Finding The Slope A Step-by-Step Guide For F(x)=(3/2)x-4

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In the realm of mathematics, linear equations stand as fundamental tools for modeling relationships between variables. These equations, characterized by their straight-line graphs, possess a unique feature known as the slope, a measure of the line's steepness and direction. Understanding the slope is crucial for interpreting and analyzing linear relationships in various fields, from physics and engineering to economics and computer science. This comprehensive guide delves into the concept of slope, exploring its definition, calculation, and significance in the context of linear equations. We will dissect the equation f(x)=32x−4f(x) = \frac{3}{2}x - 4 to pinpoint its slope, shedding light on the underlying principles that govern linear functions.

Deciphering the Slope-Intercept Form

To effectively determine the slope of a linear equation, it's essential to grasp the concept of the slope-intercept form. This form, represented as y=mx+by = mx + b, provides a clear and concise representation of a line's characteristics. In this equation, 'm' denotes the slope, while 'b' signifies the y-intercept, the point where the line intersects the vertical axis. The slope (m) dictates the rate at which the line rises or falls, while the y-intercept (b) anchors the line's position on the coordinate plane. By expressing a linear equation in slope-intercept form, we can readily identify its slope and y-intercept, facilitating a deeper understanding of its behavior and graphical representation. The beauty of the slope-intercept form lies in its ability to distill the essence of a linear equation into two key parameters, the slope and the y-intercept, which collectively define the line's position and orientation in the coordinate plane. Recognizing this form is the first step towards unlocking the secrets hidden within linear equations.

Applying the Slope-Intercept Form to Our Equation

Now, let's apply our understanding of the slope-intercept form to the equation at hand, f(x)=32x−4f(x) = \frac{3}{2}x - 4. Recognizing that f(x)f(x) is simply another way of representing 'y', we can readily see that this equation is already presented in the slope-intercept form. By comparing it to the general form y=mx+by = mx + b, we can directly identify the slope and y-intercept. The coefficient of the 'x' term, which is 32\frac{3}{2}, corresponds to 'm', the slope. The constant term, -4, corresponds to 'b', the y-intercept. This direct correspondence allows us to confidently state that the slope of the line represented by the equation f(x)=32x−4f(x) = \frac{3}{2}x - 4 is indeed 32\frac{3}{2}. This straightforward approach highlights the power of the slope-intercept form in readily extracting key information about a linear equation. It transforms the task of finding the slope from a potentially complex calculation to a simple observation, making it an invaluable tool for anyone working with linear functions.

The Significance of Slope: A Visual and Practical Perspective

The slope, represented by the value 32\frac{3}{2} in our equation, carries significant meaning both visually and practically. Visually, the slope dictates the steepness and direction of the line. A positive slope, as we have here, indicates that the line rises from left to right. The magnitude of the slope, 32\frac{3}{2}, tells us that for every 2 units we move horizontally along the x-axis, the line rises 3 units vertically along the y-axis. This rise-over-run interpretation provides a tangible understanding of the line's inclination. Practically, the slope represents the rate of change of the dependent variable (y) with respect to the independent variable (x). In real-world scenarios, this rate of change can represent various quantities, such as the speed of an object, the growth rate of a population, or the cost per unit of a product. Understanding the slope allows us to predict how the dependent variable will change in response to changes in the independent variable, making it a powerful tool for analysis and decision-making. For instance, in a cost function represented by a linear equation, the slope would represent the marginal cost, the cost of producing one additional unit.

Evaluating the Answer Choices: Identifying the Correct Slope

Now, let's turn our attention to the answer choices provided and evaluate them in light of our understanding of the slope-intercept form and the equation f(x)=32x−4f(x) = \frac{3}{2}x - 4.

  • A. -4: This option represents the y-intercept of the equation, not the slope. While the y-intercept is an important characteristic of the line, it does not define the steepness or direction. Therefore, option A is incorrect.
  • B. −32-\frac{3}{2}: This option represents the negative of the slope. While a negative slope would indicate a line that falls from left to right, the equation clearly shows a positive coefficient for the 'x' term, indicating a rising line. Thus, option B is incorrect.
  • C. 32\frac{3}{2}: This option perfectly matches the coefficient of the 'x' term in the equation, which we identified as the slope. This indicates that for every 2 units moved horizontally, the line rises 3 units vertically. Therefore, option C is the correct answer.
  • D. 4: This option is a positive number, but it does not correspond to the coefficient of the 'x' term in the equation. Therefore, option D is incorrect.

By systematically evaluating each option and comparing it to our understanding of the equation and the concept of slope, we can confidently conclude that option C, 32\frac{3}{2}, is the correct answer.

Conclusion: Mastering the Slope of Linear Equations

In conclusion, determining the slope of a linear equation is a fundamental skill in mathematics with far-reaching applications. By understanding the slope-intercept form, y=mx+by = mx + b, we can readily identify the slope (m) and y-intercept (b) of a line, gaining valuable insights into its behavior and graphical representation. In the specific case of the equation f(x)=32x−4f(x) = \frac{3}{2}x - 4, the slope is 32\frac{3}{2}, indicating a line that rises from left to right, with a steepness of 3 units vertically for every 2 units horizontally. This comprehensive guide has not only provided a step-by-step approach to solving the given problem but has also delved into the underlying concepts and significance of slope in the broader context of linear equations. By mastering these principles, you can confidently tackle a wide range of problems involving linear relationships and their applications in various fields. The ability to decipher the slope of a line is more than just a mathematical skill; it's a key to unlocking a deeper understanding of the world around us, where linear relationships abound, from the simple motion of objects to the complex dynamics of economic systems. Embracing this concept empowers you to analyze, interpret, and predict outcomes in a variety of real-world scenarios. Remember, the slope is not just a number; it's a powerful descriptor of change and a fundamental building block in the language of mathematics.

What is the slope of the line represented by the equation f(x)=32x−4f(x)=\frac{3}{2} x-4 ?

A. -4 B. −32-\frac{3}{2} C. 32\frac{3}{2} D. 4

Correct Answer: C. 32\frac{3}{2}