Finding X And Y Intercepts: Y = Log(7x + 3) - 2

by Alex Johnson 48 views

Understanding how to find the x and y intercepts of a given equation is a fundamental skill in algebra and calculus. In this article, we will dive deep into the process of finding these intercepts for the equation y = log(7x + 3) - 2. We'll break down the steps, explain the underlying concepts, and provide clear examples to ensure you grasp the method thoroughly. Whether you're a student tackling homework or someone brushing up on their math skills, this guide will help you master the art of intercept identification. Remember, finding intercepts is not just about plugging in numbers; it’s about understanding the behavior of functions and their graphical representations. Let's embark on this mathematical journey together!

Understanding Intercepts

Before we tackle our specific equation, let's define what x and y intercepts actually are. The x-intercept is the point where the graph of an equation crosses the x-axis. At this point, the y-coordinate is always zero. Conversely, the y-intercept is the point where the graph crosses the y-axis, and at this point, the x-coordinate is always zero. Finding these intercepts gives us key information about the graph of the equation, such as where it starts and where it potentially changes direction. The intercepts serve as anchors when sketching the graph, providing crucial reference points. Moreover, understanding intercepts is vital in real-world applications, from calculating break-even points in business to determining the initial conditions in physics problems. When you visualize the graph, the intercepts are the most immediate points you can identify, making them invaluable for a quick analysis.

Why are Intercepts Important?

Intercepts are more than just points on a graph; they provide crucial insights into the behavior and characteristics of a function. For instance, the x-intercepts, also known as roots or zeros, tell us where the function's value is zero. This information is essential in solving equations and understanding the function's solutions. In real-world contexts, x-intercepts can represent significant milestones, such as the point at which a company breaks even or the time when a population reaches a certain level. Similarly, the y-intercept reveals the value of the function when the input is zero, often representing an initial condition or starting point. For example, in a financial model, the y-intercept might represent the initial investment or the starting balance. The y-intercept is critical because it provides a baseline understanding of the function's state before any changes occur. Analyzing intercepts helps in sketching graphs accurately and interpreting results meaningfully, making them fundamental in mathematical analysis and problem-solving.

Finding the Y-Intercept

To find the y-intercept of the equation y = log(7x + 3) - 2, we need to set x = 0 and solve for y. This is because, as we discussed earlier, the y-intercept is the point where the graph intersects the y-axis, which occurs when x is zero. Plugging x = 0 into our equation, we get:

y = log(7(0) + 3) - 2

Simplifying the expression inside the logarithm:

y = log(3) - 2

Now, we need to evaluate log(3). Here, we assume the logarithm is base 10 unless otherwise specified. Using a calculator, we find that log(3) ≈ 0.4771. Therefore:

y ≈ 0.4771 - 2

y ≈ -1.5229

So, the y-intercept is approximately -1.5229. This tells us that the graph of the equation crosses the y-axis at the point (0, -1.5229). The y-intercept gives us a starting point on the y-axis and helps in visualizing the function’s behavior. It's a key anchor point when sketching the graph. Understanding how to find the y-intercept provides a solid foundation for analyzing more complex functions and their graphs.

Step-by-Step Guide to Finding the Y-Intercept

To recap, here's a step-by-step guide on finding the y-intercept, ensuring clarity and ease of understanding:

  1. Set x = 0: In the given equation, replace every instance of x with 0. This is the fundamental step as the y-intercept is defined as the point where the graph crosses the y-axis, which occurs when x is zero.
  2. Substitute: For the equation y = log(7x + 3) - 2, substituting x = 0 gives us y = log(7(0) + 3) - 2.
  3. Simplify the Expression: Simplify the equation by performing the arithmetic operations. In this case, 7(0) equals 0, so the equation becomes y = log(0 + 3) - 2, which simplifies to y = log(3) - 2.
  4. Evaluate the Logarithm: Use a calculator to find the value of log(3). Assuming the logarithm is base 10, log(3) ≈ 0.4771. Note that if the logarithm has a different base, you would use the appropriate logarithmic function or change-of-base formula.
  5. Final Calculation: Subtract 2 from the logarithmic value: y ≈ 0.4771 - 2, which gives y ≈ -1.5229.
  6. State the Y-Intercept: The y-intercept is the point where the graph crosses the y-axis, represented as (0, y). Therefore, for this equation, the y-intercept is approximately (0, -1.5229).

By following these steps, you can confidently find the y-intercept of any equation, providing a crucial reference point for graphing and analysis.

Finding the X-Intercept

Finding the x-intercept of the equation y = log(7x + 3) - 2 involves setting y = 0 and solving for x. The x-intercept is the point where the graph intersects the x-axis, and this always occurs when y is zero. So, we start with:

0 = log(7x + 3) - 2

First, we add 2 to both sides of the equation to isolate the logarithm:

2 = log(7x + 3)

Since we're dealing with a base 10 logarithm, we can rewrite the equation in exponential form. The equation log(a) = b is equivalent to 10^b = a. Applying this to our equation, we get:

10^2 = 7x + 3

Simplifying the left side:

100 = 7x + 3

Next, subtract 3 from both sides:

97 = 7x

Finally, divide both sides by 7 to solve for x:

x = 97 / 7

x ≈ 13.8571

Therefore, the x-intercept is approximately 13.8571. This means the graph of the equation crosses the x-axis at the point approximately (13.8571, 0). The x-intercept is an essential point that helps us understand the zeros or roots of the function, providing critical information about where the function’s value is zero.

Detailed Steps to Solve for X-Intercept

Let's break down the process of finding the x-intercept into a series of clear, actionable steps to ensure a comprehensive understanding:

  1. Set y = 0: Replace y with 0 in the equation. This is because the x-intercept is the point where the graph crosses the x-axis, which occurs when the y-coordinate is zero. For the equation y = log(7x + 3) - 2, this gives us 0 = log(7x + 3) - 2.
  2. Isolate the Logarithm: Add 2 to both sides of the equation to isolate the logarithmic term. This step is crucial for converting the equation into exponential form. Adding 2 to both sides, we get 2 = log(7x + 3).
  3. Convert to Exponential Form: Rewrite the logarithmic equation in exponential form. If the logarithm is base 10, as is the assumption when no base is specified, the equation log(a) = b is equivalent to 10^b = a. Applying this, 2 = log(7x + 3) becomes 10^2 = 7x + 3.
  4. Simplify and Solve for x: Simplify the exponential term: 10^2 equals 100, so the equation is 100 = 7x + 3. Next, subtract 3 from both sides to get 97 = 7x. Finally, divide both sides by 7 to solve for x: x = 97 / 7.
  5. Approximate the Value: Divide 97 by 7 to get the decimal approximation: x ≈ 13.8571. This gives us a precise location of the x-intercept on the x-axis.
  6. State the X-Intercept: The x-intercept is represented as a point (x, 0). Therefore, the x-intercept for this equation is approximately (13.8571, 0).

By meticulously following these steps, you can confidently determine the x-intercept of any logarithmic equation, which is an essential skill in graphing and analyzing functions.

Summary of Intercepts for y = log(7x + 3) - 2

To summarize, we have found the y-intercept and the x-intercept for the equation y = log(7x + 3) - 2. This provides us with two crucial points that can help us better understand and graph the function. Let’s reiterate our findings:

  • Y-Intercept: We found the y-intercept by setting x = 0 and solving for y. This gave us the point approximately (0, -1.5229). The y-intercept indicates where the graph crosses the y-axis and provides a starting point for visualizing the function’s behavior.
  • X-Intercept: We found the x-intercept by setting y = 0 and solving for x. This gave us the point approximately (13.8571, 0). The x-intercept is where the graph crosses the x-axis and represents the root or zero of the function, a key value in mathematical analysis.

Having both the x and y intercepts allows us to sketch a more accurate graph of the function. These intercepts act as anchor points, helping us understand the overall shape and position of the graph in the coordinate plane. Additionally, knowing the intercepts is vital for practical applications, such as understanding the initial conditions and break-even points in real-world scenarios. These two points are fundamental in analyzing and interpreting the function y = log(7x + 3) - 2.

Conclusion

In this comprehensive guide, we have explored how to find the x and y intercepts of the equation y = log(7x + 3) - 2. We began by defining what intercepts are and why they are crucial in understanding the behavior of a function. We then walked through the detailed steps of finding the y-intercept by setting x = 0 and solving for y, and the x-intercept by setting y = 0 and solving for x. By summarizing our findings, we highlighted the importance of these intercepts in graphing and analyzing functions.

The ability to find intercepts is a fundamental skill in mathematics, with wide-ranging applications in various fields. Whether you are a student learning algebra or someone applying mathematical concepts in your profession, understanding how to find x and y intercepts is invaluable. These points provide essential information about the function’s graph and behavior, enabling you to solve equations, sketch graphs, and interpret results with confidence.

For further learning and practice on logarithmic functions and their graphs, consider exploring resources like Khan Academy's section on Logarithmic Functions, which offers detailed explanations and practice exercises.