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Chapter 4: Problem 43

Graph the line containing the given point and with the given slope. $$(0,-3) ; m=-1$$

Short Answer

Expert verified
The short answer is: Using the point-slope form with the point \((0, -3)\) and slope \(-1\), we find the equation of the line to be \(y = -x - 3\). To graph the line, choose three x-values, calculate their corresponding y-values, and plot the points on a coordinate plane. The table of values is: \[ \begin{array}{c|c} x & y \\ \hline -1 & -2 \\ 0 & -3 \\ 1 & -4 \end{array} \] Plot the points \((-1, -2)\), \((0, -3)\), and \((1, -4)\) and connect them with a straight line to represent the graph of the given line.

Step by step solution

01

Use point-slope form to find the equation of the line

Substitute the given point \((0, -3)\) and the slope \(-1\) into the point-slope form of a linear equation: \(y - (-3) = -1(x - 0)\).
02

Simplify the equation

After substitution, the equation is \(y + 3 = -1x\). To rewrite it in the slope-intercept form, solve for y: 1. Subtract 3 from both sides: \(y = -1x - 3\). Now, we have the equation of the line in the slope-intercept form: \(y = -x - 3\).
03

Create a table of values for x and y

Using the equation \(y = -x - 3\), choose at least three values for x (for example, -1, 0, and 1) and calculate the corresponding values for y: \[ \begin{array}{c|c} x & y \\ \hline -1 & -2 \\ 0 & -3 \\ 1 & -4 \end{array} \]
04

Plot the points on a coordinate plane

Plot the points from the table of values, \((-1, -2)\), \((0, -3)\), and \((1, -4)\), on a coordinate plane. Connect the points with a straight line, which represents the graph of the given line. Now you have successfully graphed the line containing the point \((0, -3)\) with a slope of \(-1\).

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Key Concepts

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

Point-Slope Form
Point-slope form is a way to write the equation of a line when you know one point on the line and the slope. The formula is: \( y - y_1 = m(x - x_1) \) where \( m \) is the slope, and \((x_1, y_1)\) is a point on the line. This form is particularly useful when you are given a specific point and a slope, allowing you to quickly find the equation of the line. Remember, the goal is to substitute your given point and slope into this formula, then simplify to find an equation that's easier to work with.
Slope-Intercept Form
The slope-intercept form is another way to represent the equation of a line. It is written as: \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept, or where the line crosses the y-axis. This form is popular because it gives you immediate information about the slope and starting point of the line. Once you transform an equation into slope-intercept form, graphing it becomes simple: you start by plotting the y-intercept on the y-axis, then use the slope to determine the direction and steepness of the line.
Coordinate Plane
The coordinate plane is a two-dimensional space where we graph points, lines, and curves. It consists of two axes that intersect at a right angle: - The horizontal axis, called the x-axis.- The vertical axis, called the y-axis.Each point in this plane is represented by a pair of numbers \((x, y)\), which show its position relative to the axes. The point where the axes cross is known as the origin. Understanding how to plot points on a coordinate plane is fundamental when graphing equations, as it allows you to visualize and locate the solutions.
Table of Values
Creating a table of values is a practical way to prepare for drawing a graph. It involves selecting various values for \( x \), then calculating the corresponding \( y \) values using the equation. This approach helps in finding specific points that lie on the line. For example, if your equation is \( y = -x - 3 \), you can choose values such as -1, 0, and 1 for \( x \) and compute the \( y \) values. This results in a set of points (e.g., \((-1, -2)\), \((0, -3)\), \((1, -4)\)) that you can plot on the coordinate plane. By connecting these points with a line, you graph the equation effectively, verifying your calculations.

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Most popular questions from this chapter

Write an equation of the line perpendicular to the given line and containing the given point. Write the answer in slope-intercept form or in standard form, as indicated. \(y=x ;(10,-4) ;\) slope-intercept form

Lines \(L_{1}\) and \(L_{2}\) contain the given points. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. $$\begin{aligned}&L_{1}:(-1,-7),(2,8)\\\&L_{2}:(10,2),(0,4)\end{aligned}$$

Graph each function by finding the \(x\) - and \(y\) -intercepts and one other point. $$f(x)=-\frac{1}{2} x+2$$

Write the slope-intercept form of the equation of the line, if possible, given the following information. \(m=1\) and \(y\) -intercept \((0,0)\)

Graph each function by finding the \(x\) - and \(y\) -intercepts and one other point. $$g(x)=3 x+3$$
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