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Chapter 4: Problem 65

In the following exercises, graph by plotting points. \(y=-x-3\)

Short Answer

Expert verified
Plot the points \((0, -3)\), \((1, -4)\), and \((-1, -2)\) on the graph and draw a line through them.

Step by step solution

01

Understand the Equation

The given equation is a linear equation in the form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In this case, \(m = -1\) and \(b = -3\).
02

Identify the Y-Intercept

The y-intercept is the point where the line crosses the y-axis. Since \(b = -3\), the y-intercept is at the point \((0, -3)\). Plot this point on the graph.
03

Calculate Another Point

To find another point, choose a value for \(x\). For example, let's select \(x = 1\). Substitute \(x = 1\) into the equation: \(y = -1(1) - 3 = -1 - 3 = -4\). So, another point is \((1, -4)\). Plot this point on the graph.
04

Calculate One More Point

Choose a different value for \(x\), let鈥檚 say \(x = -1\). Substitute \(x = -1\) into the equation: \(y = -1(-1) - 3 = 1 - 3 = -2\). Another point is \((-1, -2)\). Plot this point on the graph.
05

Draw the Line

After plotting the points \((0, -3)\), \((1, -4)\), and \((-1, -2)\), draw a straight line through these points. This line represents the graph of the equation \(y = -x - 3\).

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Key Concepts

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

plotting points
When graphing a linear equation, we begin by plotting points on a Cartesian coordinate system. To plot a point, we need a pair of coordinates (x, y). The x-value tells us how far to move horizontally from the origin (0,0), and the y-value tells us how far to move vertically. For instance, to plot the point (1, -4), we move 1 unit to the right and 4 units down. After doing this for a few points, you can see the shape of the line forming.

  • Start with the y-intercept, which is easier to identify.
  • Choose convenient values for x to find corresponding y values.
  • Plot at least two points correctly to establish the line's direction.
identifying y-intercept
The y-intercept of a line is the point where the line crosses the y-axis. This occurs when the x-value is zero. In the equation of the form \(y = mx + b\), the y-intercept is given by the constant term \(b\). For example, in the equation \(y = -x - 3\), the y-intercept is -3. Therefore, the line crosses the y-axis at the point (0, -3). This point is crucial because it provides a starting point for graphing the line.

  • Set x to 0 and solve for y to find the y-intercept.
  • This point is directly read from the equation.
calculating slope
The slope of a line indicates its steepness and direction. In the equation \(y = mx + b\), the slope, m, is the coefficient of x. It tells us how much y changes for a unit change in x. A positive slope means the line ascends, while a negative slope means it descends. To calculate the slope, we can use the formula: \( m = (y_2 - y_1) / (x_2 - x_1) \). In the equation \(y = -x - 3\), the slope is -1, meaning for every unit increase in x, y decreases by 1.

  • Understand that the slope is rise over run.
  • Use slope to plot additional points, starting from the y-intercept.
linear equation in slope-intercept form
A linear equation in slope-intercept form is written as \(y = mx + b\). This form is useful because it clearly shows both the slope (m) and the y-intercept (b). From the equation, we can quickly determine where to start graphing (the y-intercept) and how to proceed (using the slope). For \(y = -x - 3\), it reveals that the line has a slope of -1 and crosses the y-axis at -3. This makes graphing straightforward:
  • Identify y-intercept b.
  • Use slope m to find other points.
  • Draw the line through these points.
By following these steps, we can graph any linear equation efficiently.

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Most popular questions from this chapter

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