+3y = 2x + 3, the y-intercept is 3)
    • To find the x-intercept, substitute y=0y = 0 into the equation and solve for xx (e.g., 0=2x+30 = 2x + 3, x=32x = -\frac{3}{2})
  • For a linear equation in standard form, Ax+By=CAx + By = C
    • To find the x-intercept, substitute y=0y = 0 into the equation and solve for xx (e.g., 2x+3(0)=62x + 3(0) = 6, x=3x = 3)
    • To find the y-intercept, substitute x=0x = 0 into the equation and solve for yy (e.g., 2(0)+3y=62(0) + 3y = 6, y=2y = 2)

Graphing with intercept points

  • To graph a linear equation using intercepts, follow these steps:
    1. Calculate the x and y-intercepts using the methods described above
    2. Plot the intercept points on the coordinate plane
    3. Connect the two intercept points with a straight line using a ruler or straightedge
  • The line extending through the intercept points represents the graph of the linear equation (e.g., the line passing through (3,0)(3, 0) and (0,2)(0, 2) represents the graph of 2x+3y=62x + 3y = 6)

Efficiency in linear equation graphing

  • Graphing using intercepts is most efficient when:
    • The equation is in standard form, Ax+By=CAx + By = C (e.g., 2x+3y=62x + 3y = 6)
    • The x and y-intercepts have integer values (e.g., (3,0)(3, 0) and (0,2)(0, 2))
  • Graphing using slope-intercept form is most efficient when:
    • The equation is in slope-intercept form, y=mx+by = mx + b (e.g., y=2x+3y = 2x + 3)
    • The slope (mm) and y-intercept (bb) values are easily identifiable (e.g., slope = 2, y-intercept = 3)
  • Consider the complexity of the equation and the ease of calculating intercepts or slope to determine the most suitable graphing method (e.g., if the equation is in standard form with integer intercepts, use the intercept method)

Coordinate Plane Components

  • The coordinate plane consists of two perpendicular number lines called axes
  • The horizontal line is the x-axis, and the vertical line is the y-axis
  • The point where the axes intersect is called the origin, with coordinates (0, 0)
  • The coordinate plane is divided into four quadrants, numbered counterclockwise from the upper right
  • A graph is a visual representation of a mathematical relationship on the coordinate plane

Key Terms to Review (11)

Axes: Axes, in the context of graphing, refer to the horizontal and vertical reference lines that form the coordinate system used to plot points and represent mathematical relationships. These intersecting lines provide a framework for visualizing and analyzing data and functions.
Coordinate Plane: The coordinate plane is a two-dimensional grid used to represent and analyze the position and relationships of points, lines, and other geometric shapes. It consists of a horizontal x-axis and a vertical y-axis that intersect at a central point known as the origin.
Graph: A graph is a visual representation of data or relationships, often using a coordinate system to plot points and lines. It is a fundamental tool in mathematics and various scientific disciplines for analyzing and communicating information in a clear and concise manner.
Intercept Points: Intercept points refer to the points where a line or curve intersects the x-axis or y-axis on a coordinate plane. These points provide valuable information about the behavior and characteristics of the function or equation represented by the line or curve.
Linear Equation: A linear equation is a mathematical equation in which the variables are raised only to the first power and are connected by addition, subtraction, or equality. These equations represent a straight line when graphed on a coordinate plane.
Origin: The origin, in the context of the rectangular coordinate system, graphing linear equations, and graphing with intercepts, refers to the point where the x-axis and y-axis intersect. This point, denoted as (0,0), serves as the starting reference point for all coordinates and is a crucial element in understanding and working with these mathematical concepts.
Quadrants: Quadrants refer to the four distinct regions created by the intersecting x-axis and y-axis in the rectangular coordinate system. These four regions are used to organize and locate points on a coordinate plane.
Slope-Intercept Form: Slope-intercept form is a way to represent a linear equation in the format $y = mx + b$, where $m$ represents the slope of the line and $b$ represents the $y$-intercept. This form allows for easy graphing and interpretation of the relationship between the variables $x$ and $y$.
Standard Form: Standard form is a way of expressing numbers, equations, or other mathematical entities in a specific, organized, and easily recognizable format. It provides a consistent and concise way to represent these elements, making them easier to work with, compare, and manipulate across various mathematical contexts.
X-Intercept: The x-intercept of a linear equation is the point where the graph of the equation crosses the x-axis. It represents the value of x when the value of y is zero, indicating where the line intersects the horizontal axis.
Y-intercept: The y-intercept is the point where a line or graph intersects the y-axis, representing the value of the function when the independent variable (x) is equal to zero. It is a crucial concept in understanding the behavior and properties of linear equations and their graphical representations.
Axes
See definition

Axes, in the context of graphing, refer to the horizontal and vertical reference lines that form the coordinate system used to plot points and represent mathematical relationships. These intersecting lines provide a framework for visualizing and analyzing data and functions.

Term 1 of 11

Key Terms to Review (11)

Axes
See definition

Axes, in the context of graphing, refer to the horizontal and vertical reference lines that form the coordinate system used to plot points and represent mathematical relationships. These intersecting lines provide a framework for visualizing and analyzing data and functions.

Term 1 of 11

Axes
See definition

Axes, in the context of graphing, refer to the horizontal and vertical reference lines that form the coordinate system used to plot points and represent mathematical relationships. These intersecting lines provide a framework for visualizing and analyzing data and functions.

Term 1 of 11



© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.