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Chapter 4: Problem 26

Graph each equation by finding the intercepts and at least one other point. $$y=-1$$

Short Answer

Expert verified
The x-intercept does not exist for the equation \(y = -1\). The y-intercept is at point \((0, -1)\), and another point on the graph is \((1, -1)\). Plot these points on the graph and draw a horizontal line through them to obtain the graph of the equation \(y = -1\), which is a horizontal line passing through the y-intercept at y = -1.

Step by step solution

01

Find the x-intercept

To find the x-intercept, we set \(y = 0\) and solve for x. In our equation, \(y = -1\), so we substitute 0 for y and solve for x. However, \(0 \ne -1\), so there is no x-intercept for this graph.
02

Find the y-intercept

To find the y-intercept, we set \(x = 0\). In our equation, \(y = -1\), the y-intercept is already calculated because the equation indicates that y is always -1 regardless of the x value. Therefore, the y-intercept is at point \((0, -1)\).
03

Find at least one other point

Since our equation is a horizontal line with a constant value of y, we can plug in any x-value to obtain another point. Let's choose \(x = 1\). When \(x = 1\), \(y = -1\). Thus, another point on the graph is \((1, -1)\).
04

Plot the points on a graph

The points we have found are the y-intercept \((0, -1)\) and another point \((1, -1)\). Plot these points on the graph, then draw a horizontal line through them. The resulting line represents the graph of the equation \(y = -1\), which is a horizontal line passing through the y-intercept at y = -1.

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Key Concepts

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

Understanding the X-Intercept
The x-intercept of a graph is where the line crosses the x-axis. This happens when the value of y is zero. For our equation, though, it's a bit different because the line is horizontal with the equation \( y = -1 \).

Here's why this matters:
  • To find the x-intercept, you substitute \( y = 0 \) into the equation.
  • In this case, substituting gives \( 0 eq -1 \), which is impossible.
This means there are no x-intercepts for the equation \( y = -1 \).

Any graph of a horizontal line (where y is a constant) won't touch the x-axis unless \( y = 0 \). This is a key detail to remember when graphing linear equations.
Discovering the Y-Intercept
The y-intercept is the point where the graph crosses the y-axis. This occurs when the value of x is zero. For the equation \( y = -1 \), this concept is straightforward.

Here is how to identify the y-intercept:
  • Set \( x = 0 \) in the equation.
  • For \( y = -1 \), you find that y is -1 no matter what x is.
  • Thus, the y-intercept is at the point \( (0, -1) \).
Understanding the y-intercept is crucial as it gives a point to start drawing the graph. For horizontal lines like this, it shows where the line consistently sits on the y-axis.
Graphing Horizontal Lines
Horizontal lines have a special place in graphing as they represent a constant y-value across all x-values. In our case, the equation is \( y = -1 \).

Some important features of horizontal lines are:
  • The line runs left to right, parallel to the x-axis.
  • It displays that y doesn't change regardless of x; here, y always equals -1.
  • Horizontal lines only have a y-intercept, and in this exercise, it's at \( (0, -1) \).
When plotting a horizontal line:
  • Start by marking the y-intercept.
  • Find another point by choosing any x-value, like \( x = 1 \), yielding another point \( (1, -1) \).
  • Draw a straight line through these points.
Horizontal lines make graphing predictable and are easy once you grasp the behavior of such equations.

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Most popular questions from this chapter

The median hourly wage of an embalmer in Illinois in 2002 was \(\$ 17.82 .\) Seth's earnings, \(E\) (in dollars), for working \(t\) hr in a week can be defined by the function \(E(t)=17.82 t .\) (Source: www.igpa.uillinois.edu) a) How much does Seth earn if he works 30 hr? b) How much does Seth earn if he works 27 hr? c) How many hours would Seth have to work to make \(\$ 623.70 ?\) d) If Seth can work at most 40 hr per week, what is the domain of this function? e) Graph the function.

Write the slope-intercept form (if possible) of the equation of the line meeting the given conditions. parallel to \(y=0\) containing \(\left(-3,-\frac{5}{2}\right)\)

Jenelle earns \(\$ 7.50\) per hour at her part-time job. Her total earnings, \(E\) (in dollars), for working \(t\) hr can be defined by the function $$ E(t)=7.50 t $$ a) Find \(E(10),\) and explain what this means in the context of the problem. b) Find \(E(15),\) and explain what this means in the context of the problem. c) Find \(t\) so that \(E(t)=210,\) and explain what this means in the context of the problem.

Write the slope-intercept form (if possible) of the equation of the line meeting the given conditions. parallel to \(6 x+y=4\) containing \((-2,0)\)

since 1998 the population of Maine has been increasing by about 8700 people per year. In 2001 , the population of Maine was about \(1,284,000 .\) a) Write a linear equation to model this data. Let \(x\) represent the number of years after \(1998,\) and let \(y\) represent the population of Maine. b) Explain the meaning of the slope in the context of the problem. c) According to the equation, how many people lived in Maine in \(1998 ?\) in \(2002 ?\) d) If the current trend continues, in what year would the population be \(1,431,900 ?\)
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