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Chapter 4: Problem 6

Name the slope and \(y\)-intercept of the graph of each equation. \(y=2 x+6\)

Short Answer

Expert verified
Slope: 2, y-intercept: 6.

Step by step solution

01

Identify the form of the equation

The given equation is in the form of a linear equation written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Determine the slope

In the equation \( y = 2x + 6 \), the coefficient of \( x \) is \( 2 \). This is the value of \( m \), which is the slope of the line.
03

Determine the y-intercept

In the equation \( y = 2x + 6 \), the constant term is \( 6 \). This is the value of \( b \), which is the y-intercept of the line.

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Key Concepts

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

Understanding the Slope
The slope is a fundamental concept in linear equations, representing the steepness or direction of a line. In mathematical terms, slope is typically denoted by the letter \( m \) and can be found in the equation of the line, usually in the format \( y = mx + b \). Here are some key points about slope:
  • The slope of a line is calculated as the change in the \( y \)-coordinates divided by the change in the \( x \)-coordinates, often represented by the formula \( m = \frac{\Delta y}{\Delta x} \).
  • A positive slope indicates that the line is rising as it moves from left to right, while a negative slope shows that the line is falling.
  • If the slope is zero, the line is horizontal, while an undefined slope means the line is vertical.
In the exercise you provided, the slope of the equation \( y = 2x + 6 \) is \( 2 \), suggesting that for every unit increase in \( x \), \( y \) increases by 2 units.
What is the Y-Intercept?
The \( y \)-intercept is a special point where the line crosses the \( y \)-axis. It is an essential element in understanding linear equations and is usually represented by the letter \( b \). Here are a few insights about the \( y \)-intercept:
  • The \( y \)-intercept occurs at the point where \( x = 0 \) in the equation \( y = mx + b \).
  • It provides a starting value for the line and shows where the line will meet the \( y \)-axis.
  • In a graph, it's the point where the line intersects the vertical axis, giving a visual clue to the line's trajectory.
For the equation \( y = 2x + 6 \), the \( y \)-intercept is \( 6 \), which means the line intersects the \( y \)-axis at the point \( (0, 6) \). This information is crucial when graphing lines, as it serves as a reference point.
Recognizing the Equation Form
Linear equations usually take on a standard form, making it easier to identify their characteristics and graph them. The standard linear equation form is \( y = mx + b \), where:
  • \( y \) and \( x \) are variables representing coordinates on the graph.
  • \( m \) is the slope, showing how steep the line is.
  • \( b \) is the \( y \)-intercept, indicating where the line crosses the \( y \)-axis.
Recognizing this form helps to quickly find the slope and \( y \)-intercept, enabling easier plotting of these lines. For instance, the equation \( y = 2x + 6 \) fits perfectly into this structure, demonstrating a slope of \( 2 \) and a \( y \)-intercept of \( 6 \). By identifying the components of this equation form, plotting becomes straightforward and less cumbersome.

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Most popular questions from this chapter

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