Analyzing Intercepts Of F(x) = (x+3)(x+5): Student Claims

Let's dive into the fascinating world of quadratic functions and explore the claims made by students about the intercepts of the function f(x) = (x+3)(x+5). This exercise isn't just about finding the right answers; it's about understanding the underlying concepts and how different interpretations can lead to different conclusions. We'll break down each student's claim, analyze their reasoning, and determine the accuracy of their statements. By the end of this discussion, you'll have a clearer grasp of how to identify and interpret intercepts of quadratic functions.

Understanding the Function f(x) = (x+3)(x+5)

Before we jump into the students' claims, let's make sure we're all on the same page regarding the function itself. f(x) = (x+3)(x+5) is a quadratic function, which means it forms a parabola when graphed. The factored form of the equation gives us valuable clues about the function's behavior, particularly its x-intercepts. Remember, the x-intercepts are the points where the parabola crosses the x-axis, meaning f(x) = 0. To find these points, we set each factor equal to zero and solve for x.

Expanding the function, we get f(x) = x² + 8x + 15. This standard form of the quadratic equation helps us identify the y-intercept. The y-intercept is the point where the parabola crosses the y-axis, which occurs when x = 0. By substituting x = 0 into the equation, we can easily find the y-coordinate of the y-intercept. Understanding these basic concepts is crucial for evaluating the claims made by the students.

Now, let’s address each student's claim, carefully analyzing their reasoning and identifying any potential errors. Remember, mathematics is not just about getting the right answer; it’s about the process of logical thinking and problem-solving. So, let's put on our thinking caps and embark on this mathematical journey together!

Jeremiah's Claim: The y-intercept is at (15, 0)

Jeremiah claims that the y-intercept of the function f(x) = (x+3)(x+5) is at (15, 0). To evaluate this claim, we need to remember what the y-intercept represents. The y-intercept is the point where the graph of the function intersects the y-axis. This occurs when the x-coordinate is equal to 0. Therefore, to find the y-intercept, we need to substitute x = 0 into the function and calculate the corresponding y-value. Let's do that now:

f(0) = (0 + 3)(0 + 5) = (3)(5) = 15

This calculation shows that when x = 0, f(x) = 15. This means the y-intercept occurs at the point (0, 15), not (15, 0) as Jeremiah claimed. Jeremiah seems to have confused the x and y coordinates. He likely correctly calculated the y-value but incorrectly placed it as the x-coordinate in the intercept. This is a common mistake, and it highlights the importance of understanding the definition of intercepts and how they relate to the coordinate plane.

Therefore, Jeremiah's claim is incorrect. The y-intercept is at (0, 15), not (15, 0). This underscores the need to be meticulous with notation and to double-check the meaning of each coordinate within a point. Now, let's move on to the next student's claim and see if we can uncover any further mathematical insights.

Lindsay's Claim: The x-intercepts are at (-3, 0) and (5, 0)

Lindsay asserts that the x-intercepts of the function f(x) = (x+3)(x+5) are at (-3, 0) and (5, 0). To determine the validity of Lindsay's claim, we must recall the definition of x-intercepts. X-intercepts are the points where the graph of the function intersects the x-axis. At these points, the y-value, or f(x), is equal to 0. The factored form of the function, f(x) = (x+3)(x+5), is particularly helpful in finding the x-intercepts.

To find the x-intercepts, we set f(x) = 0 and solve for x:

(x + 3)(x + 5) = 0

This equation is satisfied when either (x + 3) = 0 or (x + 5) = 0. Solving these equations, we get:

x + 3 = 0 => x = -3 x + 5 = 0 => x = -5

Therefore, the x-intercepts occur at x = -3 and x = -5. This means the points where the graph intersects the x-axis are (-3, 0) and (-5, 0). Lindsay correctly identified (-3, 0) as an x-intercept, but she incorrectly stated (5, 0) as the other x-intercept. The correct x-intercept is (-5, 0). This error likely stems from a sign mistake when solving the equation (x + 5) = 0.

Lindsay's claim is partially correct. She accurately identified one x-intercept but made an error in determining the other. This highlights the importance of carefully solving equations and double-checking solutions, especially when dealing with negative signs. Let's move on to the next student and see what their claim reveals about their understanding of the function.

Stephen's Claim:

Stephen's claim is missing. Without a specific claim from Stephen, we cannot analyze his understanding of the function f(x) = (x+3)(x+5) or its intercepts. This situation underscores the importance of clear and complete communication in mathematics. A claim needs to be articulated precisely to be evaluated effectively.

If Stephen were to make a claim, we would approach it in the same systematic way we analyzed Jeremiah's and Lindsay's statements. We would first ensure we understand the underlying concepts, such as the definitions of intercepts and how they relate to the function's equation. Then, we would carefully examine Stephen's reasoning and compare it to the correct mathematical principles. This process would help us determine the validity of his claim and identify any areas where his understanding might need clarification.

Since we do not have a claim from Stephen, we can only emphasize the significance of precise communication in mathematical discussions. It's like trying to solve a puzzle with missing pieces – the complete picture remains elusive without all the necessary information.

Conclusion

In this exploration of the function f(x) = (x+3)(x+5), we've analyzed the claims made by Jeremiah and Lindsay regarding its intercepts. Jeremiah incorrectly identified the y-intercept, while Lindsay accurately found one x-intercept but missed the other due to a sign error. The absence of Stephen's claim highlights the importance of clear communication in mathematical problem-solving. This exercise demonstrates that understanding the fundamental concepts of intercepts and the ability to carefully solve equations are crucial for accurately interpreting quadratic functions.

By breaking down each claim and examining the reasoning behind it, we've gained valuable insights into the nuances of working with quadratic functions. We've also seen how common errors can arise and how to avoid them by paying close attention to detail and double-checking our work. Remember, mathematics is not just about finding the right answer; it's about the journey of logical thinking and problem-solving.

To further enhance your understanding of quadratic functions and their intercepts, I recommend exploring resources like Khan Academy's Quadratic Functions section, which provides comprehensive lessons and practice exercises. Keep exploring, keep questioning, and keep learning!