Graphing The Function H(x) = -3/2x + 1 A Step-by-Step Guide

In the realm of mathematics, visualizing functions through graphs is a fundamental concept. It allows us to understand the behavior and characteristics of a function in a clear and intuitive way. In this article, we will delve into the process of plotting the graph of the linear function h(x) = -3/2x + 1. This function, a quintessential example of a linear equation, embodies the relationship between two variables, x and h(x), and its graphical representation is a straight line. Understanding how to plot such functions is crucial for anyone studying algebra, calculus, or any field that involves mathematical modeling.

Understanding Linear Functions

To effectively plot the graph of h(x) = -3/2x + 1, it's essential to first grasp the concept of linear functions. A linear function is a mathematical function whose graph is a straight line. The general form of a linear function is y = mx + c, where m represents the slope of the line and c represents the y-intercept. The slope determines the steepness and direction of the line, while the y-intercept is the point where the line intersects the y-axis.

In our case, the function h(x) = -3/2x + 1 perfectly fits this form. Here, -3/2 is the slope, indicating that for every 1 unit increase in x, h(x) decreases by 3/2 units. The y-intercept is 1, meaning the line crosses the y-axis at the point (0, 1). Recognizing these components is the first step towards accurately plotting the graph.

Furthermore, understanding the domain and range of a linear function is important. For h(x) = -3/2x + 1, the domain is all real numbers, meaning x can take any value. Similarly, the range is also all real numbers, as the function can produce any value of h(x). This unrestricted nature of linear functions makes them versatile tools in mathematical modeling.

The negative slope in our function signifies that the line will descend from left to right. This is a crucial piece of information when we start plotting points, as it gives us a general idea of what the graph should look like. The slope also dictates the rate of change of the function; a steeper slope (larger absolute value) indicates a more rapid change in h(x) for a given change in x.

Identifying Key Features of h(x) = -3/2x + 1

Before we jump into plotting points, let's solidify our understanding of the key features of our function:

• Slope (m): -3/2. This tells us the line is decreasing (negative slope) and for every 2 units we move to the right on the x-axis, we move 3 units down on the y-axis.
• Y-intercept (c): 1. This is the point where the line crosses the y-axis, specifically at the coordinates (0, 1).
• Domain: All real numbers.
• Range: All real numbers.
• Direction: Decreasing from left to right due to the negative slope.

With these features in mind, we can proceed to plotting the graph with confidence, knowing what to expect and how to interpret the resulting line.

Steps to Plot the Graph

Plotting the graph of a linear function like h(x) = -3/2x + 1 is a straightforward process that involves a few key steps. These steps ensure accuracy and clarity in the graphical representation of the function. By following these steps diligently, anyone can create a precise graph that reflects the behavior of the linear function.

1. Create a Table of Values: The first step is to create a table of values. This involves choosing several x-values and calculating the corresponding h(x) values using the function h(x) = -3/2x + 1. Selecting a range of x-values, both positive and negative, will give a comprehensive view of the line's behavior. It's often beneficial to choose simple values like -2, -1, 0, 1, and 2 for ease of calculation. This table serves as the foundation for plotting the points on the graph.

   For example, if we choose x = -2, then h(-2) = -3/2 * (-2) + 1 = 3 + 1 = 4. So, one point on our graph will be (-2, 4). Similarly, we can calculate h(x) for other chosen x-values. The table of values provides the coordinates necessary for accurately placing points on the Cartesian plane. Remember, since it's a linear function, only two points are technically needed to draw the line, but using more points ensures accuracy and helps in identifying any calculation errors.

2. Plot the Points on the Cartesian Plane: Once you have your table of values, the next step is to plot these points on the Cartesian plane. The Cartesian plane is a two-dimensional coordinate system defined by two perpendicular lines, the x-axis (horizontal) and the y-axis (vertical). Each point in the plane is represented by an ordered pair (x, h(x)), where x is the horizontal coordinate and h(x) is the vertical coordinate.

   Carefully plot each point from your table onto the graph. Ensure that you accurately locate the x and h(x) coordinates for each point. For instance, if your table includes the point (-2, 4), find -2 on the x-axis and 4 on the y-axis, and mark the point where these two values intersect. Repeat this process for all the points in your table. This step is crucial as the accuracy of the graph depends on the correct placement of these points. The plotted points will start to reveal the linear pattern of the function, forming a straight line.

3. Draw a Straight Line: After plotting the points, the final step is to draw a straight line through them. Since h(x) = -3/2x + 1 is a linear function, the plotted points should align perfectly on a straight line. Use a ruler or a straight edge to draw a line that passes through all the plotted points. Extend the line beyond the plotted points to represent the function's behavior over its entire domain, which, in this case, is all real numbers.

   The line you draw is the graphical representation of the function h(x) = -3/2x + 1. It visually demonstrates the relationship between x and h(x), showing how h(x) changes as x changes. The direction and steepness of the line are determined by the slope of the function, and the point where the line crosses the y-axis is the y-intercept. Drawing the line completes the process of plotting the graph, providing a clear visual representation of the function.

Detailed Example of Plotting Points

Let's illustrate this process with a detailed example using our function, h(x) = -3/2x + 1. We'll create a table of values, plot the points, and then draw the line.

1. Create a Table of Values:

2. Plot the Points on the Cartesian Plane:

Now, we'll plot each of these points on the Cartesian plane. Locate the x and h(x) coordinates for each point and mark them accordingly.

• (-2, 4): Find -2 on the x-axis and 4 on the y-axis, and mark the point.
• (-1, 5/2): Find -1 on the x-axis and 2.5 (5/2) on the y-axis, and mark the point.
• (0, 1): This is the y-intercept. Mark the point where the line crosses the y-axis at 1.
• (1, -1/2): Find 1 on the x-axis and -0.5 (-1/2) on the y-axis, and mark the point.
• (2, -2): Find 2 on the x-axis and -2 on the y-axis, and mark the point.

3. Draw a Straight Line:

Finally, use a ruler to draw a straight line that passes through all the plotted points. Extend the line beyond the points to represent the function's behavior over its entire domain. The resulting line is the graph of h(x) = -3/2x + 1.

This detailed example clearly demonstrates how to plot the graph of a linear function. By following these steps carefully, you can accurately represent any linear function graphically.

Interpreting the Graph

Once the graph of h(x) = -3/2x + 1 is plotted, the next crucial step is to interpret the graph. The visual representation of the function provides valuable insights into its behavior and characteristics. Understanding how to read and interpret a graph is essential for gaining a deeper understanding of the function it represents.

The graph of h(x) = -3/2x + 1, being a straight line, immediately tells us that the function is linear. The slope of the line, which we know is -3/2, determines its steepness and direction. Since the slope is negative, the line slopes downwards from left to right, indicating that the function is decreasing. This means that as the value of x increases, the value of h(x) decreases. The absolute value of the slope, 3/2, tells us the rate of change; for every 2 units we move to the right on the x-axis, the line descends 3 units on the y-axis. This provides a quantitative measure of how quickly the function decreases.

The y-intercept, which is the point where the line crosses the y-axis, is another key feature to interpret. In our case, the y-intercept is 1, meaning the line crosses the y-axis at the point (0, 1). This point represents the value of h(x) when x is 0. The y-intercept is a fixed point on the graph and serves as a reference for understanding the function's position in the Cartesian plane.

The x-intercept, though not explicitly given in the function's equation, can also be determined from the graph. The x-intercept is the point where the line crosses the x-axis, meaning h(x) = 0. To find the x-intercept, we can either read it directly from the graph or set h(x) = 0 in the equation and solve for x. In this case, solving -3/2x + 1 = 0 gives x = 2/3, so the x-intercept is (2/3, 0). The x-intercept is another crucial point that helps define the line's position and provides insight into when the function's value is zero.

Analyzing the Behavior of h(x) = -3/2x + 1

By examining the graph, we can analyze the behavior of the function h(x) = -3/2x + 1 in detail:

• Decreasing Function: The negative slope clearly shows that the function is decreasing. As you move from left to right along the x-axis, the line goes downwards, indicating a decrease in h(x) values.
• Rate of Change: The slope of -3/2 indicates the rate at which h(x) decreases with respect to x. For every unit increase in x, h(x) decreases by 3/2 units.
• Y-intercept: The point (0, 1) shows the value of h(x) when x is 0. It is the starting point of the line on the y-axis.
• X-intercept: The point (2/3, 0) indicates the value of x for which h(x) is 0. This is where the line crosses the x-axis.
• Domain and Range: The graph extends infinitely in both directions, confirming that the domain and range of the function are all real numbers.

Furthermore, the graph allows us to visualize solutions to equations involving h(x). For example, if we want to find the value of x for which h(x) = -2, we can locate the point on the graph where the y-coordinate is -2 and read the corresponding x-coordinate. This graphical method provides a visual approach to solving equations and understanding the function's behavior across different values of x.

Conclusion

Plotting the graph of the linear function h(x) = -3/2x + 1 is a fundamental exercise in understanding linear equations and their graphical representations. By following the steps outlined – creating a table of values, plotting the points on the Cartesian plane, and drawing a straight line – we can accurately visualize the function's behavior. Interpreting the graph then allows us to gain insights into the function's slope, intercepts, and overall characteristics.

The slope of -3/2 tells us the function is decreasing, and the y-intercept of 1 shows where the line crosses the y-axis. The x-intercept, calculated as 2/3, gives us the point where the function equals zero. These features, along with the visual representation of the line, provide a comprehensive understanding of h(x) = -3/2x + 1.

Graphing linear functions is not just a mathematical exercise; it's a powerful tool for problem-solving and understanding relationships between variables. Whether you're studying algebra, calculus, or any other field that involves mathematical modeling, the ability to plot and interpret graphs is an invaluable skill. The process of graphing h(x) = -3/2x + 1 serves as a microcosm of the broader application of graphical analysis in mathematics and beyond.

In conclusion, the graph of h(x) = -3/2x + 1 is a straight line that visually represents the linear relationship between x and h(x). By mastering the techniques of plotting and interpreting such graphs, we equip ourselves with a powerful tool for mathematical analysis and problem-solving. The process illuminates the key characteristics of linear functions and provides a foundation for understanding more complex mathematical concepts and models.