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Chapter 4: Problem 121

In the following exercises, graph each equation. \(y=-\frac{1}{2} x+3\)

Short Answer

Expert verified
Graph the line passing through points (0, 3) and (2, 2).

Step by step solution

01

- Determine the slope and y-intercept

The given equation is in the slope-intercept form, which is \(y = mx + b\). Here, the slope \(m\) is \(-\frac{1}{2}\) and the y-intercept \(b\) is \(+3\).
02

- Plot the y-intercept

Locate the y-intercept on the graph. Since \(b = 3\), plot a point at \((0, 3)\). This is where the line crosses the y-axis.
03

- Use the slope to find another point

The slope \(m\) tells us how to move from the y-intercept to find another point on the line. A slope of \(-\frac{1}{2}\) means you go down 1 unit (negative direction) and right 2 units (positive direction). From the point \((0, 3)\), move down 1 unit to \((0, 2)\) and then right 2 units to \(2, 2)\). Plot this point \(2, 2)\).
04

- Draw the line

Draw a straight line through the points \((0, 3)\) and \((2, 2)\). This line represents the equation \(y = -\frac{1}{2}x + 3\).

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Key Concepts

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

Slope-Intercept Form
Understanding the slope-intercept form of a linear equation is crucial. It's presented as: \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept.
The y-intercept is where the line crosses the y-axis. The slope describes the line's steepness and direction. A positive slope means the line goes upwards as you move to the right, while a negative slope means it goes downwards.
  • In our example, the equation is \(y = -\frac{1}{2}x + 3\).
  • Here, the slope \(m\) is \(-\frac{1}{2}\) and the y-intercept \(b\) is 3.

Once we identify these, we can proceed to properly graph the equation.
Plotting Points
Plotting points on a graph accurately is the next step. First, we plot the y-intercept. In our case, \(b=3\), which means the line crosses the y-axis at point \((0, 3)\).
Now, using the slope \(-\frac{1}{2}\), we find another point on the line. The slope tells us to go down by 1 unit and to the right by 2 units.
  • Starting from \((0, 3)\), move down to point \((0, 2)\).
  • Then move right to point \((2, 2)\).
  • Plot this resulting point \(2, 2)\) on the graph.
These two points are enough to draw our line.
Drawing a Line
Now that we have our points plotted, it's time to draw the line. Use a ruler to ensure the line is straight and extends through both points:
  • Point \((0, 3)\)
  • Point \((2, 2)\)

Simply connect these two points with a straight line. This line represents our given linear equation \(y=-\frac{1}{2} x+3\).
Remember, extending the line beyond these points will help show the complete picture of how the equation behaves over a range of x-values. This method can be applied to any linear equation in slope-intercept form to visualize it on a graph!

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Most popular questions from this chapter

Determine whether each ordered pair is a solution to the inequality \(y>x-3\) : (a) (0,0) (b) (2,1) (c) (-1,-5) (d) (-6,-3) (e) (1,0)

Find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form. line \(x=-4,\) point (-3,-5)

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Graph the linear inequality \(x>-3 .\)

Find the equation of a line containing the given points. Write the equation in slope-intercept form. (-6,-3) and (-1,-3)
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