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Chapter 9: Problem 37

For Exercises \(37-40,\) graph. $$y=-5$$

Short Answer

Expert verified
Answer: The graph of the equation \(y = -5\) is a horizontal line passing through the point \((0, -5)\) on the \(y\)-axis.

Step by step solution

01

Identify the equation type

The equation is of the form \(y = c\), where \(c\) is a constant. In this case, \(c = -5\). This type of equation represents a horizontal line, which has a slope of \(0\).
02

Determine the slope and \(y\)-intercept of the line

Since the equation is of the form \(y = c\), we already know that the slope \(m = 0\), and the constant \(c = -5\) represents the \(y\)-intercept. This means the line will cross the \(y\)-axis at the point \((0, -5)\).
03

Plot the line

Start by drawing the \(x\) and \(y\)-axes on a graph paper or grid. Mark the \(y\)-intercept at the point \((0, -5)\). Since the slope is \(0\), the line is horizontal, and all of the points on the line have the same \(y\)-value \(-5\). Draw a horizontal line passing through the point \((0, -5)\) to get the graph of the equation \(y = -5\).

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Key Concepts

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

Horizontal Lines
Horizontal lines are a simple yet essential part of understanding linear equations. When we talk about horizontal lines in math, we refer to lines that go straight across from left to right.
They are parallel to the x-axis and never tilt up or down.
  • Characteristics: All points on a horizontal line have the same y-coordinate.
  • Slope: The slope of a horizontal line is zero because it doesn鈥檛 rise or fall.
Imagine drawing a line across a piece of paper without moving your pen up or down鈥攖hat鈥檚 a horizontal line. It's all about consistency in the y-direction.
Y-Intercept
The y-intercept is where a line crosses the y-axis on a graph. In equations of the form \(y = c\), the y-intercept is the constant value \(c\). For the equation \(y = -5\), the y-intercept is -5.
  • Location: To find the y-intercept on a graph, look at the point where the line meets the y-axis.
  • Understanding: The y-intercept tells us that when \(x = 0\), the value of \(y\) is the intercept value.
This point is crucial because it helps in drawing the line correctly on a graph. For example, in our exercise, the line crosses the y-axis at the point \((0, -5)\).
Slope
The slope of a line describes how steep the line is and the direction in which it moves. It's calculated by how much \(y\) changes as \(x\) changes.
  • Zero Slope: When a slope is zero, the line is horizontal. This means there is no vertical change as you move along the line.
  • Formula: For a line in the form \(y = mx + b\), the slope is \(m\). In horizontal lines like \(y = -5\), \(m = 0\).
The idea of slope clarifies why horizontal lines don't slope or tilt, giving them a flat appearance on a graph. With the line \(y = -5\), it stretches evenly across the plane.
Constant Functions
Constant functions represent a unique type of function where the output value remains the same regardless of the input value.
  • Definition: For a function \(f(x) = c\), \(c\) is the constant output for all \(x\)-values.
  • Graph: The graph of a constant function is a horizontal line. For our given function \(y = -5\), it shows this principle by not changing its value no matter the \(x\) value.
This concept reflects why such equations like \(y = -5\) are easy to graph: every possible x-value yields the same y-value, resulting in a simple but effective horizontal line on the coordinate plane. Constant functions introduce the basic idea that some relationships don't change regardless of circumstances.

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