Problem 3 - College Algebra with Corequisite Support

Question

Describe in your own words what the \(y\) -intercept of a graph is.

Step-by-Step Solution

Answer

The y-intercept is where the graph crosses the y-axis with x = 0.

Step 1: Understanding the Concept

The y-intercept is the point where a graph crosses the y-axis. This point has a very specific feature: its x-value is always 0.

Step 2: Applying the Concept to an Example

For example, consider the linear equation \( y = 2x + 3 \). To find its y-intercept, substitute \( x = 0 \) into the equation, which simplifies to \( y = 3 \). So, the y-intercept is the point \( (0, 3) \).

Step 3: Generalization

In any graph, no matter the type, the y-intercept will always have an x-coordinate of 0. It will represent the value of y when x is zero.

Key Concepts

Linear Equations

Linear equations are a fundamental concept in mathematics and form the basis for understanding lines on a graph. Simply put, a linear equation is an equation that describes a straight line when graphed on a coordinate plane. These equations are usually written in the form \( y = mx + b \), where:

• \( m \) is the slope of the line, indicating how steep the line is.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

The y-intercept is particularly important because it gives us a starting point for drawing the line on a graph. For instance, in the equation \( y = 2x + 3 \), as explained in the solution, the y-intercept \( b \) is 3. This means, regardless of the value of \( x \), the line will always intersect the y-axis at the point \( (0, 3) \). Understanding linear equations is essential to tackling more complex mathematical concepts and serves as an introduction to algebra.

Graph Interpretation

Interpreting graphs is a crucial skill in mathematics and many real-world applications. When you look at a graph, you are essentially deciphering the information it displays about relationships between variables.
When dealing with linear equations, understanding the graph not only means identifying the y-intercept but also analyzing the overall behavior of the line.A graph of a linear equation provides insights such as:

• Slope: The steepness or incline of the line, determined by \( m \) in the equation \( y = mx + b \). A positive slope means the line increases from left to right, while a negative slope means it decreases.
• Y-Intercept: The point where the line crosses the y-axis, which is where the value of \( x \) is 0.
• X-Intercept: Although not mentioned directly in the initial exercise, this is where the line crosses the x-axis and \( y \) is 0.

Understanding these elements helps you interpret what the line represents. For example, the line in the equation \( y = 2x + 3 \) shows a consistent rate of increase, as evidenced by the positive slope and starting point at the y-intercept.

Coordinate Plane

The coordinate plane is a two-dimensional space formed by two intersecting lines: the x-axis and the y-axis. These axes help us locate points on the plane, forming the basis for graphing equations and interpreting solutions like the y-intercept.Each point on this plane is represented by a pair of numbers \((x, y)\), known as coordinates. These coordinates show:

• The horizontal displacement from the origin (where the x and y axes meet) is indicated by \( x \).
• The vertical displacement from the origin is indicated by \( y \).

When considering the y-intercept, the point \((0, b)\) indicates a position directly on the y-axis, as the x-coordinate is always 0.
For any linear graph, the relationship between the x and y coordinates is determined by the equation you are graphing. The ability to plot and read these coordinates allows for understanding more complex mathematical relationships beyond simple arithmetic. This skill is foundational for students seeking to grasp algebra and calculus concepts.