Plotting Linear Function Y=(1/2)x+1 With Domain D{-8 -4 0 2 6}

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In mathematics, understanding how to represent linear functions graphically is a fundamental skill. This article will guide you through the process of plotting a linear function on a graph using a given domain. We'll consider the linear function y = (1/2)x + 1 and the domain D: {-8, -4, 0, 2, 6}. By plotting the ordered pairs corresponding to these values, we can visualize the linear relationship and gain a deeper understanding of its characteristics.

Understanding Linear Functions

Before we dive into plotting the points, let's briefly discuss what a linear function is. A linear function is a function that can be represented by a straight line on a graph. The general form of a linear function is y = mx + b, where:

  • y is the dependent variable
  • x is the independent variable
  • m is the slope of the line (the rate of change of y with respect to x)
  • b is the y-intercept (the point where the line crosses the y-axis)

In our example, the function is y = (1/2)x + 1. Here, the slope (m) is 1/2, and the y-intercept (b) is 1. This means that for every 2 units we move to the right on the x-axis, the line will move 1 unit up on the y-axis. The line will also intersect the y-axis at the point (0, 1).

Determining Ordered Pairs

To plot the linear function, we need to find the ordered pairs (x, y) that correspond to the values in the given domain D: {-8, -4, 0, 2, 6}. The domain represents the set of possible input values (x) for the function. To find the corresponding y values, we'll substitute each value from the domain into the equation y = (1/2)x + 1.

Let's calculate the ordered pairs:

  1. For x = -8:

    • y = (1/2)(-8) + 1 = -4 + 1 = -3
    • Ordered pair: (-8, -3)

  2. For x = -4:

    • y = (1/2)(-4) + 1 = -2 + 1 = -1
    • Ordered pair: (-4, -1)

  3. For x = 0:

    • y = (1/2)(0) + 1 = 0 + 1 = 1
    • Ordered pair: (0, 1)

  4. For x = 2:

    • y = (1/2)(2) + 1 = 1 + 1 = 2
    • Ordered pair: (2, 2)

  5. For x = 6:

    • y = (1/2)(6) + 1 = 3 + 1 = 4
    • Ordered pair: (6, 4)

So, we have the following ordered pairs: (-8, -3), (-4, -1), (0, 1), (2, 2), and (6, 4). These are the points we will plot on the graph to represent the linear function.

Plotting the Points on the Graph

Now that we have the ordered pairs, we can plot them on a coordinate plane. A coordinate plane has two axes: the horizontal x-axis and the vertical y-axis. The point where the axes intersect is called the origin, which has coordinates (0, 0). Each ordered pair (x, y) represents a point on the plane, where x is the horizontal distance from the origin and y is the vertical distance from the origin.

Let's plot each point:

  1. (-8, -3): Start at the origin, move 8 units to the left along the x-axis (since x is -8), and then move 3 units down along the y-axis (since y is -3). Mark this point.
  2. (-4, -1): Start at the origin, move 4 units to the left along the x-axis, and then move 1 unit down along the y-axis. Mark this point.
  3. (0, 1): Start at the origin, don't move along the x-axis (since x is 0), and move 1 unit up along the y-axis. This point lies on the y-axis. Mark this point.
  4. (2, 2): Start at the origin, move 2 units to the right along the x-axis, and then move 2 units up along the y-axis. Mark this point.
  5. (6, 4): Start at the origin, move 6 units to the right along the x-axis, and then move 4 units up along the y-axis. Mark this point.

Once you've plotted all the points, you should see that they form a straight line. This confirms that the function is indeed linear. Using a ruler or a straight edge, draw a line through these points. This line represents the graph of the linear function y = (1/2)x + 1 over the given domain.

Drawing Tools and Graphing Software

In a classroom setting or when using online resources, you may be asked to use drawing tools to plot the points and draw the line. These tools often include features for:

  • Plotting points: Clicking on the graph to place a point at the desired coordinates.
  • Drawing lines: Selecting two points and drawing a line that connects them.
  • Labeling axes: Adding labels to the x and y axes to indicate the scale.
  • Zooming and panning: Adjusting the view of the graph to focus on specific areas.

There are also various graphing software and online tools available that can help you plot linear functions. These tools often provide features like:

  • Inputting equations: Typing in the equation of the line and having the software automatically plot it.
  • Creating tables of values: Generating a table of x and y values for the function.
  • Finding intercepts: Identifying the points where the line crosses the x and y axes.
  • Analyzing the graph: Determining the slope and other properties of the line.

Some popular graphing tools include Desmos, GeoGebra, and graphing calculators. These tools can be invaluable for visualizing and analyzing linear functions and other mathematical concepts.

Analyzing the Graph

Once you've plotted the linear function, you can analyze its properties by examining the graph. Some key characteristics to look for include:

  • Slope: The slope of the line (m) indicates how steep the line is and whether it's increasing or decreasing. In our example, the slope is 1/2, which is positive, so the line slopes upward from left to right. The slope can be calculated by choosing any two points on the line (x1, y1) and (x2, y2) and using the formula:

    m = (y2 - y1) / (x2 - x1)

    For example, using the points (-8, -3) and (6, 4), we get:

    m = (4 - (-3)) / (6 - (-8)) = 7 / 14 = 1/2

  • Y-intercept: The y-intercept is the point where the line crosses the y-axis. In our example, the y-intercept is (0, 1). This is the point where x = 0. The y-intercept is also the value of b in the equation y = mx + b.

  • X-intercept: The x-intercept is the point where the line crosses the x-axis. This is the point where y = 0. To find the x-intercept, we can set y to 0 in the equation and solve for x:

    0 = (1/2)x + 1

    (1/2)x = -1

    x = -2

    So, the x-intercept is (-2, 0).

  • Domain and Range: The domain is the set of all possible input values (x), and the range is the set of all possible output values (y). In our example, the domain is given as D: {-8, -4, 0, 2, 6}. The range can be determined by looking at the y values of the plotted points, which are {-3, -1, 1, 2, 4}.

By analyzing these characteristics, you can gain a comprehensive understanding of the linear function and its graphical representation.

Conclusion

Plotting a linear function on a graph is a crucial skill in mathematics. By following the steps outlined in this article, you can confidently represent linear functions graphically. Remember to determine the ordered pairs, plot the points on the coordinate plane, and draw a straight line through them. Utilize drawing tools and graphing software to aid in the process. Finally, analyze the graph to understand its properties, such as slope, intercepts, domain, and range. With practice, you'll become proficient in visualizing and interpreting linear functions.

By mastering these concepts, you'll build a strong foundation for more advanced mathematical topics that rely on graphical representation and analysis.