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Chapter 2: Problem 43

Graph each equation. \(y=-\frac{1}{2}\)

Short Answer

Expert verified
A horizontal line through \(y = -\frac{1}{2}\).

Step by step solution

01

Understand the Equation

The equation given is in the form of a horizontal line, where \(y = -\frac{1}{2}\). It represents a constant function where every point on the line has the same y-coordinate, specifically \(-\frac{1}{2}\).
02

Plot Points on the Graph

For a horizontal line with equation \(y = -\frac{1}{2}\), select any values for \(x\), because the line will have the same y-value for any x-value. For example, if \(x = -2, 0, 2\), the points \(( -2, -\frac{1}{2}), (0, -\frac{1}{2}), (2, -\frac{1}{2})\) all lie on the line.
03

Draw the Horizontal Line

Use a ruler to draw a straight horizontal line through these plotted points. Make sure the line extends across the entire graph in both directions, remaining at \(y = -\frac{1}{2}\) the entire time. Label the line with its equation to clearly indicate what it represents.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constant Functions
A constant function is a specific type of linear function where the output or y-value never changes, regardless of the input or x-value. In simpler terms, no matter what value you plug in for x, the output for y remains constant.
This is represented by the equation of the form \(y = c\), where \(c\) is a constant number. In this exercise, the equation \(y = -\frac{1}{2}\) exemplifies a constant function. Every point on this line has a y-coordinate of \(-\frac{1}{2}\).
  • These functions can be seen as horizontal lines on a graph.
  • They do not slope upward or downward; they're perfectly flat.
  • Each point on the horizontal line has identical y-values but may have different x-values.
Understanding constant functions will help you quickly identify scenarios where the output remains the same, regardless of how the input varies.
Plotting Points
Plotting points is a fundamental skill for creating graphs. It involves identifying and marking specific points on the coordinate plane by their x and y coordinates.
In this exercise, to graph the equation \(y = -\frac{1}{2}\), you start by choosing any values for x since the y-value remains constant. For instance:
  • If \(x = -2\), the point is \((-2, -\frac{1}{2})\).
  • If \(x = 0\), the point is \((0, -\frac{1}{2})\).
  • If \(x = 2\), the point is \((2, -\frac{1}{2})\).
Once these points are plotted, you can see they all lie on a horizontal line at \(y = -\frac{1}{2}\). Plotting points correctly ensures that your graph is accurate and can be easily read by others.
Coordinate Plane
The coordinate plane is a two-dimensional surface formed by an intersecting vertical and horizontal line, referred to as the y-axis and x-axis, respectively.
Understanding the coordinate plane is crucial for graphing any function or data set. Here are some key points about the coordinate plane:
  • The point where the x-axis and y-axis intersect is called the origin, which has coordinates \((0, 0)\).
  • Horizontal movements along the x-axis represent changes in x-value, and vertical movements along the y-axis represent changes in y-value.
  • When graphing the equation \(y = -\frac{1}{2}\), you remain parallel to the x-axis, showing no change in y-value across all x-values.
This graphing surface allows for a clear and precise graphic representation of relationships like the constant function represented by a horizontal line. Understanding how to navigate the coordinate plane is foundational for studying mathematics.

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