Analyzing The Quadratic Function H(x) = X^2 + 20x - 17 Transformations, Vertex Form, And Symmetry

Introduction

In the realm of mathematics, quadratic functions hold a prominent position, serving as fundamental building blocks for various applications. Among these functions, the specific quadratic function h(x) = x^2 + 20x - 17 presents an intriguing case study, offering valuable insights into the behavior and properties of quadratic expressions. In this comprehensive exploration, we embark on a journey to dissect this function, unveiling its key characteristics and revealing the underlying principles that govern its behavior. From determining its vertex form to analyzing its graphical transformations, we delve into the intricacies of this function, equipping ourselves with the knowledge to effectively manipulate and interpret quadratic expressions.

Deconstructing the Quadratic Function: Vertex Form and Transformations

To gain a deeper understanding of the function h(x) = x^2 + 20x - 17, we first turn our attention to its vertex form. The vertex form of a quadratic function provides a clear representation of its vertex, the point at which the parabola reaches its maximum or minimum value. By transforming the function into vertex form, we can readily identify the vertex coordinates and gain valuable insights into the function's overall shape and position. The process of converting a quadratic function to vertex form involves completing the square, a technique that allows us to rewrite the expression in a form that reveals the vertex. In the case of h(x) = x^2 + 20x - 17, completing the square involves adding and subtracting a constant term to create a perfect square trinomial. This process is crucial for understanding the transformations applied to the basic quadratic function f(x) = x^2.

Transforming the Function: Completing the Square

To begin, we group the first two terms of the function: h(x) = (x^2 + 20x) - 17. Next, we take half of the coefficient of the x term (which is 20), square it (10^2 = 100), and add and subtract it within the parentheses: h(x) = (x^2 + 20x + 100 - 100) - 17. This manipulation doesn't change the value of the expression, as we're essentially adding and subtracting the same quantity. Now, we can rewrite the expression as h(x) = (x^2 + 20x + 100) - 100 - 17. The expression within the parentheses is now a perfect square trinomial, which can be factored as (x + 10)^2. Thus, the function becomes h(x) = (x + 10)^2 - 117. This is the vertex form of the function, which reveals the vertex coordinates and the vertical shift applied to the basic quadratic function.

Interpreting the Vertex Form: Shifts and Transformations

The vertex form, h(x) = (x + 10)^2 - 117, provides a wealth of information about the function's transformations. The term (x + 10)^2 indicates a horizontal shift of the basic quadratic function f(x) = x^2. Specifically, the graph of h(x) is shifted 10 units to the left compared to the graph of f(x). This is because the term (x + 10) becomes zero when x = -10, indicating that the vertex of h(x) is located at x = -10. The constant term -117 indicates a vertical shift of the graph. The graph of h(x) is shifted 117 units downward compared to the graph of f(x). This is because the vertex of h(x) has a y-coordinate of -117. Therefore, to graph the function h(x) = x^2 + 20x - 17, we can start with the graph of f(x) = x^2, shift it 10 units to the left, and then shift it 117 units down. This process provides a clear visual representation of the transformations applied to the basic quadratic function.

Unveiling the Axis of Symmetry: A Mirror to the Parabola

The axis of symmetry is a fundamental property of parabolas, acting as a mirror that reflects the two halves of the curve. This imaginary line passes through the vertex of the parabola, dividing it into two symmetrical halves. Understanding the axis of symmetry is crucial for sketching the graph of a quadratic function and identifying key features like the vertex and the overall shape of the parabola. The axis of symmetry is always a vertical line, and its equation is given by x = the x-coordinate of the vertex. In the case of h(x) = x^2 + 20x - 17, we've already determined that the vertex is at (-10, -117). Therefore, the axis of symmetry is the vertical line x = -10.

Determining the Axis of Symmetry: Vertex Coordinates and Symmetry

The axis of symmetry is intrinsically linked to the vertex of the parabola. As we've established, the vertex of the function h(x) = x^2 + 20x - 17 is located at (-10, -117). The x-coordinate of the vertex, which is -10 in this case, directly corresponds to the equation of the axis of symmetry. This is because the parabola is perfectly symmetrical around the vertical line that passes through its vertex. Every point on the parabola has a corresponding point on the opposite side of the axis of symmetry, equidistant from the axis. This symmetrical property is a defining characteristic of parabolas and is essential for understanding their graphical representation.

Visualizing the Axis of Symmetry: Graphing and Symmetry

When we graph the function h(x) = x^2 + 20x - 17, the axis of symmetry becomes visually apparent. The parabola is perfectly balanced around the line x = -10. If we were to fold the graph along this line, the two halves of the parabola would perfectly overlap. This symmetry allows us to efficiently sketch the graph of the function. Once we've identified the vertex and a few points on one side of the axis of symmetry, we can easily mirror those points across the axis to complete the graph. The axis of symmetry not only provides a visual guide but also helps us understand the function's behavior. For instance, we know that the function will have the same y-value for x-values that are equidistant from the axis of symmetry. This understanding enhances our ability to analyze and interpret quadratic functions.

Identifying the Correct Statements: A Comprehensive Analysis

Having thoroughly analyzed the function h(x) = x^2 + 20x - 17, we are now well-equipped to evaluate the given statements and identify the correct ones. Let's revisit the statements:

A. To graph the function h, shift the graph of f(x) = x^2 left 10 units and down 117 units.

B. The vertex form of the function is h(x) = (x + 20)^2 - 17.

C. The axis of symmetry of the function is x = -10.

Evaluating the Statements: A Step-by-Step Approach

Statement A aligns perfectly with our earlier analysis of the vertex form. We determined that h(x) can be obtained by shifting the graph of f(x) = x^2 10 units to the left and 117 units down. This corresponds to the vertex form h(x) = (x + 10)^2 - 117, where the +10 indicates a leftward shift and the -117 indicates a downward shift. Therefore, Statement A is correct.

Statement B presents an incorrect vertex form. We meticulously completed the square and derived the vertex form as h(x) = (x + 10)^2 - 117. The given vertex form, h(x) = (x + 20)^2 - 17, is not equivalent to the original function. Expanding this incorrect vertex form would result in a different quadratic expression, demonstrating its inaccuracy. Therefore, Statement B is incorrect.

Statement C accurately identifies the axis of symmetry. We established that the axis of symmetry is a vertical line passing through the vertex's x-coordinate. Since the vertex of h(x) is (-10, -117), the axis of symmetry is indeed x = -10. This aligns with our understanding of the relationship between the vertex and the axis of symmetry in a parabola. Therefore, Statement C is correct.

Conclusion

In conclusion, our in-depth exploration of the quadratic function h(x) = x^2 + 20x - 17 has revealed its key characteristics and properties. By converting the function to vertex form, h(x) = (x + 10)^2 - 117, we gained valuable insights into its transformations and vertex coordinates. We determined that the graph of h(x) is obtained by shifting the graph of f(x) = x^2 10 units to the left and 117 units down. Furthermore, we identified the axis of symmetry as the vertical line x = -10. Based on this comprehensive analysis, we confidently identified Statements A and C as the correct answers, while Statement B was deemed incorrect due to its inaccurate vertex form. This exercise highlights the importance of understanding vertex form, completing the square, and the relationship between the vertex and the axis of symmetry in analyzing quadratic functions.