Problem 79 - Introductory Algebra for College Students

Question

What is a \(y\)-intercept of a graph?

Step-by-Step Solution

Answer

A y-intercept of a graph is the point or points where the graph intersects with the y-axis, that is, at a point where \(x = 0\). In terms of a line or curve, it represents the value of y when \(x = 0\).

Step 1: Understanding Graph Basics

A graph is a diagram that shows a relationship between two sets of numbers as a set of points having coordinates determined by the relationship. In a two-dimensional Cartesian coordinate system, these sets of numbers are usually represented by the horizontal axis (x-axis) and the vertical axis (y-axis). Any point on the graph has coordinates \((x, y)\). The x-coordinate is the distance from the y-axis (positive to the right and negative to the left), and the y-coordinate is the distance from the x-axis (positive upwards and negative downwards)

Step 2: Understanding the concept of a y-intercept

The y-intercept of a graph is the point at which the graph crosses the y-axis. In other words, it is the point where \(x = 0\). Since at this point \(x = 0\), the y-intercept is often given simply as a y-coordinate. If a line crosses the y-axis at point A for example, and if we denote \(A(y)\) to be the y-coordinate of point A, then \(A(0)\) represents the y-intercept of that line.

Step 3: Interpreting y-intercept for different lines

For a straight line, there is one and only one y-intercept. But, non-linear graphs or curves can cross the y-axis at more than one point. Hence, they would have more than one y-intercept. Understanding the concept of y-intercept is crucial for analysing and relating the mathematical expression of the line or curve to its graphical representation.

Key Concepts

Cartesian coordinate system

The Cartesian coordinate system is a mathematical concept used to graphically represent and analyze relationships between two variables. Imagine it as a grid where you can pinpoint exact locations using two numbers. These numbers are called coordinates. In a two-dimensional plane, a point is defined by two coordinates, \(x, y\), showing its horizontal and vertical positions.

• The horizontal line in this system is known as the x-axis.
• The vertical line is referred to as the y-axis.
• Both axes intersect at the point (0, 0), known as the origin.

The coordinate system is extremely useful in mathematics for plotting graphs. By locating and connecting various points, you can visualize equations and understand the relationship between different variables. This system helps in converting algebraic equations to visual representations that are easier to comprehend and analyze.

x-axis

The x-axis is a fundamental component of the Cartesian coordinate system, represented by a horizontal line. This axis serves as a reference line along which you measure the x-coordinates of a point in the plane. The x-coordinates tell you how far a point is from the y-axis, moving left or right.

• When a point's x-coordinate is positive, it is located to the right of the y-axis.
• If it is negative, the point lies to the left of the y-axis.
• An x-coordinate of zero indicates that the point is exactly on the y-axis.

The x-axis itself is crucial to understanding the layout of graphs as it can represent different scales depending on the context. Understanding the placement of points along the x-axis is key to interpreting graph data and determining relationships between variables.

y-axis

In the Cartesian coordinate system, the y-axis is the vertical line that serves as a reference for measuring the y-coordinates of points. It helps determine how far up or down a point is from the x-axis. The y-coordinate, in conjunction with the x-coordinate, specifies the location of a point within the coordinate plane.

• A positive y-coordinate places the point above the x-axis.
• A negative y-coordinate places the point below the x-axis.
• A y-coordinate of zero means the point is located on the x-axis itself.

The y-axis is particularly important for identifying the y-intercept of a graph, which is where the graph meets or intersects this axis. Recognizing how graphs relate to the y-axis aids in interpreting and solving various mathematical problems, especially those involving functions and linear equations.