Graph The Linear Function:a. Y = 7 2 X − 2 Y = \frac{7}{2}x - 2 Y = 2 7 ​ X − 2

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Introduction

Graphing linear functions is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. A linear function is a polynomial function of degree one, which means it can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. In this article, we will focus on graphing the linear function y = (7/2)x - 2.

Understanding the Linear Function

Before we dive into graphing the linear function, let's understand the components of the function. The function y = (7/2)x - 2 has two main components: the slope (m) and the y-intercept (b).

  • Slope (m): The slope of a linear function represents the rate of change of the function with respect to the input variable (x). In this case, the slope is (7/2), which means that for every unit increase in x, the value of y increases by (7/2) units.
  • Y-intercept (b): The y-intercept of a linear function represents the point where the function intersects the y-axis. In this case, the y-intercept is -2, which means that the function intersects the y-axis at the point (0, -2).

Graphing the Linear Function

To graph the linear function y = (7/2)x - 2, we need to follow these steps:

  1. Plot the y-intercept: The first step is to plot the y-intercept, which is the point (0, -2). This point represents the starting point of the function.
  2. Determine the slope: The next step is to determine the slope of the function, which is (7/2). This means that for every unit increase in x, the value of y increases by (7/2) units.
  3. Plot additional points: To graph the function, we need to plot additional points that satisfy the equation y = (7/2)x - 2. We can do this by substituting different values of x into the equation and plotting the corresponding values of y.
  4. Draw the line: Once we have plotted several points, we can draw a line that passes through these points. This line represents the graph of the linear function.

Example

Let's graph the linear function y = (7/2)x - 2 using the steps outlined above.

  1. Plot the y-intercept: The y-intercept is the point (0, -2), which we can plot on the coordinate plane.
  2. Determine the slope: The slope of the function is (7/2), which means that for every unit increase in x, the value of y increases by (7/2) units.
  3. Plot additional points: We can plot additional points by substituting different values of x into the equation y = (7/2)x - 2. For example, if we substitute x = 1, we get y = (7/2)(1) - 2 = (7/2) - 2 = -1/2. We can plot the point (1, -1/2) on the coordinate plane.
  4. Draw the line: Once we have plotted several points, we can draw a line that passes through these points. This line represents the graph of the linear function.

Graph of the Linear Function

The graph of the linear function y = (7/2)x - 2 is a straight line that passes through the points (0, -2) and (1, -1/2). The line has a slope of (7/2) and a y-intercept of -2.

Properties of the Graph

The graph of the linear function y = (7/2)x - 2 has several properties that are worth noting:

  • Straight line: The graph is a straight line that passes through the points (0, -2) and (1, -1/2).
  • Slope: The slope of the line is (7/2), which means that for every unit increase in x, the value of y increases by (7/2) units.
  • Y-intercept: The y-intercept of the line is -2, which means that the line intersects the y-axis at the point (0, -2).

Real-World Applications

The graph of the linear function y = (7/2)x - 2 has several real-world applications, including:

  • Physics: The graph can be used to model the motion of an object that is moving at a constant rate.
  • Engineering: The graph can be used to design and optimize systems that involve linear relationships between variables.
  • Economics: The graph can be used to model the relationship between two economic variables, such as supply and demand.

Conclusion

Q&A: Graphing Linear Functions

Q: What is a linear function?

A: A linear function is a polynomial function of degree one, which means it can be written in the form of y = mx + b, where m is the slope and b is the y-intercept.

Q: What is the slope of a linear function?

A: The slope of a linear function represents the rate of change of the function with respect to the input variable (x). In the case of the function y = (7/2)x - 2, the slope is (7/2).

Q: What is the y-intercept of a linear function?

A: The y-intercept of a linear function represents the point where the function intersects the y-axis. In the case of the function y = (7/2)x - 2, the y-intercept is -2.

Q: How do I graph a linear function?

A: To graph a linear function, you need to follow these steps:

  1. Plot the y-intercept: The first step is to plot the y-intercept, which is the point where the function intersects the y-axis.
  2. Determine the slope: The next step is to determine the slope of the function, which represents the rate of change of the function with respect to the input variable (x).
  3. Plot additional points: To graph the function, you need to plot additional points that satisfy the equation of the function.
  4. Draw the line: Once you have plotted several points, you can draw a line that passes through these points. This line represents the graph of the linear function.

Q: What are some real-world applications of graphing linear functions?

A: Graphing linear functions has several real-world applications, including:

  • Physics: Graphing linear functions can be used to model the motion of an object that is moving at a constant rate.
  • Engineering: Graphing linear functions can be used to design and optimize systems that involve linear relationships between variables.
  • Economics: Graphing linear functions can be used to model the relationship between two economic variables, such as supply and demand.

Q: How do I determine the equation of a linear function from its graph?

A: To determine the equation of a linear function from its graph, you need to follow these steps:

  1. Identify the slope: The slope of the line represents the rate of change of the function with respect to the input variable (x).
  2. Identify the y-intercept: The y-intercept of the line represents the point where the function intersects the y-axis.
  3. Write the equation: Once you have identified the slope and y-intercept, you can write the equation of the linear function in the form of y = mx + b.

Q: What are some common mistakes to avoid when graphing linear functions?

A: Some common mistakes to avoid when graphing linear functions include:

  • Incorrectly identifying the slope: Make sure to accurately identify the slope of the line, as it represents the rate of change of the function with respect to the input variable (x).
  • Incorrectly identifying the y-intercept: Make sure to accurately identify the y-intercept of the line, as it represents the point where the function intersects the y-axis.
  • Not plotting enough points: Make sure to plot enough points to accurately represent the graph of the linear function.

Q: How do I use graphing linear functions in real-world applications?

A: Graphing linear functions can be used in a variety of real-world applications, including:

  • Modeling motion: Graphing linear functions can be used to model the motion of an object that is moving at a constant rate.
  • Designing systems: Graphing linear functions can be used to design and optimize systems that involve linear relationships between variables.
  • Analyzing data: Graphing linear functions can be used to analyze data and identify trends and patterns.

Conclusion

In conclusion, graphing linear functions is a fundamental concept in mathematics that has several real-world applications. By understanding the properties of linear functions and how to graph them, you can apply this knowledge to a variety of fields, including physics, engineering, and economics.