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Chapter 1: Problem 47

Find the slope and \(y\) -intercept of the line and draw its graph. $$y=4$$

Short Answer

Expert verified
The slope is 0 and the y-intercept is 4.

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 type of equation represents a horizontal line on the Cartesian plane.
02

Determine the Slope

For any horizontal line described by \( y = c \), the slope is \( 0 \). This is because horizontal lines do not rise or fall, so the rate of change in \( y \) with respect to \( x \) is zero.
03

Identify the y-intercept

The \( y \)-intercept is the point where the line crosses the \( y \)-axis. For the equation \( y = 4 \), the \( y \)-intercept is at \( (0, 4) \). This means the line crosses the \( y \)-axis at \( y = 4 \).
04

Draw the Graph

Draw a horizontal line that passes through \( y = 4 \) on the graph. This line extends infinitely in both positive and negative \( x \)-directions and remains constant vertically at \( y=4 \). The graph should be a straight horizontal line parallel to the \( x \)-axis.

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Key Concepts

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

Slope
The slope of a line is an essential concept in understanding linear equations and graphing. It tells us how steep a line is, or in mathematical terms, the rate at which one variable changes with respect to another.
  • The Definition: The slope is the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. It's often represented by the letter "m" and can be calculated using the formula: \( m = \frac{\Delta y}{\Delta x} \). In this formula, \(\Delta y\) represents the change in the \(y\)-values, and \(\Delta x\) represents the change in the \(x\)-values.
  • Horizontal Lines: For horizontal lines, such as the one given by \(y = 4\), the slope is \(0\). This is because there is no change in the \(y\)-value regardless of the \(x\)-value, meaning the rise is zero.
Understanding the slope helps us grasp why horizontal lines, like \(y = 4\), do not tilt upwards or downwards, but rather extend flatly across the plane.
y-intercept
The y-intercept is where a line crosses the y-axis on a graph. It's a critical point in understanding a line's position and behavior on the Cartesian plane.
  • Identifying the y-intercept: In any equation of the form \(y = mx + c\), the constant "c" is the \(y\)-intercept. For equations like \(y = 4\), which don't have an "x" variable, the y-intercept is simply the constant number itself. Here, the line crosses the \(y\)-axis at the point \((0, 4)\).
  • Visualizing the y-intercept: On a graph, imagine dropping a vertical line down to where the main line touches the \(y\)-axis. With the line \(y = 4\), this point is unmistakably at 4 on the \(y\)-axis.
Recognizing the y-intercept helps with sketching and interpreting graphs, providing a starting point to understand how the line behaves.
Cartesian Plane
The Cartesian plane is the backdrop against which we graph equations and visualize mathematical relationships.
  • The Layout: It is a two-dimensional surface defined by two perpendicular number lines, the \(x\)-axis (horizontal) and the \(y\)-axis (vertical). The point where they intersect is the origin, labeled \((0, 0)\).
  • Graphing Lines: On this plane, every linear equation represents a line. Horizontal lines, such as \(y = 4\), are parallel to the \(x\)-axis, and they span across this plane without changing height, showcasing a unique use of the space.
  • Plotting Points and Lines: Point coordinates are given as \((x, y)\) values, indicating their position relative to the origin. For \(y = 4\), you can plot several points like \((1, 4), (2, 4), (-3, 4)\), and connect them to form the line.
Exploring the Cartesian plane is fundamental to understanding algebra and geometry, as it allows for visualization of abstract mathematical concepts in a tangible way.

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Most popular questions from this chapter

Show that the equation represents a circle, and find the center and radius of the circle. $$x^{2}+y^{2}-\frac{1}{2} x+\frac{1}{2} y=\frac{1}{8}$$

A square field in a certain state park is mowed around the edges every week. The rest of the field is kept unmowed to serve as a habitat for birds and small animals (see the figure). The field measures \(b\) feet by \(b\) feet, and the mowed strip is \(x\) feet wide. (a) Explain why the area of the mowed portion is \(b^{2}-(b-2 x)^{2}\) (b) Factor the expression in (a) to show that the area of the mowed portion is also \(4 x(b-x)\) CAN'T COPY THE GRAPH

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(a) Find the radius of each circle in the pair, and the distance between their centers; then use this information to determine whether the circles intersect. (i) \((x-2)^{2}+(y-1)^{2}=9\) \((x-6)^{2}+(y-4)^{2}=16\) (ii) \(x^{2}+(y-2)^{2}=4\) \((x-5)^{2}+(y-14)^{2}=9\) (iii) \((x-3)^{2}+(y+1)^{2}=1\) \((x-2)^{2}+(y-2)^{2}=25\) (b) How can you tell, just by knowing the radii of two circles and the distance between their centers, whether the circles intersect? Write a short paragraph describing how you would decide this and draw graphs to illustrate your answer.
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