In Depth Analysis Of The Quadratic Function F(x) = (x+3)(x+5)

This article delves into the properties and characteristics of the quadratic function f(x) = (x+3)(x+5). We will analyze the claims made by four students regarding this function, focusing on key features such as the y-intercept, x-intercepts, the vertex, and the axis of symmetry. Through a detailed examination, we aim to provide a comprehensive understanding of quadratic functions and their graphical representations.

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

At its core, f(x) = (x+3)(x+5) represents a quadratic function. Quadratic functions, characterized by the highest power of the variable being 2, are fundamental in mathematics and have wide-ranging applications in physics, engineering, and economics. The general form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants. In our case, the function is given in factored form, which provides valuable insights into its roots or x-intercepts. To fully understand this function, it's imperative to expand it into the standard form and then analyze its components. Expanding f(x) = (x+3)(x+5), we get:

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

Now, the function is in the standard form f(x) = ax² + bx + c, where a = 1, b = 8, and c = 15. This form allows us to easily identify key features of the parabola, which is the graphical representation of a quadratic function. The coefficient 'a' determines the direction of the parabola's opening (upward if a > 0, downward if a < 0). In our case, a = 1, so the parabola opens upwards. The constant term 'c' represents the y-intercept of the parabola, which is the point where the parabola intersects the y-axis. In this case, the y-intercept is 15, corresponding to the point (0, 15). Understanding the coefficients and the expanded form of the quadratic function is crucial for analyzing its behavior and verifying the claims made by the students.

Analyzing Student Claims: Y-Intercept

Jeremiah's claim focuses on the y-intercept of the function f(x) = (x+3)(x+5). The y-intercept is a crucial characteristic of any function, representing the point where the graph of the function intersects the y-axis. This occurs when the x-coordinate is equal to zero. Therefore, to find the y-intercept, we need to evaluate f(0). According to Jeremiah, the y-intercept is at (15, 0). Let's verify this claim.

To find the y-intercept, we substitute x = 0 into the function:

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

This result indicates that the y-coordinate of the y-intercept is 15. However, a crucial distinction needs to be made. A point in the coordinate plane is represented as (x, y). The y-intercept, being a point on the y-axis, will have an x-coordinate of 0. Therefore, the correct representation of the y-intercept is (0, 15), not (15, 0) as Jeremiah claimed. Jeremiah's claim incorrectly swapped the x and y coordinates. Understanding the fundamental concept of intercepts and their representation on the coordinate plane is crucial for accurate analysis of functions. While Jeremiah correctly calculated the y-value of the y-intercept, the incorrect representation of the point reveals a misunderstanding of coordinate geometry. This highlights the importance of not only performing calculations accurately but also interpreting the results correctly within the context of the problem.

Analyzing Student Claims: X-Intercepts

Lindsay's claim pertains to the x-intercepts of the function f(x) = (x+3)(x+5). X-intercepts, also known as roots or zeros, are the points where the graph of the function intersects the x-axis. At these points, the y-coordinate is zero. Finding the x-intercepts is a fundamental aspect of analyzing any function, as they provide critical information about the function's behavior and solutions. The factored form of a quadratic function, such as the one we have, f(x) = (x+3)(x+5), makes it particularly easy to determine the x-intercepts.

To find the x-intercepts, we need to solve the equation f(x) = 0. This means we need to find the values of x for which the function equals zero. Using the factored form, we have:

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

According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x:

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

This gives us two x-intercepts: x = -3 and x = -5. As x-intercepts are points on the x-axis, their y-coordinate is 0. Therefore, the x-intercepts are the points (-3, 0) and (-5, 0). Lindsay's claim should specify both intercepts and provide them in the correct coordinate format. The factored form of the quadratic function f(x) = (x+3)(x+5) provides a straightforward way to identify the x-intercepts, which are the values of x that make each factor equal to zero. Understanding the zero-product property and its application is crucial for accurately determining the x-intercepts of a quadratic function.

Determining the Vertex and Axis of Symmetry

To fully understand the graph of the quadratic function, we need to determine the vertex and the axis of symmetry. The vertex is the point where the parabola changes direction; it's the minimum point if the parabola opens upwards (as in our case) and the maximum point if the parabola opens downwards. 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, f(x) = ax² + bx + c. In our case, f(x) = x² + 8x + 15, so a = 1 and b = 8.

Using the formula, we find the x-coordinate of the vertex:

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

Now, to find the y-coordinate of the vertex, we substitute x = -4 into the function:

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

Therefore, the vertex of the parabola is at the point (-4, -1). The axis of symmetry is a vertical line that passes through the vertex, so its equation is x = -4. The vertex and axis of symmetry are fundamental characteristics of a parabola, providing insights into its position and symmetry. The vertex represents the extreme point of the parabola (minimum or maximum), and the axis of symmetry divides the parabola into two identical halves. Understanding how to calculate these features is essential for accurately sketching the graph of a quadratic function.

Conclusion

Analyzing the quadratic function f(x) = (x+3)(x+5) and the claims made by the students provides a valuable exercise in understanding the properties of quadratic functions. We've explored key features such as the y-intercept, x-intercepts, the vertex, and the axis of symmetry. By verifying the student's claims and performing detailed calculations, we've reinforced the importance of accuracy in mathematical analysis and the correct interpretation of results. Understanding these concepts is crucial for further studies in mathematics and its applications in various fields. The process of expanding the function, identifying coefficients, and applying formulas to find key characteristics demonstrates the interconnectedness of different mathematical concepts. This comprehensive analysis not only clarifies the specific function f(x) = (x+3)(x+5) but also provides a framework for analyzing other quadratic functions and their graphical representations. Remember, a deep understanding of quadratic functions is a cornerstone of advanced mathematical studies and practical applications.