Finding The Parabola Function With Y-intercept 1 And Vertex (1,0)

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When diving into the world of parabola functions, understanding their key features is paramount. Parabolas, the U-shaped curves gracing the graphs of quadratic functions, possess distinct characteristics that dictate their form and position on the coordinate plane. One crucial aspect is the y-intercept, the point where the parabola intersects the y-axis. This point reveals the function's value when x equals zero, offering a glimpse into its behavior. The vertex, another pivotal point, marks the parabola's extreme – either its minimum or maximum value. This turning point not only defines the parabola's lowest or highest reach but also serves as the axis of symmetry, splitting the curve into two mirroring halves. Furthermore, the direction a parabola opens – either upwards or downwards – is determined by the coefficient of the squared term in its equation. A positive coefficient signals an upward-opening parabola, while a negative coefficient indicates a downward-opening one. Understanding these fundamental elements – the y-intercept, vertex, and direction – forms the cornerstone for deciphering the function a parabola represents. By carefully analyzing these features, we can navigate through the options and pinpoint the correct quadratic function that aligns with the given parabola.

Understanding the relationship between these features and the standard forms of quadratic equations is key. The vertex form, f(x) = a(x - h)² + k, is particularly insightful as it directly reveals the vertex (h, k) and the vertical stretch or compression factor a. The y-intercept can be found by substituting x = 0 into the equation. By mastering these concepts, we equip ourselves with the tools to dissect any parabola and extract the underlying function it embodies. Remember, each feature acts as a clue, guiding us towards the precise equation that paints the parabola's unique curve on the graph.

In this problem, we are presented with a parabola whose y-intercept is 1 and whose vertex resides at the point (1, 0). Our mission is to decipher which function, from the provided options, this graph represents. This requires a meticulous examination of each option, comparing its inherent features with the given characteristics of the parabola. We'll delve into how the vertex form of a quadratic equation, f(x) = a(x - h)² + k, can be our guiding light, revealing the vertex (h, k) and the stretch factor a. We'll also explore the significance of the y-intercept, which we can find by setting x = 0 in the function. By systematically analyzing these key features, we'll narrow down the possibilities and pinpoint the function that perfectly captures the essence of our parabola. This analytical journey will not only lead us to the correct answer but also deepen our understanding of how the equation of a parabola dictates its visual representation on the coordinate plane. Each option presents a potential solution, a piece of the puzzle. It's our task to carefully fit the pieces together, ensuring that the function's y-intercept and vertex align with the given information. This process involves more than just identifying the right answer; it's about understanding the underlying principles that govern the relationship between a parabola's equation and its graphical form.

To begin, we acknowledge the importance of the vertex. The vertex, located at (1, 0), immediately suggests that the equation of the parabola will likely be in the form f(x) = a(x - 1)² + 0, or simply f(x) = a(x - 1)², where a is a constant that determines the parabola's direction and stretch. This insight significantly narrows down our options, allowing us to focus on functions that exhibit this structure. Next, we consider the y-intercept. The fact that the parabola intersects the y-axis at 1 means that when x = 0, f(x) = 1. This piece of information is crucial as it provides a specific point that the function must satisfy. By substituting x = 0 and f(x) = 1 into our potential equations, we can solve for the constant a and verify which function aligns perfectly with the given characteristics. This step-by-step approach, combining the information from both the vertex and the y-intercept, will guide us towards the definitive answer.

Now, let's embark on the journey of evaluating each answer choice, meticulously comparing its attributes with the given characteristics of our parabola. This process involves a deep dive into the structure of each function, extracting key information such as the vertex and the predicted y-intercept. By systematically analyzing each option, we'll be able to eliminate those that don't align with the given conditions, gradually narrowing our focus towards the correct answer. This step-by-step approach ensures that we not only identify the solution but also understand why it's the correct one. For each function, we'll consider its vertex form, which provides immediate insights into the location of the vertex and the parabola's direction. We'll also substitute x = 0 to calculate the y-intercept and verify if it matches the given value of 1. This thorough evaluation process is crucial for solidifying our understanding of parabolas and their equations.

  • Option A: f(x) = (x - 1)². This function is in vertex form, revealing a vertex at (1, 0), which aligns with the given information. To find the y-intercept, we substitute x = 0: f(0) = (0 - 1)² = 1. This also matches the given y-intercept. Therefore, Option A is a strong contender.
  • Option B: f(x) = (x + 1)². This function has a vertex at (-1, 0), which does not match the given vertex of (1, 0). Thus, we can eliminate Option B.
  • Option C: f(x) = -1(x - 1)². This function has a vertex at (1, 0), which matches the given information. However, the negative coefficient indicates that the parabola opens downwards. To find the y-intercept, we substitute x = 0: f(0) = -1(0 - 1)² = -1. This does not match the given y-intercept of 1. Therefore, we can eliminate Option C.
  • Option D: f(x) = -1(x + 1)². This function has a vertex at (-1, 0), which does not match the given vertex of (1, 0). Thus, we can eliminate Option D.

After a meticulous evaluation of each answer choice, the path to the correct function becomes crystal clear. We've systematically dissected each option, comparing its inherent features – namely the vertex and the y-intercept – with the given characteristics of the parabola. This process has not only led us to the solution but has also reinforced our understanding of how these key features dictate the equation of a parabola. The vertex form of a quadratic equation has been our guiding light, allowing us to quickly identify the vertex and the parabola's direction. The y-intercept, found by substituting x = 0, has served as a crucial validation point, confirming whether a function truly represents the given parabola. This comprehensive approach, combining analytical techniques with a deep understanding of parabolic functions, ensures that our conclusion is not just a guess but a well-reasoned deduction.

The analysis reveals that Option A, f(x) = (x - 1)², perfectly embodies the given characteristics. Its vertex at (1, 0) aligns seamlessly with the problem statement, and its y-intercept, calculated as 1, further solidifies its correctness. The other options, while resembling parabolic functions, fall short in one or more aspects. Options B and D present vertices that deviate from the given (1, 0), while Option C, despite having the correct vertex, exhibits a y-intercept of -1, contradicting the problem's condition. Therefore, through a process of elimination and rigorous verification, we confidently declare Option A as the definitive answer. This conclusion not only solves the problem but also highlights the power of understanding the interplay between a parabola's equation and its graphical representation.

Therefore, the correct answer is A. f(x) = (x - 1)².