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Chapter 7: Problem 31

Graph each equation using the slope and \(y\) -intercept. $$y=-3$$

Short Answer

Expert verified
The graph is a horizontal line crossing the y-axis at -3.

Step by step solution

01

Identify the Equation Type

The given equation is in the form of \( y = c \), where \( c \) is a constant. This indicates a horizontal line.
02

Determine the Y-Intercept

Since the equation is \( y = -3 \), the \(y\)-intercept is \((0, -3)\). This is the point where the line crosses the y-axis.
03

Understand the Slope

In the equation \( y = -3 \), the slope \( m \) is \( 0 \) because the equation does not include a \( x \) term. A slope of zero indicates that the line is horizontal.
04

Plot the Y-Intercept

Place a point on the coordinate plane at the y-intercept \((0, -3)\).
05

Draw the Horizontal Line

From the plotted y-intercept, draw a straight horizontal line extending in both directions. This horizontal line represents all points where \( y = -3 \).

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Key Concepts

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

Slope-Intercept Form
Understanding the slope-intercept form is a great foundation for graphing linear equations. The standard form of this equation is \( y = mx + b \), where \( m \) stands for the slope and \( b \) is the \( y \)-intercept.
  • **Slope (\( m \)):** This tells us how steep the line is and in which direction it moves. A positive slope moves upward, while a negative slope goes downward.
  • **Y-Intercept (\( b \)):** This is the point where the line intersects the y-axis on a graph.
In cases where the equation is just \( y = c \), such as \( y = -3 \), there is no \( x \) term. This signifies a special case where the slope is zero. All lines parallel to the x-axis have flat, horizontal slopes.
Horizontal Line
When we encounter a graph formed by an equation like \( y = c \), we are looking at a horizontal line. This type of line has some distinct features:
  • **Constant \( y \)-value:** All points on the line share the same \( y \)-coordinate. In our example, every point has \( y = -3 \).
  • **Zero Slope:** Since the line does not rise or fall, the slope is zero.
A horizontal line is simple to graph; it just extends parallel to the x-axis and runs infinitely in both directions. It never touches or approaches the x-axis since it stays consistent with its given \( y \)-value.
Y-Intercept
The y-intercept is crucial in graphing, as it provides a starting point. For a horizontal line like \( y = -3 \), the y-intercept is easy to identify.
  • **What It Tells Us:** The y-intercept shows exactly where the line crosses the y-axis. This point is always represented as \((0, b)\).
  • **Example Case:** In our problem, the y-intercept is \((0, -3)\). It means that when the x-coordinate is zero, the y-coordinate is \(-3\).
Starting at the y-intercept helps create the basis for drawing the entire line. Once you plot this point on the graph, extending the line horizontally gives a complete picture.
Graphing Techniques
Graphing linear equations involves a few key steps to ensure accuracy. Simplifying the process can make graphing more intuitive.
  • **Identify Key Components:** Recognize the slope and y-intercept from the equation to get a general idea of the line's form.
  • **Plot the Y-Intercept:** Begin by placing a point at the determined y-intercept on the coordinate plane.
  • **Draw the Line:** For horizontal lines like \( y = -3 \), draw a straight line through the y-intercept, moving parallel to the x-axis in both left and right directions.
By following these steps, you can accurately draw not just horizontal lines, but any linear equation. Understanding each element of graphing provides a solid foundation that will simplify dealing with more complex equations in the future.

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