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Chapter 1: Problem 4

Find the slopes and \(y\) -intercepts of the following lines. \(y=6\)

Short Answer

Expert verified
Slope = 0, y-intercept = 6.

Step by step solution

01

- Identify the Form of the Equation

Identify the form of the given equation. The equation given is in the form of a horizontal line: \(y = 6\).
02

- Determine the Slope of the Line

In the equation \(y = 6\), the slope \(m\) of a horizontal line is \(0\) because the line is perfectly flat and does not rise or fall.
03

- Determine the y-intercept

The \(y\)-intercept of the line is the value of \(y\) when \(x = 0\). From the equation \(y = 6\), it shows that the line intercepts the \(y\)-axis at the point \(6\). Therefore, the \(y\)-intercept is \(6\).

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Key Concepts

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

Horizontal Lines
A horizontal line is a unique type of line in the Cartesian plane. It's called horizontal because it runs left to right and stays at the same height. For example, the equation you鈥檝e seen, like \(y = 6\), is a horizontal line.

Key characteristics of horizontal lines include:
  • They have a constant y-value, meaning their height doesn't change.
  • They never touch the x-axis unless y = 0.

    • The equation \(y = 6\) means that for any x-value, y will always be 6. This is why the line is flat and horizontal. There is no rise between any two points, which leads to an important feature of horizontal lines.
    Y-Intercept
    The y-intercept is where the line crosses the y-axis on a graph. Understanding the y-intercept is crucial because it tells us a lot about the line鈥檚 position. For horizontal lines like \(y=6\), the y-intercept is particularly clear.

    When the equation is \(y=6\), every point on the line shares the y-value of 6. This tells us that the line crosses the y-axis at the point (0, 6). Here, the x-value is zero because it鈥檚 the y-axis.

    Key points to remember about the y-intercept:
  • In any line equation \(y = mx + b\), \(b\) represents the y-intercept.
  • For horizontal lines (y = k), k itself is the y-intercept since x values don't matter.

    • So, for the equation \(y = 6\), the y-intercept is simply 6.
    Line Equation Forms
    Understanding the different forms of line equations helps in quickly identifying their characteristics. There are three common forms:
  • Slope-Intercept Form \(y = mx + b\)
  • Point-Slope Form \(y - y_1 = m(x - x_1)\)
  • Standard Form \(Ax + By = C\)

    • For the equation \(y = 6\), we focus mainly on the Slope-Intercept form because it makes it easy to read the slope (m) and y-intercept (b).

    Specific to the Slope-Intercept Form:
  • It鈥檚 written as \(y = mx + b\).
  • The slope \(m\) indicates the tilt of the line.
  • The y-intercept \(b\) indicates where the line touches the y-axis.

    • In the given equation \(y = 6\), the slope (m) is 0 because there's no x-term, and the y-intercept (b) is 6. This makes it a horizontal line.

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    Most popular questions from this chapter

    g(x) is the tangent line to the graph of \(f(x)\) at \(x=a\). Graph \(f(x)\) and \(g(x)\) and determine the value of \(a\). \(f(x)=x^{3}-12 x^{2}+46 x-50, g(x)=14-2 x\)

    An object moving in a straight line travels \(s(t)\) kilometers in \(t\) hours, where \(s(t)=2 t^{2}+4 t .\) (a) What is the object's velocity when \(t=6 ?\) (b) How far has the object traveled in 6 hours? (c) When is the object traveling at the rate of 6 kilometers per hour?

    If \(f(100)=5000\) and \(f^{\prime}(100)=10\), estimate each of the following. (a) \(f(101)\) (b) \(f(100.5)\) (c) \(f(99)\) (d) \(f(98)\) (e) \(f(99.75)\)

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    Let \(f(p)\) be the number of cars sold when the price is \(p\) dollars per car. Interpret the statements \(f(10,000)=200,000\) and \(f^{\prime}(10,000)=-3\).
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