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Chapter 4: Problem 16

Each of the following equations is in slope-intercept form. Identify the slope and the \(y\) -intercept, then graph each line using this information. $$y=4$$

Short Answer

Expert verified
The slope of the given equation \(y=-4\) is \(0\), as there is no \(x\) term present. The y-intercept is \(-4\), indicating the line intersects the y-axis at the point \((0, -4)\). To graph the line, plot the y-intercept on the vertical axis and draw a horizontal line through this point, parallel to the x-axis.

Step by step solution

01

Identify the slope

The given equation is \(y = -4\). Comparing this with the general slope-intercept form \(y = mx + b\), we can see that there is no \(x\) term present, which means the slope, \(m\), must be equal to \(0\). Therefore, the slope of the line is \(0\).
02

Identify the y-intercept

The y-intercept is the point where the line intersects the y-axis. In the given equation, we have the constant value \(-4\), which indicates the line intersects the y-axis at the point \((0, -4)\). Thus, the y-intercept is \(-4\).
03

Graph the line

To graph the line, we will use the slope and the y-intercept found in steps 1 and 2. Since the slope is \(0\), the line will be horizontal. We can begin by plotting the y-intercept \((0, -4)\) on the vertical axis. Next, draw a straight horizontal line through the plotted point that is parallel to the x-axis. This line represents the equation \(y = -4\). The graph of the line can be created using any graphing tool or software, or by drawing the graph on graph paper.

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Key Concepts

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

Graphing Linear Equations
Linear equations can be graphed to visually represent their relationships on a coordinate plane. Most linear equations can be written in the slope-intercept form, which is given by the formula \( y = mx + b \). Here, \( m \) represents the slope of the line and \( b \) is the y-intercept. By identifying these two components, you can graph any linear equation.
  • The slope \( m \) indicates how steep the line is and the direction it tilts; a positive slope rises to the right, while a negative slope falls to the right.
  • The y-intercept \( b \) is where the line crosses the y-axis, and it is a key starting point for graphing.

To graph a linear equation:
  • Spot the slope and y-intercept from the equation.
  • Plot the y-intercept on the graph.
  • Use the slope to identify another point by moving up or down and left or right based on the slope value.
  • Connect these points with a straight line to graph the equation.
Y-Intercept
The y-intercept of a linear equation is crucial because it shows where the line crosses the y-axis. In the slope-intercept form, \( y = mx + b \), the \( b \) represents the y-intercept. Graphically, it is simply the point \((0, b)\) on the coordinate plane.
If you look at an equation like \( y = -4 \), you can quickly tell that the y-intercept is -4 because there is no \( x \) term, showing that every point on the line has the same y-coordinate of -4. This is typical of horizontal lines. Therefore:
  • The y-intercept is the constant in the equation.
  • It gives you the precise spot where the line will meet the y-axis.
  • Knowing this allows for easy graphing since it serves as an anchor point on the graph.
Horizontal Lines
Horizontal lines are a special case of linear equations. They occur when the slope is zero, meaning there is no rise or fall along the graph; it stays consistent. Equations of horizontal lines look like \( y = c \) where \( c \) is any constant number.
When graphing horizontal lines, consider the following:
  • The line stays parallel to the x-axis throughout.
  • All points on the line share the same y-coordinate, like in the case of \( y = -4 \).
  • The slope \( m \) in a horizontal line is always 0, reinforcing its flatness on the graph.
So when given an equation such as \( y = -4 \), you should plot a line that crosses the y-axis at -4, then draw horizontally across the graph. This represents all points where y is constantly -4, making the task simple and straightforward.

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Most popular questions from this chapter

Lines \(L_{1}\) and \(L_{2}\) contain the given points. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. $$\begin{aligned}&L_{1}:(-3,2),(0,2)\\\&L_{2}:(1,-1),(-2,-1)\end{aligned}$$

Graph each function using the slope and \(y\) -intercept. $$g(x)=3 x+3$$

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=\frac{8}{2 x+1}$$

Write the slope-intercept form of the equation of the line, if possible, given the following information. \(m=3\) and contains \((4,5)\)

Write the slope-intercept form of the equation of the line, if possible, given the following information. \(y\) -intercept \((0,6)\) and \(m=7\)
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