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Chapter 0: Problem 44

Finding an Equation of a Line In Exercises \(39-46,\) find an equation of the line that passes through the points. Then sketch the line. $$ (1,-2),(3,-2) $$

Short Answer

Expert verified
The equation of the line passing through the points (1,-2) and (3,-2) is \(y = -2\).

Step by step solution

01

Calculate the Slope

The slope of the line passing through \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as \(m = \frac{y_2 鈥� y_1}{x_2 鈥� x_1}\). Substituting \(x_1 = 1, y_1 = -2, x_2 = 3, y_2 = -2\), we get \(m = 0\), because the y-coordinates are the same, meaning the line is horizontal.
02

Find the Y-Intercept

The y-intercept 'b' is the point at which the line crosses the y-axis. Because we have a horizontal line, each point on the line has the same y-coordinate. So the y-intercept is \(b = -2\).
03

Form the Equation of the Line

A linear equation can generally be written as \(y = mx + b\), where m is the slope and b is the y-intercept. With \(m = 0\), and \(b = -2\), the equation of the line becomes \(y = -2\).
04

Sketch the Line

To sketch the line on the graph, draw a horizontal line that intersects the y-axis at -2.

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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 of a line is a measure of its steepness and direction. It is represented by the letter "m" in the equation of a line. Slope is calculated as the change in y-coordinates over the change in x-coordinates between two points on a line. Mathematically, slope is expressed as \( m = \frac{y_2 - y_1}{x_2 - x_1} \). A positive slope indicates a line that rises from left to right, while a negative slope descends.
If the slope is zero, the line is horizontal, which means there is no vertical change as you move along the line. In our exercise, the calculated slope is zero because the y-coordinates of both points are the same. This perfectly demonstrates a horizontal line.
  • Zero slope: the line is horizontal.
  • Positive slope: the line rises.
  • Negative slope: the line falls.
The Importance of the Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. It is represented by the letter "b" in the equation of a line. The y-intercept is crucial because it provides a starting point for the line graph.
In a horizontal line, like the one from our exercise, the y-intercept is equal to the common y-coordinate of any point on the line. Thus, for the equation of a horizontal line, the value of "b" is constant.
In our example, given that all the y-values on the line are -2, the y-intercept is simply \( b = -2 \). Therefore, the line crosses the y-axis at this constant point.
Formulating a Linear Equation
A linear equation can describe a straight line on a graph. It is generally written in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
A linear equation combines the concepts of slope and y-intercept to define the line's position and angle on a graph.
For a line with a zero slope, like our exercise example, the term involving \( x \) cancels out, simplifying the equation to just \( y = b \). In this case, \( y = -2 \), indicating that the line is horizontal and remains constant along \( y = -2 \).
  • Linear equation format: \( y = mx + b \)
  • No slope means a simpler equation: \( y = b \)
The Art of Graphing Lines
Graphing lines involves drawing the straight path defined by the linear equation on a coordinate plane. Each line can be visually understood through its slope and intercepts.
For our horizontal line \( y = -2 \), the graphing process is straightforward. Since the slope is zero, the line remains untouched by any change in x-coordinates.
To plot this, start at the y-intercept, which is -2 on the y-axis, and draw a horizontal line parallel to the x-axis. This way of graphing visually confirms the line鈥檚 equation and slope properties. Always check your plotted points to ensure accuracy.

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Most popular questions from this chapter

Using the Vertical Line Test In Exercises \(43-46,\) use the Vertical Line Test to determine whether \(y\) is a function of \(x .\) To print an enlarged copy of the graph, go to MathGraphs.com. $$ y=\left\\{\begin{array}{cc}{x+1,} & {x \leq 0} \\ {-x+2,} & {x>0}\end{array}\right. $$

Finding the Domain and Range of a Function In Exercises \(11-22,\) find the domain and range of the function. $$ f(x)=x^{3} $$

Finding the Domain and Range of a Piecewise Function In Exercises \(29-32,\) evaluate the function as indicated. Determine its domain and range. $$ f(x)=\left\\{\begin{array}{ll}{2 x+1,} & {x<0} \\ {2 x+2,} & {x \geq 0}\end{array}\right. $$ $$ \begin{array}{llll}{\text { (a) } f(-1)} & {\text { (b) } f(0)} & {\text { (c) } f(2)} & {\text { (d) } f\left(t^{2}+1\right)}\end{array} $$

Boiling Temperature The table shows the temperatures \(T\) (in degrees Fahrenheit) at which water boils at selected pressures \(p\) (in pounds per square inch). (Source: Standard Handbook for Mechanical Engineers) $$ \begin{array}{|c|c|c|c|c|}\hline p & {5} & {10} & {14.696(1 \text { atmosphere) }} & {20} \\ \hline T & {162.24^{\circ}} & {193.21^{\circ}} & {212.00^{\circ}} & {227.96^{\circ}} \\ \hline\end{array} $$ $$ \begin{array}{|c|c|c|c|c|c|}\hline p & {30} & {40} & {60} & {80} & {100} \\\ \hline T & {250.33^{\circ}} & {267.25^{\circ}} & {292.71^{\circ}} & {312.03^{\circ}} & {327.81^{\circ}} \\ \hline\end{array} $$ (a) Use the regression capabilities of a graphing utility to find a cubic model for the data. (b) Use a graphing utility to plot the data and graph the model. (c) Use the graph to estimate the pressure required for the boiling point of water to exceed \(300^{\circ} \mathrm{F}\) . (d) Explain why the model would not be accurate for pressures exceeding 100 pounds per square inch.

Choosing a Job As a salesperson, you receive a monthly salary of \(\$ 2000\) , plus a commission of 7\(\%\) of sales. You are offered a new job at \(\$ 2300\) per month, plus a commission of 5\(\%\) of sales. (a) Write linear equations for your monthly wage \(W\) in terms of your monthly sales \(s\) for your current job and your job offer. (b) Use a graphing utility to graph each equation and find the point of intersection. What does it signify? (c) You think you can sell \(\$ 20,000\) worth of a product per month. Should you change jobs? Explain.
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