Analyzing Claims About The Quadratic Function F(x)=(x+3)(x+5)

Mr. Walker presented his class with the quadratic function f(x) = (x+3)(x+5). This seemingly simple function opens the door to a wealth of mathematical concepts, including intercepts, roots, the vertex, and the overall shape of the parabola it represents. Four students, Jeremiah, Lindsay, Nolan, and Parker, each made a claim about this function. Let's delve into these claims, dissect the function, and determine the accuracy of each student's statement.

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

Before we evaluate the students' claims, it's crucial to thoroughly understand the function f(x) = (x+3)(x+5) itself. This function is presented in factored form, which provides immediate insights into its roots, also known as x-intercepts. To find these roots, we set f(x) equal to zero and solve for x:

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

This equation holds true when either (x+3) = 0 or (x+5) = 0. Solving these simple equations gives us:

x = -3 or x = -5

Therefore, the x-intercepts of the function are at (-3, 0) and (-5, 0). These are the points where the parabola intersects the x-axis. The factored form of the quadratic equation directly reveals these crucial points. Understanding this relationship between the factored form and the x-intercepts is a fundamental concept in algebra. This allows for quick identification of key features of the graph without extensive calculations. Further manipulation of the function allows us to explore other characteristics, like the y-intercept and the vertex, thus giving a complete picture of the parabola's behavior. Now let's expand the function to gain further insight. Expanding the factored form gives us the standard form of the quadratic equation:

f(x) = (x+3)(x+5) = x² + 5x + 3x + 15 = x² + 8x + 15

From this standard form, we can readily identify the y-intercept. The y-intercept is the point where the parabola intersects the y-axis, and it occurs when x = 0. Substituting x = 0 into the standard form equation, we get:

f(0) = 0² + 8(0) + 15 = 15

Thus, the y-intercept is at (0, 15). We can also determine the axis of symmetry and the vertex. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. The x-coordinate of the vertex can be found using the formula x = -b / 2a, where a and b are the coefficients in the standard form of the quadratic equation (ax² + bx + c). In our case, a = 1 and b = 8, so:

x = -8 / (2 * 1) = -4

This means the axis of symmetry is the vertical line x = -4. To find the y-coordinate of the vertex, we substitute x = -4 back into the function:

f(-4) = (-4)² + 8(-4) + 15 = 16 - 32 + 15 = -1

Therefore, the vertex of the parabola is at (-4, -1). Knowing the vertex and the intercepts gives us a comprehensive understanding of the parabola's shape and position on the coordinate plane. This preliminary analysis sets the stage for evaluating the claims made by the students. Each claim touches on a specific aspect of the function, and our understanding of these aspects will guide us in determining the validity of the claims. So, let's move forward and see what Jeremiah, Lindsay, Nolan, and Parker have to say about this function.

Analyzing 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 verify this claim, we need to understand what a 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 evaluate the function at x = 0. Let's substitute x = 0 into the function:

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

This result tells us that when x = 0, f(x) = 15. This corresponds to the point (0, 15), not (15, 0) as Jeremiah claimed. Therefore, Jeremiah's claim is incorrect. The point (15, 0) represents an x-intercept, where the graph intersects the x-axis, not the y-axis. It is crucial to distinguish between the x- and y-intercepts. The x-intercept is found by setting f(x) = 0 and solving for x, while the y-intercept is found by setting x = 0 and evaluating f(x). The common misconception of confusing the coordinates of the intercepts can lead to errors in understanding the graph's behavior. The y-intercept is a crucial feature of the graph as it indicates the point where the parabola begins its upward or downward curve from the left side of the graph. The y-intercept also plays a significant role in various real-world applications, such as determining the initial value in a quadratic model. For instance, in a model representing the height of a projectile over time, the y-intercept would represent the initial height of the projectile. Therefore, understanding how to correctly identify and interpret the y-intercept is essential in mathematical problem-solving and real-world applications. In Jeremiah's case, it appears he may have confused the coordinates, perhaps thinking the y-intercept should have a y-value of 0. This highlights the importance of careful attention to the definitions and concepts related to intercepts. By correctly substituting x = 0 into the function, we can definitively determine the y-intercept, allowing us to correct Jeremiah's misconception and reinforce the understanding of this key concept.

Analyzing 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 (3, 0) and (5, 0). To assess this claim, we must recall the definition of x-intercepts. The x-intercepts are the points where the graph of the function intersects the x-axis. At these points, the y-coordinate, or the function value f(x), is equal to 0. To find the x-intercepts, we set f(x) = 0 and solve for x. This is where the factored form of the quadratic equation, (x+3)(x+5) = 0, becomes particularly useful. The equation (x+3)(x+5) = 0 holds true if either (x+3) = 0 or (x+5) = 0. Solving these equations, we find:

x + 3 = 0 => x = -3

x + 5 = 0 => x = -5

Therefore, the x-intercepts are at (-3, 0) and (-5, 0). Comparing these values to Lindsay's claim of (3, 0) and (5, 0), we see that Lindsay's claim is incorrect. She likely made an error in solving the equations (x+3) = 0 and (x+5) = 0. It's a common mistake to simply take the constants within the parentheses and assume they are the x-intercepts without changing their signs. This highlights the importance of understanding the process of solving for roots from the factored form. The negative signs are crucial in determining the correct x-intercepts. This misunderstanding can have significant implications when graphing the parabola or using the function to model real-world scenarios. The x-intercepts are key features of the parabola, indicating where the function's value changes sign. They are also the roots or solutions of the quadratic equation. In practical applications, the x-intercepts can represent important values, such as the time when a projectile hits the ground or the break-even points in a business model. A correct understanding of x-intercepts is crucial for accurate interpretation and analysis of the quadratic function's behavior. Lindsay's error serves as a valuable learning opportunity to reinforce the correct process of solving for x-intercepts from the factored form and to emphasize the importance of attention to detail in algebraic manipulations. By carefully analyzing the equation and correctly solving for the roots, we can avoid such errors and gain a solid understanding of the function's characteristics. Let’s proceed to examine the claims of the remaining students to further explore the properties of the given quadratic function.

Analyzing Nolan's Claim: The axis of symmetry is the line x = -4.

Nolan claims that the axis of symmetry for the function f(x) = (x+3)(x+5) is the line x = -4. The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. It passes through the vertex of the parabola, which is the point where the parabola reaches its minimum or maximum value. There are a couple of ways to determine the axis of symmetry for a quadratic function. One method involves finding the midpoint of the x-intercepts. We already determined that the x-intercepts of the function are (-3, 0) and (-5, 0). The x-coordinate of the midpoint is the average of the x-coordinates of the intercepts:

Midpoint x-coordinate = (-3 + (-5)) / 2 = -8 / 2 = -4

This confirms that the axis of symmetry is indeed the vertical line x = -4. Another method to find the axis of symmetry involves using the standard form of the quadratic equation. As we previously expanded the function, we have f(x) = x² + 8x + 15. In the standard form, f(x) = ax² + bx + c, the axis of symmetry is given by the formula x = -b / 2a. In this case, a = 1 and b = 8, so:

x = -8 / (2 * 1) = -4

Again, this confirms that the axis of symmetry is the line x = -4. Since both methods yield the same result, Nolan's claim is accurate. The axis of symmetry is a fundamental property of a parabola, providing a crucial reference point for understanding its shape and position. It indicates the line around which the parabola is mirrored, and it directly relates to the vertex of the parabola. The vertex, being the point on the parabola that lies on the axis of symmetry, represents either the minimum or maximum value of the function. In practical applications, the axis of symmetry can be used to solve optimization problems, such as finding the maximum height of a projectile or the minimum cost in a production process. Nolan's correct identification of the axis of symmetry demonstrates a strong understanding of quadratic functions and their properties. The ability to determine the axis of symmetry using different methods showcases a comprehensive grasp of the concepts. This skill is essential for accurately graphing quadratic functions and applying them to various problem-solving scenarios. It also highlights the connection between the intercepts, the vertex, and the overall symmetry of the parabola. Now, we move on to the last student’s claim to get a complete picture of how the students understood the function.

Analyzing Parker's Claim: The vertex is at (-4, -1).

Parker claims that the vertex of the function f(x) = (x+3)(x+5) is at (-4, -1). The vertex is the point where the parabola changes direction, representing either the minimum or maximum value of the function. It lies on the axis of symmetry, which we established in Nolan's claim to be the line x = -4. This means the x-coordinate of the vertex is -4. To find the y-coordinate of the vertex, we substitute x = -4 into the function:

f(-4) = (-4 + 3)(-4 + 5) = (-1)(1) = -1

This confirms that the y-coordinate of the vertex is -1. Therefore, the vertex is indeed at (-4, -1). Another way to find the vertex is by using the standard form of the quadratic equation, f(x) = x² + 8x + 15. We already found the x-coordinate of the vertex using the formula x = -b / 2a, which gave us x = -4. To find the y-coordinate, we substitute x = -4 into the standard form:

f(-4) = (-4)² + 8(-4) + 15 = 16 - 32 + 15 = -1

Again, this confirms that the vertex is at (-4, -1). Parker's claim is accurate. The vertex is a crucial point on the parabola, providing key information about the function's behavior. For a parabola that opens upwards (like this one, since the coefficient of x² is positive), the vertex represents the minimum value of the function. Conversely, for a parabola that opens downwards, the vertex represents the maximum value. The vertex, along with the axis of symmetry and the intercepts, allows for a complete understanding of the parabola's graph. In many real-world applications, the vertex holds significant meaning. For example, in a model representing the trajectory of a projectile, the vertex would represent the maximum height reached by the projectile. In a model representing profit as a function of production, the vertex might represent the production level that maximizes profit. Parker's correct identification of the vertex showcases a strong ability to work with quadratic functions and interpret their properties. It demonstrates an understanding of the relationship between the vertex, the axis of symmetry, and the overall behavior of the parabola. This skill is essential for solving a wide range of problems involving quadratic functions.

Conclusion: Evaluating the Students' Understanding

In conclusion, by thoroughly analyzing the function f(x) = (x+3)(x+5) and evaluating the claims made by the four students, we have gained a deeper understanding of quadratic functions and their properties. Jeremiah's claim about the y-intercept was incorrect, highlighting a potential confusion between the x and y coordinates of intercepts. Lindsay's claim about the x-intercepts was also incorrect, likely stemming from an error in solving for the roots of the equation. Nolan's claim about the axis of symmetry was accurate, demonstrating a solid understanding of this key property. Parker's claim about the vertex was also accurate, showcasing the ability to correctly determine the minimum point of the parabola. This exercise demonstrates the importance of carefully applying the definitions and concepts related to quadratic functions. While some students made errors, the process of analyzing their claims provides a valuable learning opportunity for everyone. By identifying and correcting misconceptions, we can build a stronger foundation for future mathematical endeavors. The ability to accurately determine the intercepts, axis of symmetry, and vertex of a quadratic function is crucial for graphing and understanding its behavior. These skills are not only essential in mathematics but also have applications in various fields, including physics, engineering, and economics. The key takeaway from this exercise is the significance of a thorough and accurate understanding of fundamental mathematical concepts. By carefully analyzing each step and avoiding common pitfalls, we can confidently solve problems and apply our knowledge to real-world situations.